Physics Book - User contributions [en]

Speed of Sound in Solids

2023-04-17T20:21:17Z

Claudiatiller7: /* Applications of Calculating Speed of Sounds in Solids */

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Effects on Sound Travel==
Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is highly applicable. The human ear captures sound waves through the outer cartilage of the ear, called the Pinna. The sound waves then travel up the ear canal and arrive at the ear drum which vibrates from the sound waves. After traveling through the inner ear, the vibrations arrive at the Cochlea. The Cochlea transfers the vibrations into information that auditory nerves can analyze.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
Computing the speed of sound in solids depends on a mass' interatomic properties, such as interatomic bond length. In this specific case, an object's elasticity depends on the interatomic bond length. There are many applications that connect to the ability to compute the speed of sounds.

Seismic and ultrasonic imaging are also fields that benefit greatly from the calculation of sound speeds in solids. Seismic imaging refers to capturing images of the subsurface structure of the Earth. Seismic waves can be generated by earthquakes or other sources. Engineers and scientists can then locate gas and oil reservoirs, monitor activity such as volcano eruptions, and geological formations. In ultrasonic imaging, ultrasonic waves can be sent through the body and have the time it takes to bounce back be measured. This then creates images of internal organs or tissues, used in cases such as prenatal imaging as well as medical diagnosing. In more specific fields such as in Industrial Engineering, these calculations could be applied to questions regarding how to build a soundproof area. It would therefore be optimal to select a solid with a low speed of sound velocity, with a solid that has tightly packed particles. There are many vast applications to this, as an object's ability to block or allow sound waves through it. However, some cases require contractors to build structures that allow sound to travel through. Material testing uses the calculation of sound in solids to determine mechanical properties of such materials. Engineers can then calculate the material's elasticity, stiffness, and other properties. Knowledge regarding how solids are structured and how they correlate with the speed of sounds in those solids are vital to building structures that meet the criteria.

Overall, being able to calculate the speed of sounds in solids has a wide range of applications in engineering, medicine, and science.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T20:07:04Z

Claudiatiller7: /* Connectedness */

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Effects on Sound Travel==
Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is highly applicable. The human ear captures sound waves through the outer cartilage of the ear, called the Pinna. The sound waves then travel up the ear canal and arrive at the ear drum which vibrates from the sound waves. After traveling through the inner ear, the vibrations arrive at the Cochlea. The Cochlea transfers the vibrations into information that auditory nerves can analyze.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Applications of Calculating Speed of Sounds in Solids==
Computing the speed of sound in solids depends on a mass' interatomic properties, such as interatomic bond length. In this specific case, an object's elasticity depends on the interatomic bond length. There are many applications that connect to the ability to compute the speed of sounds.

In more specific fields such as in Industrial Engineering, these calculations could be applied to questions regarding how to build a soundproof area. It would therefore be optimal to select a solid with a low speed of sound velocity, with a solid that has tightly packed particles. There are many vast applications to this, as an object's ability to block or allow sound waves through it. However, some cases require contractors to build structures that allow sound to travel through. Material testing uses the calculation of sound in solids to determine mechanical properties of such materials. Engineers can then calculate the material's elasticity, stiffness, and other properties. Knowledge regarding how solids are structured and how they correlate with the speed of sounds in those solids are vital to building structures that meet the criteria.

Seismic and ultrasonic imaging are also fields that benefit greatly from the calculation of sound speeds in solids. Seismic imaging refers to capturing images of the subsurface structure of the Earth. Seismic waves can be generated by earthquakes or other sources. Engineers and scientists can then locate gas and oil reservoirs, monitor activity such as volcano eruptions, and geological formations. In ultrasonic imaging, ultrasonic waves can be sent through the body and have the time it takes to bounce back be measured. This then creates images of internal organs or tissues, used in cases such as prenatal imaging as well as medical diagnosing.

Overall, being able to calculate the speed of sounds in solids has a wide range of applications in engineering, medicine, and science.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T02:40:33Z

Claudiatiller7: /* The Structure of Solids and Sound */

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Effects on Sound Travel==
Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is highly applicable. The human ear captures sound waves through the outer cartilage of the ear, called the Pinna. The sound waves then travel up the ear canal and arrive at the ear drum which vibrates from the sound waves. After traveling through the inner ear, the vibrations arrive at the Cochlea. The Cochlea transfers the vibrations into information that auditory nerves can analyze.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
Computing the speed of sound in solids depends on a mass' interatomic properties, such as interatomic bond length. In this specific case, an object's elasticity depends on the interatomic bond length. There are many applications that connect to the ability to compute the speed of sounds.

In more specific fields such as in Industrial Engineering, these calculations could be applied to questions regarding how to build a soundproof area. It would therefore be optimal to select a solid with a low speed of sound velocity, with a solid that has tightly packed particles. There are many vast applications to this, as an object's ability to block or allow sound waves through it. However, some cases require contractors to build structures that allow sound to travel through. Knowledge regarding how solids are structured and how they correlate with the speed of sounds in those solids are vital to building structures that meet the criteria.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T02:33:21Z

Claudiatiller7: /* Connectedness */

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Sound==
Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is highly applicable. The human ear captures sound waves through the outer cartilage of the ear, called the Pinna. The sound waves then travel up the ear canal and arrive at the ear drum which vibrates from the sound waves. After traveling through the inner ear, the vibrations arrive at the Cochlea. The Cochlea transfers the vibrations into information that auditory nerves can analyze.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
Computing the speed of sound in solids depends on a mass' interatomic properties, such as interatomic bond length. In this specific case, an object's elasticity depends on the interatomic bond length. There are many applications that connect to the ability to compute the speed of sounds.

In more specific fields such as in Industrial Engineering, these calculations could be applied to questions regarding how to build a soundproof area. It would therefore be optimal to select a solid with a low speed of sound velocity, with a solid that has tightly packed particles. There are many vast applications to this, as an object's ability to block or allow sound waves through it. However, some cases require contractors to build structures that allow sound to travel through. Knowledge regarding how solids are structured and how they correlate with the speed of sounds in those solids are vital to building structures that meet the criteria.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T02:09:41Z

Claudiatiller7: /* The Structure of Solids and Sound */

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Sound==
Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is highly applicable. The human ear captures sound waves through the outer cartilage of the ear, called the Pinna. The sound waves then travel up the ear canal and arrive at the ear drum which vibrates from the sound waves. After traveling through the inner ear, the vibrations arrive at the Cochlea. The Cochlea transfers the vibrations into information that auditory nerves can analyze.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
The computation of speed of sound in solids is dependent on a mass' interatomic properties, such as interatomic bond length. I find it interesting when a object's interatomic properties determine its functionality on a larger scale. In the case of speed of sound in solids, the objects elasticity depends on the interatomic bond length.

While the computation of speed of sound in solids may not seem related to Industrial Engineering, it has clear implications in the process of choosing building materials, which is a notable section of industrial engineering. For example, if you are planning to build something soundproof, it would be optimal to choose a solid with a very low speed of sound velocity.

The industrial applications in terms of construction are vast. An objects ability to block/allow sound waves through it is very important. Of course, insulation materials and sound proof wall materials come to thought at first. However, in some cases, contractors need to build structures that allow sound to travel through, in which they would choose solid materials that correlate with high speeds of sound.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T02:00:54Z

Claudiatiller7: /* The Structure of Solids and Sound */

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Sound==
Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is great. Everyday humans' ears capture sound waves through the Pinna: the outer cartilage piece of the ear. The sound waves then travel up the ear canal and hit the ear drum, which vibrates from the sound. The vibrations travel through the inner ear and end up at the Cochlea. The Cochlea transfers the vibrations into information that the auditory nerve can analyze. This process occurs everyday and without proper education of sound traveling through solids, such as the ear's different pieces, humans wouldn't be able to completely understand the concept completely.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
The computation of speed of sound in solids is dependent on a mass' interatomic properties, such as interatomic bond length. I find it interesting when a object's interatomic properties determine its functionality on a larger scale. In the case of speed of sound in solids, the objects elasticity depends on the interatomic bond length.

While the computation of speed of sound in solids may not seem related to Industrial Engineering, it has clear implications in the process of choosing building materials, which is a notable section of industrial engineering. For example, if you are planning to build something soundproof, it would be optimal to choose a solid with a very low speed of sound velocity.

The industrial applications in terms of construction are vast. An objects ability to block/allow sound waves through it is very important. Of course, insulation materials and sound proof wall materials come to thought at first. However, in some cases, contractors need to build structures that allow sound to travel through, in which they would choose solid materials that correlate with high speeds of sound.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T01:58:00Z

Claudiatiller7: /* The Structure of Solids and Sound */

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Sound==
Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is great. Everyday humans' ears capture sound waves through the Pinna, [[File:Blausen_0330_EarAnatomy_MiddleEar.png|thumb|border|right|baseline|The inner ear.]]or the outer cartilage piece of the ear. The sound waves then travel up the ear canal and hit the ear drum, which vibrates from the sound. The vibrations travel through the inner ear and end up at the Cochlea. The Cochlea transfers the vibrations into information that the auditory nerve can analyze. This process occurs everyday and without proper education of sound traveling through solids, such as the ear's different pieces, humans wouldn't be able to completely understand the concept completely.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
The computation of speed of sound in solids is dependent on a mass' interatomic properties, such as interatomic bond length. I find it interesting when a object's interatomic properties determine its functionality on a larger scale. In the case of speed of sound in solids, the objects elasticity depends on the interatomic bond length.

While the computation of speed of sound in solids may not seem related to Industrial Engineering, it has clear implications in the process of choosing building materials, which is a notable section of industrial engineering. For example, if you are planning to build something soundproof, it would be optimal to choose a solid with a very low speed of sound velocity.

The industrial applications in terms of construction are vast. An objects ability to block/allow sound waves through it is very important. Of course, insulation materials and sound proof wall materials come to thought at first. However, in some cases, contractors need to build structures that allow sound to travel through, in which they would choose solid materials that correlate with high speeds of sound.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T01:57:33Z

Claudiatiller7:

Claudia Tiller, Spring 2023

This page discusses the speed of sound in various solids, how to calculate them, and examples of such calculations.

==The Main Idea==

The speed of sound can be defines as the distance travelled per a unit of time by a sound wave as it travels through an elastic medium. Elasticity refers to the ability of a body to resist distorting influence and return to its original shape when that influence is removed.

Factors that control the speed that sound travels in solids is measured by the solid's density and elasticity, as they affect the vibrational energy of the sound. Overall, the way the solid is composed determines the sound's speed limit through that solid.

==The Structure of Solids and Sound==

Mediums are composed of particles that can be closely knit together or spread apart. Solids are characterized by an arrangement of atoms, ions, or molecules, where these components are generally locked in their positions. The particles can also be defined as elastic or inelastic. Particles that are closer to each other allow sound to be transferred quicker through the medium. Since particles that are compressed closer together allow sound to travel faster, it can be reasoned that sound travels slower in air.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is great. Everyday humans' ears capture sound waves through the Pinna, [[File:Blausen_0330_EarAnatomy_MiddleEar.png|thumb|border|right|baseline|The inner ear.]]or the outer cartilage piece of the ear. The sound waves then travel up the ear canal and hit the ear drum, which vibrates from the sound. The vibrations travel through the inner ear and end up at the Cochlea. The Cochlea transfers the vibrations into information that the auditory nerve can analyze. This process occurs everyday and without proper education of sound traveling through solids, such as the ear's different pieces, humans wouldn't be able to completely understand the concept completely.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
The computation of speed of sound in solids is dependent on a mass' interatomic properties, such as interatomic bond length. I find it interesting when a object's interatomic properties determine its functionality on a larger scale. In the case of speed of sound in solids, the objects elasticity depends on the interatomic bond length.

While the computation of speed of sound in solids may not seem related to Industrial Engineering, it has clear implications in the process of choosing building materials, which is a notable section of industrial engineering. For example, if you are planning to build something soundproof, it would be optimal to choose a solid with a very low speed of sound velocity.

The industrial applications in terms of construction are vast. An objects ability to block/allow sound waves through it is very important. Of course, insulation materials and sound proof wall materials come to thought at first. However, in some cases, contractors need to build structures that allow sound to travel through, in which they would choose solid materials that correlate with high speeds of sound.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Speed of Sound in Solids

2023-04-17T01:12:41Z

Claudiatiller7:

Claudia Tiller, Spring 2023

This page discusses calculating the speed of sound in various solids and provides examples of such calculations.

==The Main Idea==

The speed of sound is the speed that sound waves travel through a particular medium. The medium determines the speed limit. [[File:2743324.png|right|border]]Mediums are composed of particles that could be closely knit together or relatively spread apart. Also, these particles could act elastic or inelastic. The closer together the particles are to each other the faster the sound can be transferred through the medium. In comparison to air, sound travels considerably faster in solids. The factors that control the speed that sound travels in various solids is measured by the solid's density and elasticity, as these factors effect the vibrational energy of the sound.

With an increase in density, the space between particles in the solid decreases. The smaller distance between particles, or interatomic distance, the higher the speed. With an increase in elasticity of the atoms that make up the object, the lower the speed of sound in the object. Particles that have a high elasticity take more time to return to their place once they received vibrational energy. However, if the solid is completely inelastic then the sound cannot travel through it.

The practicality of this concept is great. Everyday humans' ears capture sound waves through the Pinna, [[File:Blausen_0330_EarAnatomy_MiddleEar.png|thumb|border|right|baseline|The inner ear.]]or the outer cartilage piece of the ear. The sound waves then travel up the ear canal and hit the ear drum, which vibrates from the sound. The vibrations travel through the inner ear and end up at the Cochlea. The Cochlea transfers the vibrations into information that the auditory nerve can analyze. This process occurs everyday and without proper education of sound traveling through solids, such as the ear's different pieces, humans wouldn't be able to completely understand the concept completely.

===A Mathematical Model===

The speed of sound in solids <math> {V_{s}} </math> can be determined by the equation. Young's Modulus is a measure of elasticity of an object, and it can be computed to solve for interatomic values, such as interatomic bond stiffness or interatomic bond length.

<math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>

Alternative speed equation:

<math> {V_{s}} = \sqrt{\frac{B}{\rho}} </math>

<math> ρ </math> = density

<math> B </math> = Bulks Modulus

Bulks Modulus = <math>{\frac {ΔP}{ΔV/V}} </math>
[[File:bulk3.gif|border|right|middle]]
<math> d </math> = interatomic bond length

<math> K_{s} </math> = Interatomic bond stiffness

Youngs Modulus = (<math> Y = K_{s}/d </math>)

Youngs Modulus: <math> Y ={\frac{Stress}{Strain}}</math>

<math> Stress = {\frac{F_{tension}}{Area_{Cross Sectional}}} </math>

<math> Strain = {\frac{ΔL_{wire}}{L_{0}}} </math>

===Speeds of Various Compositions===
[[File:speedofsound.jpg]]

==Examples==

===Theoretical===

Two metal rods are made of different elements. The interatomic spring stiffness of element A is four times larger than the interatomic spring stiffness for element B. The mass of an atom of element A is four times greater than the mass of an atom of element B. The atomic diameters are approximately the same for A and B. What is the ratio of the speed of sound in rod A to the speed of sound in rod B?

Solution: In this situation, the ratio of the speed of sound in rod A to the speed of sound in rod B is 1.

Looking at the formula for computing speed of sound in solids, <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math>, you see that velocity depends three factors, interatomic stiffness, the mass of one atom, and interatomic bond length. The two rods differences in atomic mass and interatomic stiffness offset each other when the equations are set equal, and the ratio is determined to be 1.

<math> d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} = d \cdot \sqrt{\frac{4 \cdot K_{s}}{m_{atom}}} </math> After simplification <math> V_{s_{1}} = V_{s{2}} </math>

===Numerical Example ===
The Young's Modulus value of silver is 7.75e+10, atomic mass of silver is 108 g/mole, and the density of silver is 10.5 g/cm3. Using this information, calculate the speed of sound in silver.

Solution: The key to solving this problem is to realize the micro-macro connection of Young's Modulus. You are given that Young's Modulus is equal to 7.75e+10, and we know that
Youngs Modulus = (<math> K_{s}/d </math>). In this situation, we need to calculate the interatomic bond length and use it and our Young's Modulus value to determine our interatomic stiffness.

To solve for ''d'', we use the given density of silver (10.5 g/cm3). Using the basic equation for volume in relation to density and mass (<math> V=m*d</math>), we can find ''d'', since ''d'' is equal to the cube root of volume.

Once ''d'' is solved for, it can be plugged back into the the equation <math> Y = K_{s}/d </math> to solve for <math> K_{s} </math>

Now, we have solved for both interatomic bond length and stiffness. The only quantity in the final speed of sound equation we need is the mass of one atom, which can be determined using Avogardro's number and the atomic mass. <math> m_{atom} = </math> ''atomic mass'' / <math> 6.022e23 </math>

Now that all variables are solved for, we can substitute values into our <math> {V_{s}} = d \cdot \sqrt{\frac{K_{s}}{m_{atom}}} </math> equation.

<math> {V_{s}} = 1.6 \cdot 10^{-10} \cdot \sqrt{ \frac{78534.7}{1.79 \cdot 10^{-22}}} </math>

<math> {V_{s}} = 2723</math> m/s

==Connectedness==
The computation of speed of sound in solids is dependent on a mass' interatomic properties, such as interatomic bond length. I find it interesting when a object's interatomic properties determine its functionality on a larger scale. In the case of speed of sound in solids, the objects elasticity depends on the interatomic bond length.

While the computation of speed of sound in solids may not seem related to Industrial Engineering, it has clear implications in the process of choosing building materials, which is a notable section of industrial engineering. For example, if you are planning to build something soundproof, it would be optimal to choose a solid with a very low speed of sound velocity.

The industrial applications in terms of construction are vast. An objects ability to block/allow sound waves through it is very important. Of course, insulation materials and sound proof wall materials come to thought at first. However, in some cases, contractors need to build structures that allow sound to travel through, in which they would choose solid materials that correlate with high speeds of sound.

==History==

The speed of sound in air was first measured by Sir Isaac Newton, and first correctly computed by Pierre-Simon Laplace in 1816. Before this precise measurement, attempts had been made across Europe during the 1700s, most famously Reverend William Derham's experiment in 1709 across the town of Upminister, England. Reverend Derham used a shotgun's noise and several known landmarks around time to measure the time it took for the sound of the blast to be heard from select distances.

Young's Modulus was named after English physicist Thomas Young. In actuality, the concept was developed earlier by physicists Leonhard Euler and Giordano Riccati in the 1720s.

== See also ==

Youngs Modulus: [http://www.physicsbook.gatech.edu/Young%27s_Modulus]
Interatomic Bonds: [http://www.physicsbook.gatech.edu/Length_and_Stiffness_of_an_Interatomic_Bond]

===Further reading===

Further Information can be found on the speed of sound in solids in ''Matter and Interactions, 4th Edition'' by Ruth W. Chabay & Bruce A. Sherwood
===External links===

Internet resources on this topic can be found at:

Engineering Tool Box [http://www.engineeringtoolbox.com/sound-speed-solids-d_713.html]

Hyperphysics [http://hyperphysics.phy-astr.gsu.edu/hbase/sound/souspe2.html]

Potto Project [http://www.potto.org/gasDynamics/node73.html]

NDT Resource Center [https://www.nde-ed.org/EducationResources/CommunityCollege/Ultrasonics/Physics/elasticsolids.htm]

The Engineering ToolBox [http://www.engineeringtoolbox.com/speed-sound-d_82.html]

Ear Image [https://commons.wikimedia.org/wiki/File:Blausen_0330_EarAnatomy_MiddleEar.png]

Yew Chung [http://ycis-sir.weebly.com/gigamind-a-sound-experience.html]

==References==

This section contains the the references used while writing this page

Chart from [http://www.rcgroups.com/forums/attachment.php?attachmentid=3397792]

''Matter and Interactions 4th Edition'' by Chabay and Sherwood

Wikipage created by Daiven Patel

Conservation of Energy

2023-04-17T01:10:23Z

Claudiatiller7: /* Main Ideas */

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Editor: Ethan Nguyen-Tu, Spring 2023'''</div>
'''What is the Law of Conservation of Energy?'''
<br>
The Law of Conservation of Energy states that energy cannot be created or destroyed, but it can change from one form of energy to another.
<br>

==Main Ideas==
* The law of conservation of energy states that the '''total amount of energy''' of a system '''before and after''' an interaction between objects in '''conserved'''.
* This only applies to '''isolated systems''' (no outside forces acting on the system).
** Not Isolated: An object sliding across a rough floor (system = the object). There is work being done by the floor on the object because of the frictional force. Energy lost to heat due to friction is an example of mechanical energy being converted into thermal energy.
** Isolated: An object sliding across a rough floor (system = the object AND the floor). There is no work done on the system because all the forces are contained in the system.
[[File:canonfire.jpg|800px|center]]

===What does it mean?===
Since The Law of Conservation of Energy states energy cannot be created or destroyed, this means that the total energy in the universe is constant and does not change in value, assuming there is nothing beyond the universe.
<br>
:Therefore, in equation format:
::[e.1]     <math>E_{total, universe} = \text{Some Constant}</math>
::[e.2]     <math>\Delta E_{total, universe} = 0</math>

==Mathematical Model==
===Conservation of Energy===
::[e.1]     <math> E_{total, universe} = \text{Some Constant} </math>
::[e.2]     <math> \Delta E_{total, universe} = 0 </math>

===Conservation of Energy & [[The Energy Principle]]===
::[e.3]     <math> \Delta E_{system} + \Delta E_{surroundings} = 0 </math>

===Isolated System===
* <b> ΔE = W + Q </b> (if no heat transfer indicated, Q = 0; if no external forces acting on system, W = 0)
* <b> ΔE = K + U </b> (The total energy is the sum of the kinetic and potential energies. From this, you can infer that for an isolated system, any change in kinetic energy will correspond in an equal but opposite change in the potential energy and vice versa.)
These formulas can be interchanged. For example, if you know work and heat transfer are zero, energy equals zero, so K + U will equal zero

[https://www.youtube.com/watch?v=kw_4Loo1HR4 Basic Explanation of Conservation of Energy]
<br>
[https://www.youtube.com/watch?v=EZrJNIBX2wk Skater Visualization of Transfers of Energy]

===Computational Model===
These gifs demonstrate the energy principal from a '''Conservation of Energy''' standpoint. As the ball on a spring approaches the equilibrium point, the '''kinetic energy increases''' and the '''spring potential decreases'''. These values will '''oscillate''', but the '''total energy will stay constant'''! This demonstration was written in GlowScript and '''iteratively updates the ball's momentum''' while taking into account the spring force.

[[File:Spring1.gif|300px]]
[[File:Graphspring.gif]]

===Useful Energy Formulas===

'''[[Kinetic Energy]]'''
* Kinetic Energy: <math>{E_{kinetic} = K = \frac{1}{2}m\overrightarrow{v}^2}</math>

'''[[Potential Energy]]'''
* Gravitational Potential: <math>{U_{gravity} = U_g = \frac{Gm_1m_2}{ \big{|} \overrightarrow{r} \big{|}}(-\hat{r})}</math>
** Approximation when near the Earth's Surface: <math>{\Delta U_g=mg\Delta h}</math>
* Spring (Elastic) Potential: <math>{U_{spring} = U_s = \frac{1}{2}k_ss^2}</math>

'''[[Thermal Energy]]'''
* Thermal Energy: <math>{\Delta E_{thermal} = mC \Delta T }</math>

==Examples==
To view the solution, click '''Expand''' at the top right of each question.

===Easy===

<div class="toccolours mw-collapsible mw-collapsed">

====Question 1====
A ball is at rest on a table with 50 J of potential energy. It then rolls off the table, and at one point in time as it falls, the ball has 30 J of kinetic energy.
'''What is the potential energy of the ball at that instant?'''

<div class="mw-collapsible-content">

[[File:EasyEnergyConservation.png]]
<br>
See Reference 3

E<sub>initial</sub> = E<sub>final</sub> <br>
K<sub>initial</sub> + U<sub>initial</sub> = K<sub>final</sub> + U<sub>final</sub> <br>
0 J + 50 J = 30 J + U<sub>final</sub> <br>
U<sub>final</sub> = 20 J

</div>
</div>

===Medium===
<div class="toccolours mw-collapsible mw-collapsed">

====Question 1====
A ball is at rest 50 m above the ground. You then drop the ball.
'''What is its speed before hitting the ground?'''

<div class="mw-collapsible-content">

[[File:ballcalc.jpg|400px]]

</div>
</div>

===Hard===
<div class="toccolours mw-collapsible mw-collapsed">

====Question 1====
The driver of an SUV (m = 1700 kg) isn’t paying attention and rear ends a car (m = 950 kg) on level ground at a red light. On impact, both drivers lock their brakes. The SUV and car stick together and travel a distance of 8.2 m before they come to a stop. The coefficient of friction between the tires and the road is 0.72.
'''How fast was the SUV traveling just before the collision?'''

<div class="mw-collapsible-content">

[[File:collidecars.jpg|600px]]

</div>
</div>

==Connectedness==

===How does it relate to things we want to study?===
In Physics, we separate what we are looking at into a system and its surroundings. This is a zero-sum separation where what we are interested in is included in the system and everything else in the universe is lumped into the system's surroundings.
<br>
: In equation format:    Universe = System + Surroundings     [e.4]
This division of the universe into a system and its surroundings can also be applied to the total energy in the universe.
<br>
: In equation format:    <math>E_{total, universe} = E_{system} + E_{surroundings} </math>     [e.5]
Furthermore, the Law of Conservation of Energy tells us that the total energy in the universe is constant.
: In equation format:    <math>E_{total, universe} = E_{system} + E_{surroundings} = \text{Some Constant} </math>     [e.6]
However, when considering the surroundings in [e.5] and [e.6] for systems that are not the entire universe, we run into an important problem:
<br>
: When considering the system's surroundings, is that it is highly time inefficient, and essentially impossible, to consider and factor in everything in the universe that is not included in the system. Therefore, this means that we can only find completely accurate energy values by applying the Law of Conservation of Energy to isolated systems in which the system has no net external force and does not exchange matter and energy with its surroundings. In this case, the Law of Conservation of Energy tells us that the energy within the isolated system is constant. Nevertheless, applying the Law of Conservation of Energy to closed systems, in which matter is not able to enter or leave but energy can, we can get close approximations by identifying the main actors in the surroundings that influence our system.

===How does it related to [[The Energy Principle]]?===
Since the Law of Conservation of Energy says energy cannot be created or destroyed, [[The Energy Principle]] tells us that the only way for a system to gain or lose energy is from its surroundings losing or gaining the same amount of energy. Therefore, [[The Energy Principle]] can be generalized in terms of conservation of energy.
<br>
: Written in equation form, [[The Energy Principle]] in terms of the Law of Conservation of Energy is:    <math>{\Delta E_{system}+\Delta E_{surroundings}=0}</math>     [e.3]

==History==

<b>Who:</b> Many physicists contributed to the knowledge of energy, however it is most notably attributed to Julius Robert Mayer
<br>
<b>What:</b> Most formally discovered the law of conservation of energy
<br>
<b>When:</b> 1842
<br>
<b>Where:</b> Germany
<br>
<b>Why:</b> To explain what happens to energy in an isolated system
<br>
See Reference 6

== See also ==

* [[The Energy Principle]]
* [[Kinetic Energy]]
* [[Potential Energy]]
* [[Work]]

===Additional Reading===
# Goldstein, Martin, and Inge F., (1993). The Refrigerator and the Universe. Harvard Univ. Press. A gentle introduction.
# Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.
# Nolan, Peter J. (1996). Fundamentals of College Physics, 2nd ed. William C. Brown Publishers.

===External Links===
# [https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-conservation-of-energy Khan Academy]
# [http://www.physicsclassroom.com/class/energy/Lesson-2/Application-and-Practice-Questions Practice Questions]
# [http://physics.info/energy-conservation/problems.shtml More Practice]
# [http://gilliesphysics.weebly.com/uploads/5/7/5/2/57520801/conservation_of_energy_practice_problems.pdf Basic Examples]
# [http://www.physnet.org/modules/pdf_modules/m158.pdf The First Law of Thermodynamics]

==References==

# "Conservation of Energy." Hmolpedia. Web. 1 Dec. 2015. <http://www.eoht.info/page/Conservation+of+energy>.
# "University of Wisconsin Green Bay." Speed & Stopping Distance of a Roller-Coaster. Web. 1 Dec. 2015. <http://www.uwgb.edu/fenclh/problems/energy/2/>.
# "Motion." G9 to Engineering. Web. 1 Dec. 2015. <http://www.g9toengineering.com/resources/translational.htm>.
# "Energy of Falling Object." HyperPhysics. Web. 1 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/flobj.html>.
# "Conservation of Energy & Momentum Problem: Collision of Two Cars at a Stoplight." University of Wisconsin- Green Bay Physics. Web. 2 Dec. 2015. <http://www.uwgb.edu/fenclh/problems/energy/6/>.
# "Law of Conservation of Mass Energy." Law of Conservation of Mass Energy. Web. 3 Dec. 2015. <http://www.chemteam.info/Thermochem/Law-Cons-Mass-Energy.html>.
# "Law of Conservation of Energy" New York University. Web. 18 Apr. 2018. <http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_2/node4.html>.
# "Law of Conversation of Energy" ME Mechanical. Web. 18 Apr. 2018. <https://me-mechanicalengineering.com/law-of-conservation-of-energy/>.
# "Law of conservation of energy" University of Calgary. Web. 8 Apr. 2023. <https://energyeducation.ca/encyclopedia/Law_of_conservation_of_energy>.
# "Basic Definitions" LibreTexts Chemistry. Web. 8 Apr. 2023. <https://chem.libretexts.org/Bookshelves/General_Chemistry/General_Chemistry_Supplement_(Eames)/Thermochemistry/Basic_Definitions#:~:text=A%20closed%20system%20does%20not,is%20approximately%20an%20isolated%20system.>

[[Category:Energy]]

Conservation of Energy

2023-04-17T01:10:08Z

Claudiatiller7: /* Main Ideas */

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">'''Editor: Ethan Nguyen-Tu, Spring 2023'''</div>
'''What is the Law of Conservation of Energy?'''
<br>
The Law of Conservation of Energy states that energy cannot be created or destroyed, but it can change from one form of energy to another.
<br>

==Main Ideas==
CLAUDIA TILLER
* The law of conservation of energy states that the '''total amount of energy''' of a system '''before and after''' an interaction between objects in '''conserved'''.
* This only applies to '''isolated systems''' (no outside forces acting on the system).
** Not Isolated: An object sliding across a rough floor (system = the object). There is work being done by the floor on the object because of the frictional force. Energy lost to heat due to friction is an example of mechanical energy being converted into thermal energy.
** Isolated: An object sliding across a rough floor (system = the object AND the floor). There is no work done on the system because all the forces are contained in the system.
[[File:canonfire.jpg|800px|center]]

===What does it mean?===
Since The Law of Conservation of Energy states energy cannot be created or destroyed, this means that the total energy in the universe is constant and does not change in value, assuming there is nothing beyond the universe.
<br>
:Therefore, in equation format:
::[e.1]     <math>E_{total, universe} = \text{Some Constant}</math>
::[e.2]     <math>\Delta E_{total, universe} = 0</math>

==Mathematical Model==
===Conservation of Energy===
::[e.1]     <math> E_{total, universe} = \text{Some Constant} </math>
::[e.2]     <math> \Delta E_{total, universe} = 0 </math>

===Conservation of Energy & [[The Energy Principle]]===
::[e.3]     <math> \Delta E_{system} + \Delta E_{surroundings} = 0 </math>

===Isolated System===
* <b> ΔE = W + Q </b> (if no heat transfer indicated, Q = 0; if no external forces acting on system, W = 0)
* <b> ΔE = K + U </b> (The total energy is the sum of the kinetic and potential energies. From this, you can infer that for an isolated system, any change in kinetic energy will correspond in an equal but opposite change in the potential energy and vice versa.)
These formulas can be interchanged. For example, if you know work and heat transfer are zero, energy equals zero, so K + U will equal zero

[https://www.youtube.com/watch?v=kw_4Loo1HR4 Basic Explanation of Conservation of Energy]
<br>
[https://www.youtube.com/watch?v=EZrJNIBX2wk Skater Visualization of Transfers of Energy]

===Computational Model===
These gifs demonstrate the energy principal from a '''Conservation of Energy''' standpoint. As the ball on a spring approaches the equilibrium point, the '''kinetic energy increases''' and the '''spring potential decreases'''. These values will '''oscillate''', but the '''total energy will stay constant'''! This demonstration was written in GlowScript and '''iteratively updates the ball's momentum''' while taking into account the spring force.

[[File:Spring1.gif|300px]]
[[File:Graphspring.gif]]

===Useful Energy Formulas===

'''[[Kinetic Energy]]'''
* Kinetic Energy: <math>{E_{kinetic} = K = \frac{1}{2}m\overrightarrow{v}^2}</math>

'''[[Potential Energy]]'''
* Gravitational Potential: <math>{U_{gravity} = U_g = \frac{Gm_1m_2}{ \big{|} \overrightarrow{r} \big{|}}(-\hat{r})}</math>
** Approximation when near the Earth's Surface: <math>{\Delta U_g=mg\Delta h}</math>
* Spring (Elastic) Potential: <math>{U_{spring} = U_s = \frac{1}{2}k_ss^2}</math>

'''[[Thermal Energy]]'''
* Thermal Energy: <math>{\Delta E_{thermal} = mC \Delta T }</math>

==Examples==
To view the solution, click '''Expand''' at the top right of each question.

===Easy===

<div class="toccolours mw-collapsible mw-collapsed">

====Question 1====
A ball is at rest on a table with 50 J of potential energy. It then rolls off the table, and at one point in time as it falls, the ball has 30 J of kinetic energy.
'''What is the potential energy of the ball at that instant?'''

<div class="mw-collapsible-content">

[[File:EasyEnergyConservation.png]]
<br>
See Reference 3

E<sub>initial</sub> = E<sub>final</sub> <br>
K<sub>initial</sub> + U<sub>initial</sub> = K<sub>final</sub> + U<sub>final</sub> <br>
0 J + 50 J = 30 J + U<sub>final</sub> <br>
U<sub>final</sub> = 20 J

</div>
</div>

===Medium===
<div class="toccolours mw-collapsible mw-collapsed">

====Question 1====
A ball is at rest 50 m above the ground. You then drop the ball.
'''What is its speed before hitting the ground?'''

<div class="mw-collapsible-content">

[[File:ballcalc.jpg|400px]]

</div>
</div>

===Hard===
<div class="toccolours mw-collapsible mw-collapsed">

====Question 1====
The driver of an SUV (m = 1700 kg) isn’t paying attention and rear ends a car (m = 950 kg) on level ground at a red light. On impact, both drivers lock their brakes. The SUV and car stick together and travel a distance of 8.2 m before they come to a stop. The coefficient of friction between the tires and the road is 0.72.
'''How fast was the SUV traveling just before the collision?'''

<div class="mw-collapsible-content">

[[File:collidecars.jpg|600px]]

</div>
</div>

==Connectedness==

===How does it relate to things we want to study?===
In Physics, we separate what we are looking at into a system and its surroundings. This is a zero-sum separation where what we are interested in is included in the system and everything else in the universe is lumped into the system's surroundings.
<br>
: In equation format:    Universe = System + Surroundings     [e.4]
This division of the universe into a system and its surroundings can also be applied to the total energy in the universe.
<br>
: In equation format:    <math>E_{total, universe} = E_{system} + E_{surroundings} </math>     [e.5]
Furthermore, the Law of Conservation of Energy tells us that the total energy in the universe is constant.
: In equation format:    <math>E_{total, universe} = E_{system} + E_{surroundings} = \text{Some Constant} </math>     [e.6]
However, when considering the surroundings in [e.5] and [e.6] for systems that are not the entire universe, we run into an important problem:
<br>
: When considering the system's surroundings, is that it is highly time inefficient, and essentially impossible, to consider and factor in everything in the universe that is not included in the system. Therefore, this means that we can only find completely accurate energy values by applying the Law of Conservation of Energy to isolated systems in which the system has no net external force and does not exchange matter and energy with its surroundings. In this case, the Law of Conservation of Energy tells us that the energy within the isolated system is constant. Nevertheless, applying the Law of Conservation of Energy to closed systems, in which matter is not able to enter or leave but energy can, we can get close approximations by identifying the main actors in the surroundings that influence our system.

===How does it related to [[The Energy Principle]]?===
Since the Law of Conservation of Energy says energy cannot be created or destroyed, [[The Energy Principle]] tells us that the only way for a system to gain or lose energy is from its surroundings losing or gaining the same amount of energy. Therefore, [[The Energy Principle]] can be generalized in terms of conservation of energy.
<br>
: Written in equation form, [[The Energy Principle]] in terms of the Law of Conservation of Energy is:    <math>{\Delta E_{system}+\Delta E_{surroundings}=0}</math>     [e.3]

==History==

<b>Who:</b> Many physicists contributed to the knowledge of energy, however it is most notably attributed to Julius Robert Mayer
<br>
<b>What:</b> Most formally discovered the law of conservation of energy
<br>
<b>When:</b> 1842
<br>
<b>Where:</b> Germany
<br>
<b>Why:</b> To explain what happens to energy in an isolated system
<br>
See Reference 6

== See also ==

* [[The Energy Principle]]
* [[Kinetic Energy]]
* [[Potential Energy]]
* [[Work]]

===Additional Reading===
# Goldstein, Martin, and Inge F., (1993). The Refrigerator and the Universe. Harvard Univ. Press. A gentle introduction.
# Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9.
# Nolan, Peter J. (1996). Fundamentals of College Physics, 2nd ed. William C. Brown Publishers.

===External Links===
# [https://www.khanacademy.org/science/physics/work-and-energy/work-and-energy-tutorial/a/what-is-conservation-of-energy Khan Academy]
# [http://www.physicsclassroom.com/class/energy/Lesson-2/Application-and-Practice-Questions Practice Questions]
# [http://physics.info/energy-conservation/problems.shtml More Practice]
# [http://gilliesphysics.weebly.com/uploads/5/7/5/2/57520801/conservation_of_energy_practice_problems.pdf Basic Examples]
# [http://www.physnet.org/modules/pdf_modules/m158.pdf The First Law of Thermodynamics]

==References==

# "Conservation of Energy." Hmolpedia. Web. 1 Dec. 2015. <http://www.eoht.info/page/Conservation+of+energy>.
# "University of Wisconsin Green Bay." Speed & Stopping Distance of a Roller-Coaster. Web. 1 Dec. 2015. <http://www.uwgb.edu/fenclh/problems/energy/2/>.
# "Motion." G9 to Engineering. Web. 1 Dec. 2015. <http://www.g9toengineering.com/resources/translational.htm>.
# "Energy of Falling Object." HyperPhysics. Web. 1 Dec. 2015. <http://hyperphysics.phy-astr.gsu.edu/hbase/flobj.html>.
# "Conservation of Energy & Momentum Problem: Collision of Two Cars at a Stoplight." University of Wisconsin- Green Bay Physics. Web. 2 Dec. 2015. <http://www.uwgb.edu/fenclh/problems/energy/6/>.
# "Law of Conservation of Mass Energy." Law of Conservation of Mass Energy. Web. 3 Dec. 2015. <http://www.chemteam.info/Thermochem/Law-Cons-Mass-Energy.html>.
# "Law of Conservation of Energy" New York University. Web. 18 Apr. 2018. <http://www.nyu.edu/classes/tuckerman/adv.chem/lectures/lecture_2/node4.html>.
# "Law of Conversation of Energy" ME Mechanical. Web. 18 Apr. 2018. <https://me-mechanicalengineering.com/law-of-conservation-of-energy/>.
# "Law of conservation of energy" University of Calgary. Web. 8 Apr. 2023. <https://energyeducation.ca/encyclopedia/Law_of_conservation_of_energy>.
# "Basic Definitions" LibreTexts Chemistry. Web. 8 Apr. 2023. <https://chem.libretexts.org/Bookshelves/General_Chemistry/General_Chemistry_Supplement_(Eames)/Thermochemistry/Basic_Definitions#:~:text=A%20closed%20system%20does%20not,is%20approximately%20an%20isolated%20system.>

[[Category:Energy]]