Physics Book - User contributions [en]

Electric Polarization

2017-11-30T03:02:47Z

Kdarisipudi3: An example of polarization through dielectric material

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

[[File:Phys.jpg]]

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

Given this property, polarization in insulators happens very rapidly, in a matter of nanoseconds. In contrast to conductors, which have their excess charges on the surface, all of the excess charges in an insulator stay in the interior.

The distinguishing feature of conductors that makes them susceptible to polarization is that that conductors have a "floating sea of electrons" that move around. This property is especially predominant in metals. Because electrons move around in conductors, we can measure their speed through ''drift speed''.

'''Drift speed: V=uE'''

Where
''V'' is drift speed,
''u'' is mobility of the charge,
and ''E'' is the magnitude of the electric field,

When a conductor is at equilibrium, the drift speed will reach 0.

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

While there is no direct correlation of electric polarization to my major (industrial engineering), there is one indirect parallel. By looking at the history of electric polarization, and how the understanding we have came to be, we can infer that the concept went through several iterations before it became widely accepted. In that same vein, many theorems and principles in stochastics, optimization, and simulations similarly were expounded on over the years before they were accepted into practice.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

Experimental studies in the 18th century made it possible to distinguish positive and negative charges. The properties of charges in a condenser were studied and reported by Faraday in 1837, which he called the dielectric. He also introduced the concept of permittivity/dielectric constant, which is represented by ε. Simultaneously, Maxwell presented this theory of electromagnetic phenomena, which tied electric field intensity and dielectric displacement.

In the 19th century, Debye published a theory that extended the Clausius-Mossotti equation, which consequently came to be known as the Debye equation. This equation shed light upon the fact that the dielectric constant depended on both molecular polarizability as well as the permanent moment of the molecules.

In 1936, Onsager modified Debye's work and found that Debye had included torque in his calculations when he shouldn't have. Onsager furthered the relation of the dilelectric constant and the molecular dipole moment by describing their relationship in polar liquids and non-polar solvents.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

File:Phys.jpg

2017-11-30T03:01:52Z

Kdarisipudi3: Kdarisipudi3 uploaded a new version of "File:Phys.jpg"

Electric Polarization

2017-11-24T05:03:37Z

Kdarisipudi3: /* Connectedness */

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

Given this property, polarization in insulators happens very rapidly, in a matter of nanoseconds. In contrast to conductors, which have their excess charges on the surface, all of the excess charges in an insulator stay in the interior.

The distinguishing feature of conductors that makes them susceptible to polarization is that that conductors have a "floating sea of electrons" that move around. This property is especially predominant in metals. Because electrons move around in conductors, we can measure their speed through ''drift speed''.

'''Drift speed: V=uE'''

Where
''V'' is drift speed,
''u'' is mobility of the charge,
and ''E'' is the magnitude of the electric field,

When a conductor is at equilibrium, the drift speed will reach 0.

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

While there is no direct correlation of electric polarization to my major (industrial engineering), there is one indirect parallel. By looking at the history of electric polarization, and how the understanding we have came to be, we can infer that the concept went through several iterations before it became widely accepted. In that same vein, many theorems and principles in stochastics, optimization, and simulations similarly were expounded on over the years before they were accepted into practice.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

Experimental studies in the 18th century made it possible to distinguish positive and negative charges. The properties of charges in a condenser were studied and reported by Faraday in 1837, which he called the dielectric. He also introduced the concept of permittivity/dielectric constant, which is represented by ε. Simultaneously, Maxwell presented this theory of electromagnetic phenomena, which tied electric field intensity and dielectric displacement.

In the 19th century, Debye published a theory that extended the Clausius-Mossotti equation, which consequently came to be known as the Debye equation. This equation shed light upon the fact that the dielectric constant depended on both molecular polarizability as well as the permanent moment of the molecules.

In 1936, Onsager modified Debye's work and found that Debye had included torque in his calculations when he shouldn't have. Onsager furthered the relation of the dilelectric constant and the molecular dipole moment by describing their relationship in polar liquids and non-polar solvents.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

Electric Polarization

2017-11-24T05:00:29Z

Kdarisipudi3: /* History */

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

Given this property, polarization in insulators happens very rapidly, in a matter of nanoseconds. In contrast to conductors, which have their excess charges on the surface, all of the excess charges in an insulator stay in the interior.

The distinguishing feature of conductors that makes them susceptible to polarization is that that conductors have a "floating sea of electrons" that move around. This property is especially predominant in metals. Because electrons move around in conductors, we can measure their speed through ''drift speed''.

'''Drift speed: V=uE'''

Where
''V'' is drift speed,
''u'' is mobility of the charge,
and ''E'' is the magnitude of the electric field,

When a conductor is at equilibrium, the drift speed will reach 0.

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

Experimental studies in the 18th century made it possible to distinguish positive and negative charges. The properties of charges in a condenser were studied and reported by Faraday in 1837, which he called the dielectric. He also introduced the concept of permittivity/dielectric constant, which is represented by ε. Simultaneously, Maxwell presented this theory of electromagnetic phenomena, which tied electric field intensity and dielectric displacement.

In the 19th century, Debye published a theory that extended the Clausius-Mossotti equation, which consequently came to be known as the Debye equation. This equation shed light upon the fact that the dielectric constant depended on both molecular polarizability as well as the permanent moment of the molecules.

In 1936, Onsager modified Debye's work and found that Debye had included torque in his calculations when he shouldn't have. Onsager furthered the relation of the dilelectric constant and the molecular dipole moment by describing their relationship in polar liquids and non-polar solvents.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

Electric Polarization

2017-11-16T06:05:16Z

Kdarisipudi3: /* Conductors and Insulators */

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

Given this property, polarization in insulators happens very rapidly, in a matter of nanoseconds. In contrast to conductors, which have their excess charges on the surface, all of the excess charges in an insulator stay in the interior.

The distinguishing feature of conductors that makes them susceptible to polarization is that that conductors have a "floating sea of electrons" that move around. This property is especially predominant in metals. Because electrons move around in conductors, we can measure their speed through ''drift speed''.

'''Drift speed: V=uE'''

Where
''V'' is drift speed,
''u'' is mobility of the charge,
and ''E'' is the magnitude of the electric field,

When a conductor is at equilibrium, the drift speed will reach 0.

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

Electric Polarization

2017-11-16T05:59:44Z

Kdarisipudi3: /* Conductors and Insulators */

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

The distinguishing feature of conductors that makes them susceptible to polarization is that that conductors have a "floating sea of electrons" that move around. This property is especially predominant in metals. Because electrons move around in conductors, we can measure their speed through ''drift speed''.

'''Drift speed: V=uE'''

Where
''V'' is drift speed,
''u'' is mobility of the charge,
''E'' is the magnitude of the electric field,

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

Electric Polarization

2017-11-16T05:58:58Z

Kdarisipudi3: /* Drift speed: V=uE */

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

The distinguishing feature of conductors that makes them susceptible to polarization is that that conductors have a "floating sea of electrons" that move around. This property is especially predominant in metals. Because electrons move around in conductors, we can measure their speed through ''drift speed''.

'''Drift speed: V=uE'''

Where
''V'' is drift speed
''u'' is mobility of the charge
''E'' is the magnitude of the electric field

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

Electric Polarization

2017-11-16T05:58:41Z

Kdarisipudi3: /* Conductors and Insulators */

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

The distinguishing feature of conductors that makes them susceptible to polarization is that that conductors have a "floating sea of electrons" that move around. This property is especially predominant in metals. Because electrons move around in conductors, we can measure their speed through ''drift speed''.

== '''Drift speed: V=uE''' ==

Where
''V'' is drift speed
''u'' is mobility of the charge
''E'' is the magnitude of the electric field

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

Electric Polarization

2017-11-16T05:50:41Z

Kdarisipudi3: /* Cause of Polarization */

'''''Claimed Edit by Kamesh Darisipudi (FALL 2017)'''''

''Claimed Edit by Eleanor Thomas Fall 2016''
'''Claimed by Annalise Irby (Spring 2017)
'''== The Main Idea ==

Polarization, used broadly, is the act of dividing into opposites. Electric polarization is the process of separating opposite charges inside an object. This occurs when an electric field, let's say created by a charged object A, induces the electrons to move in object B. This electron movement causes one portion of object B to have an excess negative charge and the other to have an excess positive charge. Object B could be a neutral object with a net charge of zero, but can still be polarized and attracted to object A. If A were positively charged, the electrons in object B would be attracted to the side closest to A (since opposite charges attract) which would create an induced dipole. This dipole is not permanent; if object A were to be removed, B would return to its neutral state.

=== Cause of Polarization ===
The cause of polarization can be found when the structure of an atom is closely examined. Atoms have a positively charged nucleus consisting of protons and neutrons tightly clumped together. Surrounding this nucleus is a negatively charged electron cloud which is not rigidly connected to the nucleus. Since the negative electron cloud does not have to be centered at its corresponding positive nucleus, if an electric field is applied on the atom, the two opposite charges can move relative to one another. This applied electric field induced the polarization of atoms.

We can see this exemplified when we look at a hydrogen atom for instance. From a microscopic perspective, it is hard to observe a single electron during polarization so we can look at an atom's electric cloud as a whole. When a positive charge was applied to a hydrogen atom, the electron cloud as shifted to the left due to the attraction to the opposite charge, while the atom's nucleus also shifted to the right. If we were observing a single electron, there is a greater chance that a single electron will be found to the right of the nucleus as opposed to the left.

=== Conductors and Insulators ===
Conductors are materials that allow electric charge to pass through it since it contains freely moving charged particles. When an electric field is applied to a conductor, electrons are transferred across the surface of the object and it becomes polarized.
An insulator is a material that does not conduct very much electric charge because all the electrons are rigidly bound to the atoms or molecules; they do not permit the free flow of electrons. When an insulator is subjected to an electric field, individual atoms or molecules are polarized rather than the whole object.

An example of a polarized insulator:

[[File:Screen_Shot_2016-11-27_at_10.52.28_PM.png]]

=== Misconception ===
Neutral objects (with a net charge of zero) CAN be attracted to charged ones because induced dipoles are formed and create an electric field at the location of the object. However, repulsion of an induced object cannot happen as it always brings unlike signs closer.

=== Mathematical Model ===

The amount of induced polarization is directly proportional to the magnitude of the applied electric field.

Induced Polarization : <math>\vec{P} = \alpha \vec{E}</math> .
'''"P"''' is the dipole moment of polarized atoms, '''"α"''' is the polarizability of a particular material, and '''"E"''' is the applied electric field. The value of '''"α"''' is dependent on many factors, has been recorded through experiments, and can be obtained through reference manuals.

Applied electric field is equal to:
[[File:Electricfieldet.jpg]]

== Examples ==
===Simple===
Question: A positively charged object is placed near a neutral atom. Draw the polarization of the atom.

[[File:etpolarizedatom.png]]
===Middling===
Question: A negatively charged object is placed above a metal sphere in equilibrium. Draw the polarization of the sphere.

[[File:etplasticsphere.png]]

===Difficult===
Question: A negatively charged metal object is placed in a region with an electric field going in the +y direction. Draw the polarization of the object.

[[File:etmetalblock.png]]

== Connectedness ==
Electric polarization is a key component of insulators and conductors. These materials and their uses are incredibly important for every day life. For example, lightbulb filaments are conductors which carry electrons from a negatively charged area to a positively charged area. Insulators, which don't conduct electric charges, can be used to insulate buildings. This is related to my major of study, environmental engineering, because we work to make buildings more sustainable by reducing the amount of energy required to keep them running.

== History ==
Benjamin Franklin was the first person to use "positive" and "negative" in describing charges and came up with principle of conservation of charges. In 1897, J.J. Thomson experimented with cathode rays and found out that electrons exist in the rays. In 1911, Ernest Rutherford discovered that atoms have a concentrated positive center with protons fixed inside nucleus.

== See Also ==
*[[Charge]]

*[[Electric Field]]

*[[Benjamin Franklin]]

*[[J.J. Thomson]]

*[[Ernest Rutherford]]

=== Further Reading ===
Physics of Dielectrics for the Engineer by Roland Coelho

==== Application ====
http://www.nrcresearchpress.com/doi/abs/10.1139/t89-067?journalCode=cgj#.VmFE97mFPIU

=== External Links ===
https://courses.cit.cornell.edu/ece303/Lectures/lecture7.pdf

== Reference ==
http://www.physicsclassroom.com/class/estatics/Lesson-1/Polarization

https://www.youtube.com/watch?v=3xSIA5UVAo8

http://scienceline.ucsb.edu/getkey.php?key=408

https://www.aip.org/history/gap/Franklin/Franklin.html

Matter and Interactions Fourth Edition by Ruth W. Chabay, Bruce A. Sherwood

Electric Polarization

2017-11-15T20:41:46Z

Kdarisipudi3: /* Mathematical Model */

Electric Polarization

2017-11-15T20:39:10Z

Kdarisipudi3: /* Mathematical Model */

Electric Polarization

2017-11-15T20:31:51Z

Kdarisipudi3: /* Mathematical Model */

Electric Polarization

2017-11-10T07:13:50Z

Kdarisipudi3:

Power (Mechanical)

2017-04-10T03:48:20Z

Kdarisipudi3: /* History */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

A weightlifter does 5000 J of work over a time span of 5 seconds. How much power does the weightlifter exert?

:<math>power = \frac{W}{\Delta t} = \frac{5000 J}{5 s} = 1000 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

A pump can potentially look like this:

[[File:pump2.jpg]]

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:47:58Z

Kdarisipudi3: /* History */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

A weightlifter does 5000 J of work over a time span of 5 seconds. How much power does the weightlifter exert?

:<math>power = \frac{W}{\Delta t} = \frac{5000 J}{5 s} = 1000 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

A pump can potentially look like this:

[File:pump2.jpg]

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:46:07Z

Kdarisipudi3: /* History */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

A weightlifter does 5000 J of work over a time span of 5 seconds. How much power does the weightlifter exert?

:<math>power = \frac{W}{\Delta t} = \frac{5000 J}{5 s} = 1000 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

[[File:pump2.jpg]]

<IMG SRC="<{[[File:pump2.jpg]]" HEIGHT="20" WIDTH="20" BORDER="0">

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:44:26Z

Kdarisipudi3: /* History */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

A weightlifter does 5000 J of work over a time span of 5 seconds. How much power does the weightlifter exert?

:<math>power = \frac{W}{\Delta t} = \frac{5000 J}{5 s} = 1000 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

[[File:pump2.jpg]]

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

File:Pump2.jpg

2017-04-10T03:43:52Z

Kdarisipudi3: pump2

pump2

File:Pump.png

2017-04-10T03:39:32Z

Kdarisipudi3: Kdarisipudi3 uploaded a new version of "File:Pump.png"

File:Pump.png

2017-04-10T03:38:27Z

Kdarisipudi3:

Power (Mechanical)

2017-04-10T03:37:48Z

Kdarisipudi3: /* History */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

A weightlifter does 5000 J of work over a time span of 5 seconds. How much power does the weightlifter exert?

:<math>power = \frac{W}{\Delta t} = \frac{5000 J}{5 s} = 1000 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

[[File:pump.png]]

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:33:24Z

Kdarisipudi3: /* Simple */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

A weightlifter does 5000 J of work over a time span of 5 seconds. How much power does the weightlifter exert?

:<math>power = \frac{W}{\Delta t} = \frac{5000 J}{5 s} = 1000 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:30:16Z

Kdarisipudi3: /* Middling */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:29:47Z

Kdarisipudi3: /* Middling */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103 kg and has a constant acceleration of 4.6m/s2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

:<math> P = {\frac{290950}{5}}</math>
:<math> P = 58190W</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:27:54Z

Kdarisipudi3: /* Middling */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math> 

A man starting at rest accelerates in the positive x direction. It has a mass of 1.1 x 103kg and has a constant acceleration of 4.6m/2 for 5 seconds. Determine in the average power generated by this vehicle:

:<math> V_o = 0 m/s &emsp; m=1,100kg &emsp; a=4.6m/s^2 &emsp; t=5s </math>

:<math>P={\frac{W}{t}} &emsp; W=F*S &emsp; W=\Delta KE</math>

:<math> x=V_ot+{\frac{1}{2}}at</math>
:<math> x=0+{\frac{1}{2}}(4.6)*(5)^2</math>
:<math> x=57.5m</math>

:<math> W = mas</math>
:<math> W = (1,100)*(4.6)*(57.5)</math>
:<math> W = 290950J</math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:02:25Z

Kdarisipudi3: /* A Mathematical Model */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>. Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T03:01:46Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>.

Given these conditions, we can arrive to the mechanical advantage:

:<math>MA=\frac{F_A}{F_B}=\frac{v_A}{v_B}</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:59:22Z

Kdarisipudi3: /* A Mathematical Model */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

With P being the input force, where <math>F_A</math> is a force that moves with velocity <math>v_A</math> and <math>F_B</math> is a force that moves with velocity <math>v_B</math>.

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:56:37Z

Kdarisipudi3: /* A Mathematical Model */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)</math>

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A</math>

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:56:00Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity. The SI unit for power is watts (J/s)

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math> 

If a force is conservative (meaning that it holds the property where the work done in moving a particle from two locations is independent of the path it takes), we can derive the following formula after applying the gradient theorem:

:<math>W_c = U(B) - U(A)

Where A and B represent the final and initial locations of along the path in which the work was done. 

In the case where we assume that the mechanical system has no losses, we can draw the conclusion that the input power must equal the output power. Knowing this provides us a formula for mechanical advantage, which is a metric for the factor in which a force is amplified through a tool:

:<math>P = F_B v_B=F_A v_A

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:47:24Z

Kdarisipudi3: /* A Mathematical Model */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work. 

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity. 

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product. 

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v&ensp;dt </math>

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:43:32Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v dt </math>

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:43:03Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v dt </math>

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:42:04Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int_c F * v dt </math>

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:40:34Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W_c = \int x </math>

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:38:46Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>Work along a curve = {integral} </math>

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:36:22Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>Wc = </math>

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:35:21Z

Kdarisipudi3: /* A Mathematical Model */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The work done by a certain force that moves along a curve can be calculated through the following integral:
:<math>W =

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:30:18Z

Kdarisipudi3: /* The Main Idea */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time. In the context of mechanical systems, mechanical power is an aggregation of both forces and movement. Specifically, power is an object's velocity multiplied by the object's force. In the context of a shaft, power is calculated as the product of a shaft's torque with its angular velocity.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:26:31Z

Kdarisipudi3: /* Connectedness */

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been fascinated with the evolution of power generation over time as society itself evolves. As a result, I have been interested in the ways power is stored, generated, and applied.

2. How is it connected to your major?

My major of industrial and systems engineering deals with making systems efficient and as optimized as possible. With mechanical power applied in the context of industrial engineering, we can approach and address questions pertaining to the efficient transfer of power within mechanical systems.

3. Is there an interesting industrial application?

An interesting application of mechanical power is seen in hydraulic systems where hydroelectric generators, canals, and other power generations facilities use water to pump and push, consequently storing and transferring mechanical energy.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-10T02:11:02Z

Kdarisipudi3:

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been interested in the topic of energy sustainability and this topic can be related to the use of "green power" - the generation of electric energy from renewable resources - as opposed to consumption of fossil fuels.

2. How is it connected to your major?

My major is environmental engineering, so the subject of power is very much connected to line of study, in regards to reducing air pollution, managing waste and water supply, etc. all with sparing use of power.

3. Is there an interesting industrial application?

It's safe to say that this topic can be applied to every breadth of industry, but one specific example is the construction of dams to produce hydroelectric power from water-propelled turbines.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels. This example is seen further in the late 18th century in German towns where power is transmitted from water wheels to salt wells. This "primitive" is still present in today's day and age, as oil fields transmitting power from pumping engines to pump jacks follow the same principle. Other mediums to transfer mechanical power include hydraulic systems, pneumatic systems, and solid structures such as gears and driveshafts.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>. 
<https://en.wikipedia.org/wiki/Power_transmission#Mechanical_power>

[[Category:Energy]]

Power (Mechanical)

2017-04-09T06:41:47Z

Kdarisipudi3:

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been interested in the topic of energy sustainability and this topic can be related to the use of "green power" - the generation of electric energy from renewable resources - as opposed to consumption of fossil fuels.

2. How is it connected to your major?

My major is environmental engineering, so the subject of power is very much connected to line of study, in regards to reducing air pollution, managing waste and water supply, etc. all with sparing use of power.

3. Is there an interesting industrial application?

It's safe to say that this topic can be applied to every breadth of industry, but one specific example is the construction of dams to produce hydroelectric power from water-propelled turbines.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine. We can observe this concept of power in our every day life through a measure of horsepower, comparing to the power of a horse.

The transmission of mechanical power has evolved over the course of the past few centuries. In the 16th century, mechanical power transmission involved systems of push-rods that linked pumps to water wheels.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>.

[[Category:Energy]]

Power (Mechanical)

2017-04-09T05:58:56Z

Kdarisipudi3:

EDITING CLAIMED BY KAMESH DARISIPUDI (SPRING 2017)

==The Main Idea==

Power is the rate of doing work or the amount of energy consumed over an interval of time.

===A Mathematical Model===

When a force is applied over a distance in a unit of time, power is calculated by

[[File:power(1).jpg |border|right]]

:<math>power = \frac{F \Delta r}{\Delta t} = \frac{W}{\Delta t}</math>
where '''F''' is force, '''Δr''' is displacement, '''Δt''' is the duration of time and '''W''' is work.

It then follows that instantaneous power is

:<math>power = F\cdot v</math>
where '''v''' is velocity.

In rotational systems, power is the product of the [[torque]] <var>τ</var> and [[Angular Velocity]] <var>ω</var>,

:<math>power = \boldsymbol{\tau} \cdot \boldsymbol{\omega}, \,</math>

where '''ω''' is measured in radians per second. The <math> \cdot </math> represents scalar product.

The SI unit for power is watts (J/s)

==Examples==

===Simple===

A certain motor is capable of doing 3000 J of work in 12 s
What is the power output of this motor?

:<math>power = \frac{W}{\Delta t} = \frac{3000 J}{12 s} = 250 J/s </math>

===Middling===

Here are questions dealing with human power. '''(a)''' If you follow a diet of 2000 food calories per day (2000 kC), what is your average rate of energy consumption in watts (power input)? (A food or “large” calorie is a unit of energy equal to 4.2 J; a regular or “small” calorie is equal to 4200 J.) '''(b)''' How many days of a diet of 2000 large calories are equivalent to the gravitational energy change from sea level to the top of Mount Everest, 8848 m above sea level? Assume your weight is 58 kg. (The body is not anywhere near 100% efficient in converting chemical energy into change in altitude. Also note that this is in addition to your basal metabolism.)

'''(a)''' <math>power = \frac{W}{\Delta t} = \frac{2000 kC}{day} \cdot \frac{4200 J}{1 kC} \cdot \frac{1 day}{24 h} \cdot \frac{1 h}{3600 s} = 97.2 J/s </math>

'''(b)''' <math> {\frac{97.2 J}{s}} \cdot \frac{3600 s}{1 h} \cdot \frac{24 h}{1 day} = 8398080 J/day </math>

:<math> \Delta U_g = mg\Delta y = (58 kg)(9.8 m/s^2)(8848 m - 0 m) = 5029203.2 J </math>

:<math> \Delta t = \frac{W}{power} = \frac{5029203.2 J}{8398080 J/day} = 0.599 days </math>

===Difficult===

A bicyclist is going up an inclined slope with an angle <math> \alpha </math> = 2.9 degrees by a uniform speed of 27 km/h. The magnitude of the air resistance force is given by <math> F_{odp} = kv^2 </math> kgs/m where the numerical value of <math> k = 0.3 </math> if the unit of the speed is m/s and the unit of the resistance force is the newton (N). The mass of the bicyclist including the bike is 70 kg. Do not consider the rolling resistance. '''(a)''' What forward force exerted on the bike by the road is needed to make the bicyclist move with constant speed? '''(b)''' How much work does the bicyclist do when riding a distance of 1200 m? '''(c)''' What is the power of the bicyclist during the ride? Assume there is no loss of mechanical energy.
[[File:wikipic1.png |border|right]]
:<math> F_g </math> = weight
:N = normal force exerted on the bike by the road
:<math> F_{odp} </math> = air resistance
:F = unknown forward force exerted on the bike by the road

'''(a)''' <math> x-components: -F_gsin\alpha -F_{odp} + F = 0 </math>

:<math> y-components: N -F_gcos\alpha = 0 </math>

:<math> F = F_gsin\alpha + F_{odp} = mgsin\alpha + kv^2 </math> [[File:wikipic2.png |border|right]]

:<math> v = \frac{27 km}{h} \cdot \frac{1000 m}{1 km} \cdot \frac{1 h}{3600 s} = 7.5 m/s </math>

:<math> F = (70 kg)(9.8 m/s^2)sin2.9 + (0.3 kgs/m)(7.5 m/s)^2 = 51.6 N </math>

'''(b)''' <math> W = F\Delta r = (51.6 N)(1200 m) = 61920 J </math>

'''(c)''' <math> power = F\cdot v = (51.6 N)(7.5 m/s) = 387 J/s </math>

==Connectedness==

1. How is this topic connected to something that you are interested in?

I've always been interested in the topic of energy sustainability and this topic can be related to the use of "green power" - the generation of electric energy from renewable resources - as opposed to consumption of fossil fuels.

2. How is it connected to your major?

My major is environmental engineering, so the subject of power is very much connected to line of study, in regards to reducing air pollution, managing waste and water supply, etc. all with sparing use of power.

3. Is there an interesting industrial application?

It's safe to say that this topic can be applied to every breadth of industry, but one specific example is the construction of dams to produce hydroelectric power from water-propelled turbines.

==History==

Electrical power transmission has replaced mechanical power transmission in all but the very shortest distances. From the 16th century through the industrial revolution to the end of the 19th century mechanical power transmission was the norm. The oldest long-distance power transmission technology involved systems of push-rods connecting waterwheels to distant mine-drainage and brine-well pumps. The unit of power, the watt, was named after the mechanical engineer [[James Watt]], for his contributions to the development of the steam engine.

== See also ==

*[[Work]]
*[[Gravitational Potential Energy]]
*[[Newton's Second Law of Motion]]
*[[System & Surroundings]]

===Further reading===

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. Section 7.6.

===External links===

<http://www.physicsclassroom.com/class/energy/Lesson-1/Power> 
<http://hyperphysics.phy-astr.gsu.edu/hbase/pow.html>

==References==

Chabay, Ruth W.; Sherwood, Bruce A. Matter and Interactions, 4th Edition: 1-2. Wiley. 
"General Mechanics/Work and Power." - Wikibooks, Open Books for an Open World. Web. [https://en.wikibooks.org/wiki/General_Mechanics/Work_and_Power General Mechanics/Work and Power] 
<https://physicstasks.eu/280/bicyclist-going-uphill>. 
<https://en.wikipedia.org/wiki/Power_(physics)>.

[[Category:Energy]]