Translational, Rotational and Vibrational Energy - Revision history

Slakshmisha3: /* References */

2026-04-29T04:14:47Z

References

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	*https://trinket.io/glowscript/		*https://trinket.io/glowscript/

			*https://ocw.mit.edu/courses/physics/

2026-04-29T04:12:31Z

References

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

	*~~http~~://~~www~~.~~robjorstad.com~~/~~Phys161~~/~~161Lab~~/~~161RotationalKinematicsSim.pdf~~		*https://openstax.org/details/books/university-physics-volume-1

	*~~http~~://~~p3server~~.pa.~~msu.edu~~/~~coursewiki~~/~~doku.php?id=183_notes:energy_sep~~		*https://www.khanacademy.org/science/physics/work-and-energy

	*https://~~cnx~~.org/~~contents/1Q9uMg_a@6.4:V7Fr-AEP@3~~/~~103-Relating-Angular-and-Trans~~		*https://en.wikipedia.org/wiki/Kinetic_energy

	*https://en.wikipedia.org/wiki/~~Gaspard-Gustave_de_Coriolis~~		*https://en.wikipedia.org/wiki/Moment_of_inertia

	*https://~~newton~~.~~ph.msstate.edu~~/~~fox~~/~~public_html/ph2213/examples10-core.pdf~~		*https://trinket.io/glowscript/

2026-04-29T04:11:18Z

External Links

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	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.		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.

	~~===External Links===~~
	*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==

2026-04-29T04:08:01Z

Calculation Example

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

	Step 1: Translational Energy		Step 1: Translational Energy

	\[		::<math>K_{trans} = \cfrac{1}{2}mv^2 = 9 \, J</math>
	K_{trans} = \~~frac~~{1}{2}mv^2 = 9 \, J
	\]

	Step 2: Rotational Energy		Step 2: Rotational Energy

	\[ K_{~~rot~~} = \~~frac~~{1}{2}~~I\omega~~^2		::<math>K_{trans} = \cfrac{1}{2}mv^2 = 9 \, J</math>
	\]

	Using \( \omega = \~~frac~~{v}{r} \): \[ K_{rot} = \~~frac~~{1}{4}mv^2 = 4.5 \, J \]		Using <math>\omega = \cfrac{v}{r}</math>:

			::<math>K_{rot} = \cfrac{1}{4}mv^2 = 4.5 \, J</math>

	Total Energy:		Total Energy:

	\[ K_{total} = 13.5 \, J		::<math>K_{total} = 13.5 \, J</math>
	\]

	---		---

2026-04-29T04:05:44Z

Moment of Inertia

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	====Moment of Inertia====		====Moment of Inertia====
	\[
	I = \sum ~~m_ir_i~~^2		::<math>I = \sum m_i r_i^2</math>
	\]

	This measures how difficult it is to rotate an object.		This measures how difficult it is to rotate an object.
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	Mass farther from the axis increases rotational energy because of the \(r^2\) term.		Mass farther from the axis increases rotational energy because of the \(r^2\) term.

	[[File:~~Moment_of_inertia_diagram~~.svg\|~~300px\|right\|thumb\|Mass farther from the axis increases rotational inertia.~~]]
			[[File:Moment of inertia solid sphere.svg\|Moment_of_inertia_solid_sphere]]

	---		---
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	=====Angular Speed and Velocity=====		=====Angular Speed and Velocity=====

	\[		::<math>\omega = \cfrac{2\pi}{T}</math>
	\omega = \~~frac~~{2\pi}{T}
	\]

	\[		::<math>v = \omega r</math>
	v = \omega r
	\]

	Points farther from the center move faster.		Points farther from the center move faster.

2026-04-29T04:04:11Z

Rotational Kinetic Energy

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	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.		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.

	\[		::<math>K_{rot} = \cfrac{1}{2}I\omega^2</math>
	K_{rot} = \~~frac~~{1}{2}I\omega^2
	\]

	* \(I\): moment of inertia
	* \(\omega\): angular velocity

			* <math>I</math>: moment of inertia
			* <math>\omega</math>: angular velocity

	====Moment of Inertia====		====Moment of Inertia====

2026-04-29T04:03:46Z

Translational Kinetic Energy

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

	[[File:~~Center_of_mass_diagram.svg\|300px\|center\|thumb\|The~~ center of mass ~~represents the average position~~ of ~~mass in a system~~.]]		[[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]]

	---		---

2026-04-29T04:01:11Z

Translational Kinetic Energy

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

	\[		::<math>K_{trans} = \cfrac{1}{2}Mv_{CM}^2</math>
	K_{trans} = \~~frac~~{1}{2}Mv_{CM}^2
	\]

	* \(M\): total mass		* <math>M</math>: total mass
	* \(v_{CM}\): velocity of the center of mass		* <math>v_{CM}</math>: velocity of the center of mass


	The center of mass is calculated as:		The center of mass is calculated as:

	\[		::<math>r_{CM} = \cfrac{\sum m_ir_i}{\sum m_i}</math>
	r_{CM} = \~~frac~~{\sum m_ir_i}{\sum m_i}		::<math>v_{CM} = \cfrac{\sum m_iv_i}{\sum m_i}</math>
	~~\quad\quad~~
	v_{CM} = \~~frac~~{\sum m_iv_i}{\sum m_i}
	\]

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

2026-04-29T03:59:53Z

Total Kinetic Energy

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	=== Total Kinetic Energy ===		=== Total Kinetic Energy ===

	\[K_{total} = K_{translational} + K_{rotational} + K_{vibrational}\]		<math>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.		This equation shows that energy must be considered in multiple forms when objects both move and rotate.

2026-04-29T03:59:08Z

Total Kinetic Energy

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	=== Total Kinetic Energy ===		=== Total Kinetic Energy ===

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

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