Is Rotational Kinetic Energy Analogous to Work in Torque Analysis?

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Discussion Overview

The discussion revolves around the relationship between torque and rotational kinetic energy, specifically whether the integral of torque times angular displacement corresponds to a variation in rotational kinetic energy. Participants explore the terminology and conceptual parallels between work and this integral, as well as the implications of these relationships in different physical scenarios.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants propose that integrating torque over angular displacement yields a result analogous to work, suggesting it could be termed "worque."
  • Others argue that while both torque and force can be considered forms of work, they refer to different physical manifestations, particularly in scenarios like a disk rolling down an incline.
  • A participant notes that the numerical results for translational and rotational kinetic energy are not the same, raising questions about the terminology used.
  • Some participants clarify that the grouping of terms into "rotational" and "translational" components is for convenience, as a rigid body consists of many particles, each contributing to the total kinetic energy.
  • There is a challenge regarding the mathematical understanding of the relationship between torque and kinetic energy, with some participants emphasizing the need for calculations to clarify these concepts.
  • One participant expresses confusion about the relationship between the center of mass component of kinetic energy and the rotational component, seeking clarification on whether the integral of torque corresponds to the net variation of the rotational kinetic energy.

Areas of Agreement / Disagreement

Participants express differing views on the terminology and conceptual understanding of torque and work, with no clear consensus reached on whether the integral of torque corresponds to a specific variation in rotational kinetic energy. Some agree on the equivalence of the concepts, while others highlight the distinctions.

Contextual Notes

There are unresolved mathematical steps and assumptions regarding the definitions of work and energy in the context of rotational motion, as well as the implications of integrating torque with respect to angular displacement.

Who May Find This Useful

This discussion may be of interest to students and professionals in physics, engineering, and related fields who are exploring the concepts of rotational dynamics and the relationships between torque, work, and kinetic energy.

DaTario
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Hi all,

If we take the integral of tau (the resultant torque) times the angular displacement theta, with respect to the angular variable theta, are we to expect the result to correspond to the variation in rotational kinetic energy?

Is is this the case, does this integral have any particular name (analogous to work) ? "WORQUE" :)

Thank you

DaTario
 
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DaTario said:
If we take the integral of tau (the resultant torque) times the angular displacement theta, with respect to the angular variable theta, are we to expect the result to correspond to the variation in rotational kinetic energy?
Yes.
Is is this the case, does this integral have any particular name (analogous to work) ?
It's just called work.
 
It is also called work? But it reffers to different physical manifestation. Considering for example a disk rolling down an incline. The variation in kinetic energy is dou to the work done by the resutant force while the variation in rotational kinetic energy is produced by this "worque" ...

best wishes

DaTario
 
Realize that integrating torque over an angle is equivalent to integrating force over a distance. Both give a change in kinetic energy and can be called work. It requires work to increase translational KE and it requires work to increase rotational KE.
 
Ok, I know both energies are energies, but the numerical results in each calculation are not the same.
But if you are saying that both expressions are called work, it is ok to me.
In a certain way I was expecting something like rotational work or something like that.

Thank you

Best wishes

DaTario
 
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Just to add to Doc Al's comment...
recall that a rigid body is a many-particle system.
The grouping of certain terms into a "rotational" part is just for convenience.
 
DaTario said:
Is is this the case, does this integral have any particular name (analogous to work) ? "WORQUE" :)

I like the name "WORQUE", I think I'll start using it instead.

My boss is from NY, he tells me to get to "worque" all the time! :smile:
 
stewartcs said:
I like the name "WORQUE", I think I'll start using it instead.

My boss is from NY, he tells me to get to "worque" all the time! :smile:

Should that be New Yorque?
 
robphy said:
Just to add to Doc Al's comment...
recall that a rigid body is a many-particle system.
The grouping of certain terms into a "rotational" part is just for convenience.

I guess I understand this part.

I am a little bit confused, as in the case of a disk rolling down an incline 1/2 * M V^2 is the kinetic energy of the CM while 1/2* I Omega^2 is related to the rotational motion.
 
  • #10
DaTario said:
I am a little bit confused, as in the case of a disk rolling down an incline 1/2 * M V^2 is the kinetic energy of the CM while 1/2* I Omega^2 is related to the rotational motion.
As robphy indicated, the real deal is that each piece of the rigid body--each tiny element of mass--is moving with some speed. So each mass element has some KE. Add them all up, and that's the total KE.

It's just easier to take advantage of the fact that it's a rigid body and separate out a component of kinetic energy associated with the translational motion of the center of mass and a component associated with rotation motion about the center of mass. But that sum is equal to what you'd get if you just added the KE of each mass element.
 
  • #11
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  • #12
Doc Al said:
It's just easier to take advantage of the fact that it's a rigid body and separate out a component of kinetic energy associated with the translational motion of the center of mass and a component associated with rotation motion about the center of mass. But that sum is equal to what you'd get if you just added the KE of each mass element.

Very nicely said. But now comes my question: the work of the resultant is the net variation of the CM component of the total KE.

Will the integral of Tau (resultant torque) with respect to Theta (angular displacement) correspond to the net variation of the rotational component of the total KE ?

best wishes

DaTario
 
  • #13
Can't you do the maths for once, DaTario?

For a point on a rigid body, acted upon by an external force [itex]\vec{f}[/tex], with distance [itex]\vec{r}[/itex] from the mass centre, with velocity [tex]\vec{v}=\vec{v}_{C.M}+\vec{\omega}\times\vec{r}[/tex] its contribution to the power of the rigis body is:<br /> [tex]\vec{f}\cdot\vec{v}=\vec{f}\cdot\vec{v}_{C.M}+\vec{f}\cdot(\vec{\omega}\times\vec{r})=\vec{f}\cdot\vec{v}_{C.M}+(\vec{r}\times\vec{f})\cdot\vec{\omega}[/tex].<br /> <br /> See if you can glean the result from this.[/itex]
 
  • #14
I am not asking you to make calculations. You misinterpreted me. I am asking if someone has this complementary information about the work-energy theorem.

If it is clear to you I would be pleased to know if the integral of Tau (resultant torque) with respect to Theta (angular displacement) corresponds to the net variation of the rotational component of the total KE?

Could you explain what is the purpose of this forum, if it isn't for people to interact, to exchange information and to debate?
 
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  • #15
DaTario said:
I am not asking you to make calculations.
That is your main mistake.
You misinterpreted me. I am asking if someone has this complementary information about the work-energy theorem.
Which is given by..the calculations.

Could you explain what is the purpose of this forum, if it isn't for people to interact, to exchange information and to debate?
I just gave you the information, I'm not into hand-wavy airiness, neither should you, if you actually want to learn some physics, which is the main point of this forum.
 
  • #16
DaTario said:
If it is clear to you I would be pleased to know if the integral of Tau (resultant torque) with respect to Theta (angular displacement) corresponds to the net variation of the rotational component of the total KE?

This is something you should be able to answer yourself. Take a simple, two-dimensional case, so you don't have to worry about vectors. Write down an equation relating the torque and angular displacement to the force, velocity and linear displacement of each small part of the object. Use this to set up two integrals, one for [itex]F dr[/itex], and one for [itex]\tau d\theta[/itex]. Check whether the results are equal.
 
  • #17
DaTario said:
But now comes my question: the work of the resultant is the net variation of the CM component of the total KE.
Yes, if by work you mean [itex]\int \vec{F}_{net} \cdot d\vec{r}_{cm}[/itex].

Will the integral of Tau (resultant torque) with respect to Theta (angular displacement) correspond to the net variation of the rotational component of the total KE ?
Yes, as long as the torque is taken with respect to the center of mass.
 
  • #18
Doc Al, Thanks.
Ben Niehoff, welcome and thanks.
Aridno, you have talents.

best wishes

DaTario
 
  • #19
DaTario said:
Is is this the case, does this integral have any particular name (analogous to work) ? "WORQUE" :)

robphy said:
Should that be New Yorque?

(Groan) You are both dorques. Actually, I liked both, but now put a corque in it.
 

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