Translational, Rotational and Vibrational Energy: Difference between revisions

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SHREYA LAKSHMISHA SPRING 2026
==Main Idea==
==Main Idea==
In many cases, analyzing the kinetic energy of an object is in fact more difficult than just applying the formula <math> K = \cfrac{1}{2}mv^2 </math>. An example of this is when throwing a basketball because not only does it move through the air, but it is also rotating around its own axis. When analyzing more complicated movements like this one, it is necessary to break kinetic energy into different parts, such as rotational, translational, and vibrational, and analyze each one separately to give a more accurate picture.
In many real-world situations, analyzing the kinetic energy of an object is more complex than just applying the formula:


Translational kinetic energy is the kinetic energy associated with the motion of the center of mass of an object. This would be the basketball traveling in the air from one location to another. While relative kinetic energy is the kinetic energy associated to the rotation or vibration of the atoms of the object around its center or axis. Relative kinetic energy would be the rotation of the basketball around it's axis. Later on this page, we go into more depth about the different types of kinetic energy.
<math> K = \cfrac{1}{2}mv^2 </math>


Here is a link to a video which explains kinetic energy in detail: [https://youtu.be/Cobhu3lgeMg]
For example when a basketball is thrown, it is not only moving through space, but also rotating about its own axis. Because of this, the total kinetic energy must be broken into components.
 
The total kinetic energy of a system can be separated into:
* Translational energy (motion of the center of mass)
* Rotational energy (spinning motion)
* Vibrational energy (internal motion of particles)
 
This breakdown allows us to more accurately analyze motion in physical systems.
 
 
[[File:Rolling Racers - Moment of inertia.gif|Rolling_Racers_-_Moment_of_inertia]]


===Mathematical Model===
===Mathematical Model===
=== Total Kinetic Energy ===
=== Total Kinetic Energy ===
As we just saw, the total kinetic energy of a multi particle system can be divided into the energy associated with motion of the center of mass and the motion relative to the center of mass.


::<math> K_{total} = K_{translational} + K_{relative} </math>
<math>K_{total} = K_{translational} + K_{rotational} + K_{vibrational}</math>


The relative kinetic energy is composed of motion due to rotation about the center of mass and vibrations/oscillations of the object.
This equation shows that energy must be considered in multiple forms when objects both move and rotate.


::<math> K_{total} = K_{translational} + K_{rotational} + K_{vibrational} </math>


====Translational Kinetic Energy====
====Translational Kinetic Energy====


"Translation" means:
::<math>K_{trans} = \cfrac{1}{2}Mv_{CM}^2</math>
 
* <math>M</math>: total mass 
* <math>v_{CM}</math>: velocity of the center of mass
 
 
The center of mass is calculated as:
 
::<math>r_{CM} = \cfrac{\sum m_ir_i}{\sum m_i}</math>
::<math>v_{CM} = \cfrac{\sum m_iv_i}{\sum m_i}</math>
 
'''Key Idea:''' Translational energy depends only on how the object moves.
 
[[File:Beam with pivot P, center of mass S and center of percussion C.svg|Beam_with_pivot_P,_center_of_mass_S_and_center_of_percussion_C]]
---
 
====Rotational Kinetic Energy====
The total energy due to vibrations is the sum of the potential energy associated with interactions causing the vibrations and the kinetic energy of the vibrations.
 
::<math>K_{rot} = \cfrac{1}{2}I\omega^2</math>
 
* <math>I</math>: moment of inertia 
* <math>\omega</math>: angular velocity
 
====Moment of Inertia====
 
::<math>I = \sum m_i r_i^2</math>


::''To move from one location to another location''
This measures how difficult it is to rotate an object.


By calculating translational kinetic energy, we can track how one object moves from one location to another. Since the translational kinetic energy is associated with the movement of the center of mass of the object, it is important to know how to calculate the location of the center of mass and the velocity of the center of mass which is shown in the two equations below:
'''Important Insight:'''
Mass farther from the axis increases rotational energy because of the \(r^2\) term.


::<math>r_{CM} = \cfrac{m_1r_1 + m_2r_2+m_3r_3 + ...}{m_1 + m_2 +m_3}</math>


::<math>v_{CM} = \cfrac{m_1v_1 + m_2v_2+m_3v_3 + ...}{m_1+ m_2 +m_3}</math>
[[File:Moment of inertia solid sphere.svg|Moment_of_inertia_solid_sphere]]


:Here is a link to a video if you want to refresh your knowledge on center of mass: [https://youtu.be/5qwW8WI1gkw]
---


The motion of the center of mass is described by the velocity of the center of mass. Using the total mass and the velocity of the center of mass, we define the translational kinetic energy as:
=====Angular Speed and Velocity=====


::<math>K_{translational} = \cfrac{1}{2}M_{total}v_{CM}^2</math>
::<math>\omega = \cfrac{2\pi}{T}</math>


====Vibrational Kinetic Energy====
::<math>v = \omega r</math>
The total energy due to vibrations is the sum of the potential energy associated with interactions causing the vibrations and the kinetic energy of the vibrations.  
 
Points farther from the center move faster.
 
---
 
=====Vibrational Kinetic Energy=====
 
Vibrational energy comes from internal motion of particles within an object.
 
* Important in molecules and thermal systems 
* Usually not directly calculated in introductory physics problems 
 
---


::<math>E_{vibrational} = K_{vibrational} + U_{s}</math>
===Physical Intuition===
Consider a rolling wheel:


The easiest way to find vibrational kinetic energy is by knowing the other energy terms and isolating the vibrational kinetic energy. This is when there is no rotational kinetic energy:
* Moves forward -> translational energy
* Spins -> rotational energy
* Internal atoms vibrate -> vibrational energy


::<math>E_{total} = K_{trans} + K_{vibrational} + U_{s} +E_{rest}</math>
==Examples==
::<math>K_{vibrational} = E_{total} - (K_{trans} + U_{s} +E_{rest})</math>


====Rotational Kinetic Energy====
===Conceptual Example===
[[File:Kinetic_energy.png|300px|right|thumb|Here are links to two videos that cover rotational kinetic energy and moment of interia: [https://youtu.be/craljBk-E5g][https://youtu.be/XlFlZHfAZeE]]]
Rotational kinetic energy is the energy due to the rotation about the center of mass. It can be calculated by finding the angular momentum and inertia of the system, which will be discussed in greater detail in the next two sections. The equation used to find kinetic rotational energy is below:


::<math>K_{rotational} =\frac{1}{2} I_{cm}{\omega}^2</math>
A bowling ball rolls without slipping.


Another important rotational equation is:
'''Which energies are present?'''


::<math></math>
* Translational ✔ 
* Rotational  ✔ 
* Vibrational ✖ (ignored at this level)


=====Moment of Inertia=====
---
The moment of inertia of an object shows the difficulty of rotating an object, since the larger the moment of inertia the more energy is required to rotate the object at the same angular velocity as an object with a smaller moment of inertia. The moment of inertia of an object is defined as the sum of the products of the mass of each particle in the object with the square of their distance from the axis of rotation. The general formula for calculating the moment of inertia of an object is:


::<math>I = m_1{r_{1}}^2 + m_2{r_{2}}^2 + m_3{r_{3}}^2 +...</math>
===Calculation Example===


:::Here <math> r_1, r_2, r_3 </math> represent the perpendicular distance from the point/axis of rotation.
A solid disk rolls without slipping.


:or
Given:
* <math>m = 2 \, kg</math> 
* <math>v = 3 \, \frac{m}{s}</math> 
* <math>I = \cfrac{1}{2}mr^2</math>


::<math>I = \sum_{i} m_{i}{r_{i}}^2</math>
Step 1: Translational Energy


:For a body with a uniform distribution of mass this can be turned into an integral:
::<math>K_{trans} = \cfrac{1}{2}mv^2 = 9 \, J</math>


::<math>I = \int r^2 \ dm</math>
Step 2: Rotational Energy


The units of rotational inertia are <math> kg \cdot m^2 </math>  
::<math>K_{trans} = \cfrac{1}{2}mv^2 = 9 \, J</math>


[[File:4c906c92cebe30d9486deb2a682acf561d23c9c1.png|900px|center]]
Using <math>\omega = \cfrac{v}{r}</math>:


=====Angular Speed and Acceleration=====
::<math>K_{rot} = \cfrac{1}{4}mv^2 = 4.5 \, J</math>
The angular speed is the rate at which the object is rotating. It is given in the following formula:
[[File:Angularvelocity.png|right|200px]]


::<math>\omega = \cfrac{2\pi}{T}</math>, where
Total Energy:  


:::<math>T =</math> the period of the rotation
::<math>K_{total} = 13.5 \, J</math>  


The angle in which a disk turns is <math>2 \pi</math> in a time <math>T</math>. It is measured in radians per second. The tangential velocity of an object is related to its radius r at the angular speed because the tangential velocity increases when the distance from the center of an object increases. It is shown in the equation below:
---


::<math>v(r)= \omega r</math>
===Common Mistakes===


The angular acceleration a rotating object goes through to change its angular speed is given by:
* Forgetting rotational energy in rolling problems 
* Using incorrect relationship between \(v\) and \(\omega\) 
* Ignoring moment of inertia differences 
* Assuming only translational motion matters


::<math>a(r) = \alpha r</math>
---


===Computational Model===
==Computational Model==
:'''Insert Computational Model here'''


==Examples==
GlowScript simulation:


===Simple===
https://trinket.io/glowscript/31d0f9ad9e


===Middling===
This model helps visualize rotational motion and energy changes.


===Difficult===
---


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


*This topic connected to me because I used to dance when I was younger. This section focused on kinetic energy and the different parts of kinetic energy. You could break up different parts of dance and compare it to kinetic energy.
'''Personal Connection:'''
Dance and sports like tennis involve rotation and motion, similar to energy concepts discussed here.  
 
'''Academic Connection:'''
Important in physics, engineering, and chemistry for analyzing motion and energy systems.
 
'''Industrial Applications:'''
* Flywheels for energy storage
* Rotating machinery
* Engines and turbines
 
---


'''2. How is it connected to your major?'''
===History===


*In Chemical Engineering, we will focus on the kinetic energy on the microscopic level and determining the energy of the particles by looking at the translational, rotational, and vibrational energies of the atom, and how they allow chemical reactions to precess.  
The concept of kinetic energy developed over time through contributions from scientists such as Aristotle, Leibniz, Bernoulli, and Gaspard-Gustave Coriolis. The term “kinetic energy” was later coined by Lord Kelvin.


'''3. Is there an interesting industrial application?'''
===Why This Matters for Exams===


*There are many machines that use kinetic energy for power, and we will probably see in a few years from now the use of rotational, translational, and vibrational energy to power anything from phones to computers.
Most physics problems:
* Combine translation and rotation
* Require identifying ALL forms of energy


==History==
Missing one energy component often leads to incorrect answers.
Kinetic energy was first set apart from potential energy by Aristotle. Later, in the 1600's, Leibniz and Bernoulli developed the idea that <math>E \propto mv^2</math>, and they called it the 'living force.' However, it wasn't until 1829 that Gaspard-Gustave Coriolis showed the first signs of understanding kinetic energy the way that we do today by focusing on the transfer on energy in rotating water wheels. Finally, in 1849, Lord Kelvin is said to have coined the term 'kinetic energy.'


==See also==
---


===Further Reading===
===Summary===
*[[Point Particle Systems]]<br>
*[[Real Systems]]<br>
*[[Conservation of Energy]]<br>
*[[Potential Energy]]<br>
*[[Thermal Energy]]<br>
*[[Internal Energy]]<br>
*[[Center of Mass]]<br>


===External Links===
Kinetic energy in real systems consists of multiple components. By separating it into translational, rotational, and vibrational parts, we can more accurately understand and analyze motion.
*https://www.youtube.com/watch?v=5qwW8WI1gkw&feature=youtu.be<br>
*https://youtu.be/Cobhu3lgeMg<br>
*https://www.youtube.com/watch?v=craljBk-E5g&feature=youtu.be<br>
*https://youtu.be/XlFlZHfAZeE<br>
*https://youtu.be/vL5yTCyRMGk<br>
*https://en.wikipedia.org/wiki/Kinetic_energy<br>
*https://en.wikipedia.org/wiki/Moment_of_inertia<br>
*https://www.khanacademy.org/science/physics/torque-angular-momentum/torque-tutorial/a/rotational-inertia<br>
*https://youtu.be/vL5yTCyRMGk


==References==
==References==
All problem examples, youtube videos, and images are from the websites referenced below:


*http://p3server.pa.msu.edu/coursewiki/doku.php?id=183_notes:energy_sep
*https://openstax.org/details/books/university-physics-volume-1
 
*https://www.khanacademy.org/science/physics/work-and-energy
 
*https://en.wikipedia.org/wiki/Kinetic_energy
 
*https://en.wikipedia.org/wiki/Moment_of_inertia


*https://cnx.org/contents/1Q9uMg_a@6.4:V7Fr-AEP@3/103-Relating-Angular-and-Trans
*https://trinket.io/glowscript/


*https://en.wikipedia.org/wiki/Gaspard-Gustave_de_Coriolis
*https://ocw.mit.edu/courses/physics/

Latest revision as of 00:14, 29 April 2026

SHREYA LAKSHMISHA SPRING 2026

Main Idea

In many real-world situations, analyzing the kinetic energy of an object is more complex than just applying the formula:

[math]\displaystyle{ K = \cfrac{1}{2}mv^2 }[/math]

For example when a basketball is thrown, it is not only moving through space, but also rotating about its own axis. Because of this, the total kinetic energy must be broken into components.

The total kinetic energy of a system can be separated into:

  • Translational energy (motion of the center of mass)
  • Rotational energy (spinning motion)
  • Vibrational energy (internal motion of particles)

This breakdown allows us to more accurately analyze motion in physical systems.


Rolling_Racers_-_Moment_of_inertia

Mathematical Model

Total Kinetic Energy

[math]\displaystyle{ K_{total} = K_{translational} + K_{rotational} + K_{vibrational} }[/math]

This equation shows that energy must be considered in multiple forms when objects both move and rotate.


Translational Kinetic Energy

[math]\displaystyle{ K_{trans} = \cfrac{1}{2}Mv_{CM}^2 }[/math]
  • [math]\displaystyle{ M }[/math]: total mass
  • [math]\displaystyle{ v_{CM} }[/math]: velocity of the center of mass


The center of mass is calculated as:

[math]\displaystyle{ r_{CM} = \cfrac{\sum m_ir_i}{\sum m_i} }[/math]
[math]\displaystyle{ v_{CM} = \cfrac{\sum m_iv_i}{\sum m_i} }[/math]

Key Idea: Translational energy depends only on how the object moves.

Beam_with_pivot_P,_center_of_mass_S_and_center_of_percussion_C ---

Rotational Kinetic Energy

The total energy due to vibrations is the sum of the potential energy associated with interactions causing the vibrations and the kinetic energy of the vibrations.

[math]\displaystyle{ K_{rot} = \cfrac{1}{2}I\omega^2 }[/math]
  • [math]\displaystyle{ I }[/math]: moment of inertia
  • [math]\displaystyle{ \omega }[/math]: angular velocity

Moment of Inertia

[math]\displaystyle{ I = \sum m_i r_i^2 }[/math]

This measures how difficult it is to rotate an object.

Important Insight: Mass farther from the axis increases rotational energy because of the \(r^2\) term.


Moment_of_inertia_solid_sphere

---

Angular Speed and Velocity
[math]\displaystyle{ \omega = \cfrac{2\pi}{T} }[/math]
[math]\displaystyle{ v = \omega r }[/math]

Points farther from the center move faster.

---

Vibrational Kinetic Energy

Vibrational energy comes from internal motion of particles within an object.

  • Important in molecules and thermal systems
  • Usually not directly calculated in introductory physics problems

---

Physical Intuition

Consider a rolling wheel:

  • Moves forward -> translational energy
  • Spins -> rotational energy
  • Internal atoms vibrate -> vibrational energy

Examples

Conceptual Example

A bowling ball rolls without slipping.

Which energies are present?

  • Translational ✔
  • Rotational ✔
  • Vibrational ✖ (ignored at this level)

---

Calculation Example

A solid disk rolls without slipping.

Given:

  • [math]\displaystyle{ m = 2 \, kg }[/math]
  • [math]\displaystyle{ v = 3 \, \frac{m}{s} }[/math]
  • [math]\displaystyle{ I = \cfrac{1}{2}mr^2 }[/math]

Step 1: Translational Energy

[math]\displaystyle{ K_{trans} = \cfrac{1}{2}mv^2 = 9 \, J }[/math]

Step 2: Rotational Energy

[math]\displaystyle{ K_{trans} = \cfrac{1}{2}mv^2 = 9 \, J }[/math]

Using [math]\displaystyle{ \omega = \cfrac{v}{r} }[/math]:

[math]\displaystyle{ K_{rot} = \cfrac{1}{4}mv^2 = 4.5 \, J }[/math]

Total Energy:

[math]\displaystyle{ K_{total} = 13.5 \, J }[/math]

---

Common Mistakes

  • Forgetting rotational energy in rolling problems
  • Using incorrect relationship between \(v\) and \(\omega\)
  • Ignoring moment of inertia differences
  • Assuming only translational motion matters

---

Computational Model

GlowScript simulation:

https://trinket.io/glowscript/31d0f9ad9e

This model helps visualize rotational motion and energy changes.

---

Connectedness

Personal Connection: Dance and sports like tennis involve rotation and motion, similar to energy concepts discussed here.

Academic Connection: Important in physics, engineering, and chemistry for analyzing motion and energy systems.

Industrial Applications:

  • Flywheels for energy storage
  • Rotating machinery
  • Engines and turbines

---

History

The concept of kinetic energy developed over time through contributions from scientists such as Aristotle, Leibniz, Bernoulli, and Gaspard-Gustave Coriolis. The term “kinetic energy” was later coined by Lord Kelvin.

Why This Matters for Exams

Most physics problems:

  • Combine translation and rotation
  • Require identifying ALL forms of energy

Missing one energy component often leads to incorrect answers.

---

Summary

Kinetic energy in real systems consists of multiple components. By separating it into translational, rotational, and vibrational parts, we can more accurately understand and analyze motion.

References

Retrieved from "http://www.physicsbook.gatech.edu/index.php?title=Translational,_Rotational_and_Vibrational_Energy&oldid=48316"