Physics Book - User contributions [en]

File:Resistors1.jpg

2016-11-28T00:11:38Z

Bmduffy5:

Resistivity

2016-11-28T00:11:19Z

Bmduffy5: /* A Mathematical Model */

Claimed by Brian Duffy -- Fall 2016

Resistivity is the measure of a specific materials ability to impede the flow of an electric current.
The SI unit of resistivity is measured in Ohms per meter,
(<math>({Ohm}⋅{Meter})</math> or <math>({Ω}⋅{m})</math>) and is used to determine the resistance of a given conductor.
Resistivity of an object is almost entirely dependent on two specific factors: temperature and material. Resistivity is totally dependent of the shape or size of whatever material, which is different from overall resistance (which depends on resistivity, length and cross-sectional area of the object).

==The Main Idea==
Resistivity is essentially a constant that describes the resistability of a specific material with respect to the current that passes through it. Some materials will more readily allow the flow of current in comparison to others. For instance, copper has half the resistivity as that of aluminum. Thus, most wires are made out of copper instead of aluminum -- as aluminum impedes the flow of electrical current. Materials like glass, which are poor conductors usually have very high resistivity, and metals which conduct electricity well have much lower resistivity.

===A Mathematical Model===

Resistance is often calculated from resistivity using the following equation <math>R = \frac{\rho L}{A} </math> where R is the resistance <math>\rho</math> is the resistivity, L is the length, and A is the cross-sectional area. While the area of the wire or object may be variable, as well as the length, resistivity remains constant because the material remains constant.

In a circuit the Electrical Resistance is then used to calculate the current in a circuit using Ohm's Law and the following equation <math>I = \frac{|\Delta V|}{R} </math> where '''V''' is the voltage, '''I''' is the current, and '''R''' is the resistance. In this equations voltage and resistance are independent variables, whereas the Current is the dependent variable. This law, while useful, only works for ohmic resistors.

[[File:Ohmic_Resistor1.jpg]]

[[File:Resistors1.jpg]]

The definition provided above is specific to ohmic resistors, as stated. These resistors have a uniform cross-section, where current flows uniformly through them. Instead, a more general definition starts with the idea that an electric field inside a specific material is responsible for the electric current flowing within it. Because most of the resistors we will deal with are uniform, a simple ratio of field to current can be used in regards to resistivity. Thus, the electrical resistivity, or "p" can be defined as the ratio of the electric field to the density of the current it creates:

:<math>\rho=\frac{E}{J}, \,\!</math>

where "ρ" is the resistivity (ohm⋅meter), "E" is the magnitude of the electric field (volts per meter), and J is the magnitude of the current density (amperes per square meter).

Note, remember above where objects with higher resistance are worse conductors and vice versa; this can be seen when ''E'' and ''J'' are inside the conductor.
Conductivity is the inverse of resistivity:

:<math>\sigma=\frac{1}{\rho} = \frac{J}{E}. \,\!</math>

===Water Analogy===
The relationship between resistivity and resistance can be thought of as a series of pipes. Electrical Resistance in a particular material is similar to an analogy of pipes of varying diameter. The larger the pipe the easier it is for water to get through. The resistivity of the "pipes" never change, but the cross sectional area does, in order for the facilitation of "water" flow (current). If you think about sucking water through a thick straw and a coffee stirrer, it is much more difficult to suck the water using the stirrer. Though the materials both are made of may be the same and the water is the same in each instance (i.e. resistivity), it is harder to use the coffee stirrer because of the smaller cross-sectional area.

[[File:Thickness.gif]]

===Resistivity of Materials===
Every conductor has a natural resistivity that is relatively consistent at a given temperature. This number is calculated through experimentation. Here is a list of common conductors and their resistivities.

[[File:Resistivity.jpg]]
(http://avstop.com/ac/Aviation_Maintenance_Technician_Handbook_General/images/fig10-41.jpg)

As you can see, silver has a lower resistivity than copper but because silver is the more expensive of the two, copper is the material of choice when making wires for both construction and classroom use.

===Temperature===
In addition to each material having a different resistivity. The same material at different temperatures may exhibit different resistivities. As materials heat up they become less facilitative of current flow. This is due to the fact that nuclei are moving faster at a sub-atomic level, when this occurs it creates more of a wall which is difficult for electrons to pass through. Because the nuclei do not move as much when the material is colder, it creates less of a "sheet" of nuclei for electrons to pass through.

[[File:TemperatureHot.gif]] [[File:TemperatureCold.gif]]

==Examples==

Here are a few questions involving resistance and resistivity of differing difficulty.

==='''Simple'''===
====Problem 1====

An unknown ohmic resistor is attached to a 3V battery and the current is measured at 1 amp. Calculate the resistance of the unknown resistor.

Also, theoretically, if this resistor has a diameter of 0.03m, and length of 0.06m, what is its resistivity?

====Solution====

Using the equation <math>I = \frac{dV}{R} </math> we can substitute is 1 for I and 3 for dV leaving us with the equation <math>1 = \frac{3}{R} </math>. Solving for R we come to the conclusion that the resistance must be 3 ohms.

To calculate resistivity, we put 7.06E-4m^2 for cross-sectional area, 0.06m for length and 3 ohms for resistance into our equation, getting <math>3 = \frac{\rho 0.06}{7.06E-4} </math>. When you solve for resitivity, you get the answer of 0.0353 ohms/meter

==='''Middling'''===

====Problem 2====

A cylinder of an unknown material has a resistance of 30 ohms. Another cylinder made of the exact same material is twice as long and has a radius that is twice as large. What is the resistance of this cylinder?

====Solution====
Given the equation <math>R = \frac{\rho L}{A} </math> we know that when the length is doubled the resistance must also double. In addition we know that when the radius is doubled, the cross section area must go up by a factor of 4. This means that the resistance would go down by a factor of 1/4. Putting both of those facts together know that R2 = R1 * 2 * 1/4 or R2 = 15 ohms.

==='''Difficult'''===

====Problem 3====

A battery and resistor circuit is connected to a very sensitive ohmmeter and the two are then taken outside and left in the sun on a very hot day. What, if anything, will happen to its reading after being outside for a few minutes and why? Assume the battery is unaffected.

====Solution====

The current reading would be less than what it was while inside. Since the resistor circuit is outside, and hotter than usual the resistor will heat up. As discussed before, resistivity of a material goes up when it is hotter than normal, due to the increase motion of nuclei. The resistance goes up, but because the battery remains unaffected and holds the same amount of charge, the value of the current must go down (due to the inverse relationship of resistance and current). This is all based off of the very simple Ohms Law; <math>I = \frac{V}{R} </math>

==Scope==

Resistivity is a very important topic for all types of engineers, especially electrical and mechanical. It is important to know how circuits operate outside of laboratory conditions, and one thing that changes a lot in the real world is resistivity due to normal and extreme temperature conditions. Also, when building things like phones and other electronics, the total cost is always important. Though gold and silver may have very low resistivity, designers have to look at other metals that offer similar attributes but for lower costs. So overall, it is an important fundamental concept for engineers to understand the idea of resistance, but also play a large role in large scale manufacturing of different devices.

== See also ==

*[[Ohm's Law]]
*[[Thin and Thick Wires]]
*[[Current]]

===Further reading===

1. Matter and Interactions by Ruth Chabay and Bruce Sherwood

===External links===

Helpful Links

1. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html

2. http://www.britannica.com/technology/resistance-electronics

3. http://www.cleanroom.byu.edu/Resistivities.phtml

4. http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

5. http://www.regentsprep.org/Regents/physics/phys03/bresist/default.htm

Helpful Videos

1. https://www.youtube.com/watch?v=-PJcj1TCf_g

2. https://www.youtube.com/watch?v=J4Vq-xHqUo8

==References==

1. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html

2. http://www.britannica.com/technology/resistance-electronics

3. http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

4. http://www.regentsprep.org/Regents/physics/phys03/bresist/default.htm

5. http://forums.extremeoverclocking.com/showthread.php?p=4144637

[[Category:Simple Circuits]]

Resistivity

2016-11-28T00:09:36Z

Bmduffy5: /* A Mathematical Model */

Claimed by Brian Duffy -- Fall 2016

Resistivity is the measure of a specific materials ability to impede the flow of an electric current.
The SI unit of resistivity is measured in Ohms per meter,
(<math>({Ohm}⋅{Meter})</math> or <math>({Ω}⋅{m})</math>) and is used to determine the resistance of a given conductor.
Resistivity of an object is almost entirely dependent on two specific factors: temperature and material. Resistivity is totally dependent of the shape or size of whatever material, which is different from overall resistance (which depends on resistivity, length and cross-sectional area of the object).

==The Main Idea==
Resistivity is essentially a constant that describes the resistability of a specific material with respect to the current that passes through it. Some materials will more readily allow the flow of current in comparison to others. For instance, copper has half the resistivity as that of aluminum. Thus, most wires are made out of copper instead of aluminum -- as aluminum impedes the flow of electrical current. Materials like glass, which are poor conductors usually have very high resistivity, and metals which conduct electricity well have much lower resistivity.

===A Mathematical Model===

Resistance is often calculated from resistivity using the following equation <math>R = \frac{\rho L}{A} </math> where R is the resistance <math>\rho</math> is the resistivity, L is the length, and A is the cross-sectional area. While the area of the wire or object may be variable, as well as the length, resistivity remains constant because the material remains constant.

In a circuit the Electrical Resistance is then used to calculate the current in a circuit using Ohm's Law and the following equation <math>I = \frac{|\Delta V|}{R} </math> where '''V''' is the voltage, '''I''' is the current, and '''R''' is the resistance. In this equations voltage and resistance are independent variables, whereas the Current is the dependent variable. This law, while useful, only works for ohmic resistors.

[[File:Ohmic_Resistor1.jpg]]

[[File:Res2.jpg]]

The definition provided above is specific to ohmic resistors, as stated. These resistors have a uniform cross-section, where current flows uniformly through them. Instead, a more general definition starts with the idea that an electric field inside a specific material is responsible for the electric current flowing within it. Because most of the resistors we will deal with are uniform, a simple ratio of field to current can be used in regards to resistivity. Thus, the electrical resistivity, or "p" can be defined as the ratio of the electric field to the density of the current it creates:

:<math>\rho=\frac{E}{J}, \,\!</math>

where "ρ" is the resistivity (ohm⋅meter), "E" is the magnitude of the electric field (volts per meter), and J is the magnitude of the current density (amperes per square meter).

Note, remember above where objects with higher resistance are worse conductors and vice versa; this can be seen when ''E'' and ''J'' are inside the conductor.
Conductivity is the inverse of resistivity:

:<math>\sigma=\frac{1}{\rho} = \frac{J}{E}. \,\!</math>

===Water Analogy===
The relationship between resistivity and resistance can be thought of as a series of pipes. Electrical Resistance in a particular material is similar to an analogy of pipes of varying diameter. The larger the pipe the easier it is for water to get through. The resistivity of the "pipes" never change, but the cross sectional area does, in order for the facilitation of "water" flow (current). If you think about sucking water through a thick straw and a coffee stirrer, it is much more difficult to suck the water using the stirrer. Though the materials both are made of may be the same and the water is the same in each instance (i.e. resistivity), it is harder to use the coffee stirrer because of the smaller cross-sectional area.

[[File:Thickness.gif]]

===Resistivity of Materials===
Every conductor has a natural resistivity that is relatively consistent at a given temperature. This number is calculated through experimentation. Here is a list of common conductors and their resistivities.

[[File:Resistivity.jpg]]
(http://avstop.com/ac/Aviation_Maintenance_Technician_Handbook_General/images/fig10-41.jpg)

As you can see, silver has a lower resistivity than copper but because silver is the more expensive of the two, copper is the material of choice when making wires for both construction and classroom use.

===Temperature===
In addition to each material having a different resistivity. The same material at different temperatures may exhibit different resistivities. As materials heat up they become less facilitative of current flow. This is due to the fact that nuclei are moving faster at a sub-atomic level, when this occurs it creates more of a wall which is difficult for electrons to pass through. Because the nuclei do not move as much when the material is colder, it creates less of a "sheet" of nuclei for electrons to pass through.

[[File:TemperatureHot.gif]] [[File:TemperatureCold.gif]]

==Examples==

Here are a few questions involving resistance and resistivity of differing difficulty.

==='''Simple'''===
====Problem 1====

An unknown ohmic resistor is attached to a 3V battery and the current is measured at 1 amp. Calculate the resistance of the unknown resistor.

Also, theoretically, if this resistor has a diameter of 0.03m, and length of 0.06m, what is its resistivity?

====Solution====

Using the equation <math>I = \frac{dV}{R} </math> we can substitute is 1 for I and 3 for dV leaving us with the equation <math>1 = \frac{3}{R} </math>. Solving for R we come to the conclusion that the resistance must be 3 ohms.

To calculate resistivity, we put 7.06E-4m^2 for cross-sectional area, 0.06m for length and 3 ohms for resistance into our equation, getting <math>3 = \frac{\rho 0.06}{7.06E-4} </math>. When you solve for resitivity, you get the answer of 0.0353 ohms/meter

==='''Middling'''===

====Problem 2====

A cylinder of an unknown material has a resistance of 30 ohms. Another cylinder made of the exact same material is twice as long and has a radius that is twice as large. What is the resistance of this cylinder?

====Solution====
Given the equation <math>R = \frac{\rho L}{A} </math> we know that when the length is doubled the resistance must also double. In addition we know that when the radius is doubled, the cross section area must go up by a factor of 4. This means that the resistance would go down by a factor of 1/4. Putting both of those facts together know that R2 = R1 * 2 * 1/4 or R2 = 15 ohms.

==='''Difficult'''===

====Problem 3====

A battery and resistor circuit is connected to a very sensitive ohmmeter and the two are then taken outside and left in the sun on a very hot day. What, if anything, will happen to its reading after being outside for a few minutes and why? Assume the battery is unaffected.

====Solution====

The current reading would be less than what it was while inside. Since the resistor circuit is outside, and hotter than usual the resistor will heat up. As discussed before, resistivity of a material goes up when it is hotter than normal, due to the increase motion of nuclei. The resistance goes up, but because the battery remains unaffected and holds the same amount of charge, the value of the current must go down (due to the inverse relationship of resistance and current). This is all based off of the very simple Ohms Law; <math>I = \frac{V}{R} </math>

==Scope==

Resistivity is a very important topic for all types of engineers, especially electrical and mechanical. It is important to know how circuits operate outside of laboratory conditions, and one thing that changes a lot in the real world is resistivity due to normal and extreme temperature conditions. Also, when building things like phones and other electronics, the total cost is always important. Though gold and silver may have very low resistivity, designers have to look at other metals that offer similar attributes but for lower costs. So overall, it is an important fundamental concept for engineers to understand the idea of resistance, but also play a large role in large scale manufacturing of different devices.

== See also ==

*[[Ohm's Law]]
*[[Thin and Thick Wires]]
*[[Current]]

===Further reading===

1. Matter and Interactions by Ruth Chabay and Bruce Sherwood

===External links===

Helpful Links

1. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html

2. http://www.britannica.com/technology/resistance-electronics

3. http://www.cleanroom.byu.edu/Resistivities.phtml

4. http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

5. http://www.regentsprep.org/Regents/physics/phys03/bresist/default.htm

Helpful Videos

1. https://www.youtube.com/watch?v=-PJcj1TCf_g

2. https://www.youtube.com/watch?v=J4Vq-xHqUo8

==References==

1. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html

2. http://www.britannica.com/technology/resistance-electronics

3. http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

4. http://www.regentsprep.org/Regents/physics/phys03/bresist/default.htm

5. http://forums.extremeoverclocking.com/showthread.php?p=4144637

[[Category:Simple Circuits]]

File:Ohmic Resistor1.jpg

2016-11-28T00:09:18Z

Bmduffy5:

Resistivity

2016-11-28T00:07:13Z

Bmduffy5:

Claimed by Brian Duffy -- Fall 2016

Resistivity is the measure of a specific materials ability to impede the flow of an electric current.
The SI unit of resistivity is measured in Ohms per meter,
(<math>({Ohm}⋅{Meter})</math> or <math>({Ω}⋅{m})</math>) and is used to determine the resistance of a given conductor.
Resistivity of an object is almost entirely dependent on two specific factors: temperature and material. Resistivity is totally dependent of the shape or size of whatever material, which is different from overall resistance (which depends on resistivity, length and cross-sectional area of the object).

==The Main Idea==
Resistivity is essentially a constant that describes the resistability of a specific material with respect to the current that passes through it. Some materials will more readily allow the flow of current in comparison to others. For instance, copper has half the resistivity as that of aluminum. Thus, most wires are made out of copper instead of aluminum -- as aluminum impedes the flow of electrical current. Materials like glass, which are poor conductors usually have very high resistivity, and metals which conduct electricity well have much lower resistivity.

===A Mathematical Model===

Resistance is often calculated from resistivity using the following equation <math>R = \frac{\rho L}{A} </math> where R is the resistance <math>\rho</math> is the resistivity, L is the length, and A is the cross-sectional area. While the area of the wire or object may be variable, as well as the length, resistivity remains constant because the material remains constant.

In a circuit the Electrical Resistance is then used to calculate the current in a circuit using Ohm's Law and the following equation <math>I = \frac{|\Delta V|}{R} </math> where '''V''' is the voltage, '''I''' is the current, and '''R''' is the resistance. In this equations voltage and resistance are independent variables, whereas the Current is the dependent variable. This law, while useful, only works for ohmic resistors.

[[File:Ohmic_Resistor.jpg]]

[[File:Res2.jpg]]

The definition provided above is specific to ohmic resistors, as stated. These resistors have a uniform cross-section, where current flows uniformly through them. Instead, a more general definition starts with the idea that an electric field inside a specific material is responsible for the electric current flowing within it. Because most of the resistors we will deal with are uniform, a simple ratio of field to current can be used in regards to resistivity. Thus, the electrical resistivity, or "p" can be defined as the ratio of the electric field to the density of the current it creates:

:<math>\rho=\frac{E}{J}, \,\!</math>

where "ρ" is the resistivity (ohm⋅meter), "E" is the magnitude of the electric field (volts per meter), and J is the magnitude of the current density (amperes per square meter).

Note, remember above where objects with higher resistance are worse conductors and vice versa; this can be seen when ''E'' and ''J'' are inside the conductor.
Conductivity is the inverse of resistivity:

:<math>\sigma=\frac{1}{\rho} = \frac{J}{E}. \,\!</math>

===Water Analogy===
The relationship between resistivity and resistance can be thought of as a series of pipes. Electrical Resistance in a particular material is similar to an analogy of pipes of varying diameter. The larger the pipe the easier it is for water to get through. The resistivity of the "pipes" never change, but the cross sectional area does, in order for the facilitation of "water" flow (current). If you think about sucking water through a thick straw and a coffee stirrer, it is much more difficult to suck the water using the stirrer. Though the materials both are made of may be the same and the water is the same in each instance (i.e. resistivity), it is harder to use the coffee stirrer because of the smaller cross-sectional area.

[[File:Thickness.gif]]

===Resistivity of Materials===
Every conductor has a natural resistivity that is relatively consistent at a given temperature. This number is calculated through experimentation. Here is a list of common conductors and their resistivities.

[[File:Resistivity.jpg]]
(http://avstop.com/ac/Aviation_Maintenance_Technician_Handbook_General/images/fig10-41.jpg)

As you can see, silver has a lower resistivity than copper but because silver is the more expensive of the two, copper is the material of choice when making wires for both construction and classroom use.

===Temperature===
In addition to each material having a different resistivity. The same material at different temperatures may exhibit different resistivities. As materials heat up they become less facilitative of current flow. This is due to the fact that nuclei are moving faster at a sub-atomic level, when this occurs it creates more of a wall which is difficult for electrons to pass through. Because the nuclei do not move as much when the material is colder, it creates less of a "sheet" of nuclei for electrons to pass through.

[[File:TemperatureHot.gif]] [[File:TemperatureCold.gif]]

==Examples==

Here are a few questions involving resistance and resistivity of differing difficulty.

==='''Simple'''===
====Problem 1====

An unknown ohmic resistor is attached to a 3V battery and the current is measured at 1 amp. Calculate the resistance of the unknown resistor.

Also, theoretically, if this resistor has a diameter of 0.03m, and length of 0.06m, what is its resistivity?

====Solution====

Using the equation <math>I = \frac{dV}{R} </math> we can substitute is 1 for I and 3 for dV leaving us with the equation <math>1 = \frac{3}{R} </math>. Solving for R we come to the conclusion that the resistance must be 3 ohms.

To calculate resistivity, we put 7.06E-4m^2 for cross-sectional area, 0.06m for length and 3 ohms for resistance into our equation, getting <math>3 = \frac{\rho 0.06}{7.06E-4} </math>. When you solve for resitivity, you get the answer of 0.0353 ohms/meter

==='''Middling'''===

====Problem 2====

A cylinder of an unknown material has a resistance of 30 ohms. Another cylinder made of the exact same material is twice as long and has a radius that is twice as large. What is the resistance of this cylinder?

====Solution====
Given the equation <math>R = \frac{\rho L}{A} </math> we know that when the length is doubled the resistance must also double. In addition we know that when the radius is doubled, the cross section area must go up by a factor of 4. This means that the resistance would go down by a factor of 1/4. Putting both of those facts together know that R2 = R1 * 2 * 1/4 or R2 = 15 ohms.

==='''Difficult'''===

====Problem 3====

A battery and resistor circuit is connected to a very sensitive ohmmeter and the two are then taken outside and left in the sun on a very hot day. What, if anything, will happen to its reading after being outside for a few minutes and why? Assume the battery is unaffected.

====Solution====

The current reading would be less than what it was while inside. Since the resistor circuit is outside, and hotter than usual the resistor will heat up. As discussed before, resistivity of a material goes up when it is hotter than normal, due to the increase motion of nuclei. The resistance goes up, but because the battery remains unaffected and holds the same amount of charge, the value of the current must go down (due to the inverse relationship of resistance and current). This is all based off of the very simple Ohms Law; <math>I = \frac{V}{R} </math>

==Scope==

Resistivity is a very important topic for all types of engineers, especially electrical and mechanical. It is important to know how circuits operate outside of laboratory conditions, and one thing that changes a lot in the real world is resistivity due to normal and extreme temperature conditions. Also, when building things like phones and other electronics, the total cost is always important. Though gold and silver may have very low resistivity, designers have to look at other metals that offer similar attributes but for lower costs. So overall, it is an important fundamental concept for engineers to understand the idea of resistance, but also play a large role in large scale manufacturing of different devices.

== See also ==

*[[Ohm's Law]]
*[[Thin and Thick Wires]]
*[[Current]]

===Further reading===

1. Matter and Interactions by Ruth Chabay and Bruce Sherwood

===External links===

Helpful Links

1. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html

2. http://www.britannica.com/technology/resistance-electronics

3. http://www.cleanroom.byu.edu/Resistivities.phtml

4. http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

5. http://www.regentsprep.org/Regents/physics/phys03/bresist/default.htm

Helpful Videos

1. https://www.youtube.com/watch?v=-PJcj1TCf_g

2. https://www.youtube.com/watch?v=J4Vq-xHqUo8

==References==

1. http://hyperphysics.phy-astr.gsu.edu/hbase/electric/resis.html

2. http://www.britannica.com/technology/resistance-electronics

3. http://www.nist.gov/data/PDFfiles/jpcrd155.pdf

4. http://www.regentsprep.org/Regents/physics/phys03/bresist/default.htm

5. http://forums.extremeoverclocking.com/showthread.php?p=4144637

[[Category:Simple Circuits]]

2016-04-14T04:39:03Z

Bmduffy5: /* Main Idea */

'''CLAIMED BY BRIAN DUFFY'''
This topic covers Newton's Law of Universal Gravitation.

Created by Ankit Khanal.
[[File:Newtonlawofgravitation.jpg|250px|thumb|right|Newton's Law of Universal Gravitation]]
==Main Idea==
Gravitational Force, defined in Newton's law of Universal Gravitation is the attractive force between any two bodies of any mass, at any distance. Gravity is one of the essential forces that hold the Universe together. The law states that every object in the universe is exerting a pulling force on each other that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Gravitational Force is considered to be the weakest of the four fundamental forces of nature: strong nuclear force, electromagnetic force, weak nuclear force and finally gravity. The first test of this law was performed in the laboratory by the British scientist Henry Cavendish in 1798, 72 years after the death of Sir Isaac Newton.

===A Mathematical Model===

The gravitational law states that every point mass attracts other point mass by a force pointing along the line intersecting both points. The force is directly proportional to the product of the masses and inversely proportional to the square distance between them.
[[File:1232.png|200px|thumb|right|Two masses at a certain distance attracting each other]]
::<math>|\vec{\mathbf{F}}_{grav}|= G \frac{m_1 m_2}{r^2}\ </math>

:where,
::* ''F'' is the force between the masses;
::* ''G'' is the [https://en.wikipedia.org/wiki/Gravitational_constant gravitational constant] <math>6.674×10^{−11} \frac{N m^2}{kg^2}\ </math>;
::* ''m''1 is the first mass;
::* ''m''2 is the second mass;
::* ''r'' is the distance between the centers of the masses.

====Vector Form====

Newton's law of universal gravitation can be expressed in vector form to account for the direction of the gravitational force.
::<math>\vec{\mathbf{F}}_{grav}= -G \frac{m_1 m_2}{r^2}\ </math><math>\mathbf{\hat{r}}</math>

[[File:Relative_position.png|200px|thumb|right|Distance between two objects, object 2 and object 1 used to calculate the relative position:"final minus initial" and its direction]]
The relative position vector <math>\vec{\mathbf{r}}</math> is the distance from the center of object 1 to the center of object 2. The relative position between these two object can be represented as <math>\vec{\mathbf{r}}_{2-1}</math>, which means that the object 1 is the initial location and object 2 is the final location. The magnitude of <math>\vec{\mathbf{r}}</math>, represented as <math>r^2</math>, is the distance between the center of two objects.

The direction of the gravitational force on object 2 by object 1 is specified by a unit vector,<math>-\mathbf{\hat{r}}</math>, with a minus sign in front of it. The minus sign shows that the force on object 2 by 1 is in the opposite direction to <math>\mathbf{\hat{r}}</math>.
[[File:Gravitational_field.png|200px|thumb|right|Gravitational Field around the Earth. Near the surface of the Earth an object m experience a force of mg. The g of the field decreases with increase in distance from the Earth]]
Steps for calculating Gravitational Force
::* Calculate <math>\vec{\mathbf{r}}</math> by using the position of the center of object 2 relative to the center of object 1 i.e. <math>\vec{\mathbf{r}} = \vec{\mathbf{r}}_2-\vec{\mathbf{r}}_1</math>
::* Calculate center to center distance between the objects, <math>|\vec{\mathbf{r}}|</math>
::* Calculate <math>|\vec{\mathbf{F}}_{grav}|</math> using the gravitational force formula
::* Calculate the unit vector <math>-\mathbf{\hat{r}}</math> =<math>\vec{\mathbf{r}}</math> /<math>|\vec{\mathbf{r}}|</math>
::* Multiply the Magnitude, <math>|\vec{\mathbf{F}}_{grav}|</math>, by the unit vector.

Gravitational Force near the surface of the earth is calculated by using:
::<math>\vec{\mathbf{F}}_{grav} = mg</math>
:where,
::* ''F'' is the force;
::* ''g'' is the magnitude of gravitational field near the Earth's surface;
::* ''m'' is the mass of the object.

===A Computational Model===

A computational representation of Gravitational force can be created using VPython.

The code below show how we can find the net force, momentum and final position in Python.

<code>
while t < 235920:
rate(5000)
'''#Calculate the Gravitational Force acting on the craft due to Earth and Moon'''</code>
[[File:Spacecraf.png|300px|thumb|right|Gravitational Interaction between the space craft, Earth and the Moon. The red arrow represents the momentum of the spacecraft at the moment]]

'''''#Craft to Earth Calculations'''''
r = craft.pos - Earth.pos
rmag = sqrt(r.x**2 + r.y**2 + r.z**2)
fgravmag = (G * mEarth * mcraft) / (rmag**2
rhat = r / rmag
fgrav = -1 * fgravmag * rhat
'''''#Craft to Moon Calculations'''''
rmc = craft.pos - Moon.pos
rmcmag = sqrt(rmc.x**2 + rmc.y**2 + rmc.z**2)
fgravmoonmag = (G * mMoon * mcraft) / (rmcmag**2)
rmchat = rmc / rmcmag
fgravmoon = -1 * fgravmoonmag * rmchat
'''''#Fnet is sum of these two forces'''''
fnet = fgrav + fgravmoon

From this we can calculate the change in momentum and the new positions using the net force.

'''''#Update the position of the spacecraft by calculating the new momentum and the new velocity'''''
pcraft = pcraft + (fnet*deltat)
vcraft = pcraft / mcraft
craft.pos = craft.pos + (vcraft*deltat)

==Units==
In Si Unit, Gravitational Force F is measured in Newtons (N), the two masses, m1 and m2 are measures in kilograms (Kg), the distance is measured in meters (m), and the gravitational constant G is measured in N m2 kg-2 and has a value of 6.674×10−11 N m2kg−2. The gravitation constant G is universal because no matter how big or small the masses are it is same for any interacting masses. The value of constant G appeared in Newton's law of universal gravitation, but it was not measured until seventy two years after Newton's death by Henry Cavendish with his Cavendish experiment in 1798. The value of gravitational constant was also the first test of Newton's law between two masses in Laboratory.

==Examples==

* Problems taken from Textbook and old Test on T-square

===Simple===

''Determine the force of gravitational attraction between the earth (m = <math>6 x 10^{24} kg</math>) and a 70 kg physics student if the student is in an airplane at 40,000 feet above earth's surface. This would place the student a distance of <math>6.38 x 10^6 m </math> from earth's center.''

The solution of the problem involves substituting known values of G (<math>6.673 x 10^{-11}</math> N m2/kg2), m1 (<math>6 x 10^{24}</math> kg), m2 (70 kg) and d (<math>6.38 x 10^6 m </math>) into the universal gravitation equation and solving for Fgrav. The solution is as follows:

::<math> |\mathbf{F_{g}}| = G \frac{m_E m_M}{r^2} </math>
::<math> |\mathbf{F_{g}}| = 6.7x10^{-11} \frac{6e24 * 70}{6.38 x 10^6} </math>
::<math> |\mathbf{F_{g}}| =686 N </math>

===Middling===
''The mass of the Earth is <math>6 x 10^ {24}</math> kg, and the mass of the Moon is <math>7 x 10^ {22}</math> kg. At a particular instance the moon is at location <math><2.8 x 10^8,0,-2.8 x 10^8></math> m, in a coordinate system whose origin is at the center of the earth. '''(a)''' What is <math>\vec{\mathbf{r}}</math>, the relative position vector from the Earth to the Moon? '''(b)''' What is <math>|\vec{\mathbf{r}}|</math>? '''(c)''' What is the unit vector <math>\vec{\mathbf{r}}</math>? '''(d)''' What is the gravitation force exerted by the Earth on the Moon? Your answer should be in vector.''

The solution of the problem involves substituting known values of G (<math>6.673 x 10^{-11}</math>N m2/kg2), m1 <math>6 x 10^ {24}</math> kg, m2 <math>7 x 10^ {22}</math> kg and d <math><2.8 x 10^8,0,-2.8 x 10^8></math> m m into the universal gravitation equation and solving for Fgrav. The solution is as follows:
[[File:Gravitation_force_earth_moon.gif|700px|thumb|right|Gravitational Force Between Moon and Earth]]

: '''(a)''' The position vector of Moon relative to Earth is,
:: <math>\vec{\mathbf{r}}</math> = <math><2.8 x 10^8,0,-2.8 x 10^8> - <0,0,0></math>
:: <math>\vec{\mathbf{r}}</math> = <math><2.8 x 10^8,0,-2.8 x 10^8></math> m

: '''(b)''' The magnitude of position vector of Moon relative to Earth is,
:: <math>|\vec{\mathbf{r}}|</math> = <math>\sqrt{(2.8 x 10^8)^2+0^2+(-2.8 x 10^8)^2}</math>
:: <math>|\vec{\mathbf{r}}|</math> = <math> 4.0 x 10^8 </math> m

: '''(c)''' The unit vector of Moon relative to Earth is,
:: <math>\mathbf{\hat{r}} = \frac {\vec{\mathbf{r}}}{|\vec{\mathbf{r}}|}</math>
:: <math>\mathbf{\hat{r}} = \frac{<2.8 x 10^8,0,-2.8 x 10^8>}{4.0 x 10^8} </math>
:: <math>\mathbf{\hat{r}} = <0.7,0,-0.7></math>

: '''(d)''' The expression for the gravitational force on the Moon by the Earth is,
::<math> |\vec{\mathbf{F}}_{g}| = G \frac{m_E m_M}{|\vec{\mathbf{r^2}}|} </math>
::<math> |\vec{\mathbf{F}}_{g}| = 6.7x10^{-11} \frac{6 x 10^ {24} * 7 x 10^ {22}}{(4.0 x 10^8)^2 }</math>
::<math> |\vec{\mathbf{F}}_{g}| = 1.76x10^{20}N</math>

::<math> \vec{\mathbf{F}}_{g}</math> = <math> -|\mathbf{F_{grav}}|*{\mathbf{\hat{r}}}</math>
::<math> \vec{\mathbf{F}}_{g}</math> = <math>-1.76x10^{20}*<0.7,0,-0.7> N </math>
::<math> \vec{\mathbf{F}}_{g}</math> = <math><-1.232x10^{20},0,1.232x10^{20}> N </math>

===Difficult===

''In the following problems you will be asked to calculate the net gravitational force acting on the Moon.To do so, please use the following variables:''
:::'''Mass'''
:::''mS - Mass of the Sun''
:::''mE - Mass of the Earth''
:::''mM - Mass of the Moon''

:::'''Initial Positions'''
:::''<math>\vec{\mathbf{r_{S}}} = <0,0,0> m </math>- Position of the Sun''
:::''<math>\vec{\mathbf{r_{E}}}</math> = <<math>L,0,0></math> m- Position of the Earth''
:::''<math>\vec{\mathbf{r_{M}}}</math> = <<math>L,h,0></math> m - Position of the Moon''

'''(a)''' ''Calculate the gravitational force on the Moon due to the Earth''
'''(b)''' ''Calculate the gravitational force on the Moon due to the Sun''
'''(c)''' ''Calculate the net gravitational force on the Moon''

'''(a)''' The gravitational force on the Moon due to the Earth is,
::<math>\vec{\mathbf{r}} = \vec{\mathbf{r_M}}-\vec{\mathbf{r_E}}</math>
::<math>\vec{\mathbf{r}}</math> = <<math>L,h,0></math>-<<math>L,0,0></math>
::<math>\vec{\mathbf{r}}</math> = <<math>0,h,0> m</math>

:: <math>|\vec{\mathbf{r}}| = \sqrt{(0^2+h^2+0^2)}</math>
:: <math>|\vec{\mathbf{r}}| = h </math> ''m''

::<math>\mathbf{\hat{r}} = \frac {\vec{\mathbf{r}}}{|\vec{\mathbf{r}}|}</math>
::<math>\mathbf{\hat{r}} = \frac{<0,h,0>}{h} </math>
::<math>\mathbf{\hat{r}} = <0,1,0> </math>

::<math> |\vec{\mathbf{F}}_{g_1}| = G \frac{m_E m_M}{|\vec{\mathbf{r^2}|}} </math>
::<math> |\vec{\mathbf{F}}_{g_1}| = G \frac{m_E m_M}{h^2} </math>

::<math> \vec{\mathbf{F}}_{g_1} = -|\vec{\mathbf{F}}_{g_1}|*{mathbf{\hat{r}}}</math>
::<math> \vec{\mathbf{F}}_{g_1} = <0,-G \frac{m_E m_M}{h^2},0> N </math>

'''(b)''' The gravitational force on the Moon due to the Sun is,

::<math>\vec{\mathbf{r}} = \vec{\mathbf{r_M}}-\vec{\mathbf{r_S}}</math>
::<math>\vec{\mathbf{r}}</math> = <<math>L,h,0>-<0,0,0></math>
::<math>\vec{\mathbf{r}}</math> = <<math>L,h,0> m </math>

::<math>|\vec{\mathbf{r}}| = \sqrt{(L^2+h^2+0^2)}</math>
:: <math>|\vec{\mathbf{r}}| = \sqrt{(L^2+h^2)}</math>

::<math>\mathbf{\hat{r}} = \frac {\vec{\mathbf{r}}}{|\vec{\mathbf{r}}|}</math>
::<math>\mathbf{\hat{r}} = \frac {<(L,h,0)>}{\sqrt{(L^2+h^2)}} </math>

::<math> |\vec{\mathbf{F}}_{g_2}| = G \frac{m_S m_M}{|\vec{\mathbf{r^2}|}} </math>
::<math> |\vec{\mathbf{F}}_{g_2}| = G \frac{m_S m_M}{L^2+h^2} </math>

::<math> \vec{\mathbf{F}}_{g_2} = -|\vec{\mathbf{F}}_{g_2}|*{\mathbf{\hat{r}}}</math>
::<math> \vec{\mathbf{F}}_{g_2} = <-G \frac{m_S m_M L}{(L^2+h^2)^{3/2}},-G \frac{m_S m_M h}{(L^2+h^2)^{3/2}},0> N </math>

'''(c)''' The net gravitational force on the Moon is,
::<math>\vec{\mathbf{F}}_{net} = \vec{\mathbf{F}}_{g_1}+\vec{\mathbf{F}}_{g_2}</math>
::<math>\vec{\mathbf{F}}_{net} = <0,-G \frac{m_E m_M}{h^2},0> +<-G \frac{m_S m_M L}{(L^2+h^2)^{3/2}},-G \frac{m_S m_M h}{(L^2+h^2)^{3/2}},0></math>
::<math>\vec{\mathbf{F}}_{net} = <-G \frac{m_S m_M L}{(L^2+h^2)^{3/2}},-G \frac{m_E m_M}{h^2}-G \frac{m_S m_M h}{(L^2+h^2)^{3/2}},0> N </math>

==Connectedness==
:* The very first thing I remember learning in physics was gravity, how it worked and why it was so important. This is the reason why I am very interested in physical interactions. Newton's Law of Universal Gravitation is very important when it comes to describing and understanding the interactions between planets and there moons to how entire galaxies move.

:* This topic has nothing to do with my major.

:* Newton's Law of Universal Gravitation is related to every physical that we see around us. Every time you jump, we experience gravity. It pulls us back down to the ground. Without gravity, we'd float off into the atmosphere, along with all of the other matter on earth. It can be used to discover new planets, find the escape velocity needed to escape from any planet, and it can also used to launch satellites in the space and keep it spinning in an circular path. [http://sandyabouttechnology.blogspot.com/2012/09/gravitation-force-and-application.html See] to learn more about gravitational Force and its application.

==History==
[[File:SirIsaacNewton.jpg ‎|150px|thumb|right|Portrait of Isaac Newton]]
Sir Isaac Newton, the 17th century English scientist, is popularly associated with gravity but scientists had been studying gravity for centuries before Newton. Based on Galileo’s and Kepler’s works, Newton published “Principia” in 1687, in which he explained that the force that makes planets move around the Sun is the same force that makes objects fall in the Earth. Newton explained that gravity is a force that attracts different masses and is inversely proportional to the square distance between the objects. However Newtons did not fully understand how gravitational force acted from one object to another over vast distances. Despite this flaw, this law has been used for more than three centuries and is still used by scientists. Albert Einstein rejected in theory and explained it in his general theory of relativity, in which he stated that gravity is not a force but it is a consequence of the curvature of space-time. Even Einstein did not fully explain gravitational force.

== See also ==

Relevant materials

===Further reading===

Matter and Interactions By Ruth W. Chabay, Bruce A. Sherwood - Chapter 3

===External links===

:'''1:''' http://www.physicsclassroom.com/Class/circles/u6l3c.cfm
:'''2:''' https://www.youtube.com/watch?v=0hOuNtRMSAI
:'''3:''' http://csep10.phys.utk.edu/astr161/lect/history/newtongrav.html
:'''4:''' http://www.animations.physics.unsw.edu.au/jw/gravity.htm
:'''5:''' http://sandyabouttechnology.blogspot.com/2012/09/gravitation-force-and-application.html

==References==

:'''1:''' "Newton's Law of Universal Gravitation." Newton's Law of Universal Gravitation. N.p., n.d. Web. 27 Nov. 2015.<http://www.physicsclassroom.com/Class/circles/u6l3c.cfm>

:'''2:''' Rankin, Alan. "What Is a Gravitational Force?" WiseGeek. Conjecture, n.d. Web. 28 Nov. 2015. <http://www.wisegeek.com/what-is-a-gravitational-force.htm>

:'''3:''' "History of Gravity." History of Gravity. N.p., n.d. Web. 28 Nov. 2015. <https://www.physics.wisc.edu/museum/Exhibits-1/Mechanics/GravPit/index_HistGrav-2.html>

:'''4:''' <https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation>

:'''5:''' <https://amabute.files.wordpress.com/2012/01/applenewton1.jpg>

[[Category:Interactions]]