Question about the definition of Torque Equilibrium

[rmln]

TL;DR

torque equilibrium clarification

Hi, I have the following question:
Suppose you have an ideal rod and two forces of equal magnitude are applied to its ends, in such a way:

Now it is obvious that the rod is in rotational equilibrium but my textbook says that for rotational equilibrium, the net torque must be 0 about any axis, however in this case it is only 0 about its midpoint. So what am I missing here?

Thanks in advance!

 

[Lnewqban]

So what am I missing here?

The downward force of magnitude 2F that is applied at the midpoint pivot.

Welcome, rmin!

 

[A.T.]

TL;DR Summary: torque equilibrium clarification

Hi, I have the following question:
Suppose you have an ideal rod and two forces of equal magnitude are applied to its ends, in such a way:

View attachment 361558


Now it is obvious that the rod is in rotational equilibrium but my textbook says that for rotational equilibrium, the net torque must be 0 about any axis, however in this case it is only 0 about its midpoint. So what am I missing here?

Thanks in advance!

You simply define 'rotational equilibrium' differently than your textbook.

It's true that your rod, if initially at rest, will not change its orientation, and thus will not start rotating in a reference point invariant way.

But for reference points, for which the net torque is not zero, it will gain angular momentum. To prevent this, the net force must be zero as well.

 

[rmln]

The downward force of magnitude 2F that is applied at the midpoint pivot.

Welcome, rmin!

Thanks, but I'm curious about this specific case where there are only these two forces. I can see that in the situation you're talking about, there's clearly torque equilibrium.

 

[rmln]

You simply define 'rotational equilibrium' differently than your textbook.

It's true that your rod, if initially at rest, will not change its orientation, and thus will not start rotating in a reference point invariant way.

But for reference points, for which the net torque is not zero, it will gain angular momentum. To prevent this, the net force must be zero as well.

Thanks for your answer! Does this imply that it is impossible to achieve rotational equilibrium (not just about certain point(s) but for a body) without the net force acting on the body being zero?

 

[jbriggs444]

Thanks, but I'm curious about this specific case where there are only these two forces. I can see that in the situation you're talking about, there's clearly torque equilibrium.

If there are only those two forces then there is a non-zero net linear force. That means that there is a non-zero rate of linear momentum change. ($\sum F = ma = \frac{dp}{dt}$).

That, in turn, means that if we take our rotational axis somewhere other than on the line traversed by the center of mass that the non-zero rate of change of linear momentum results in a non-zero rate of change of angular momentum. $L = r \times p + I \omega$ and $\frac{dL}{dt} = r \times \frac{dp}{dt} + I \frac{d \omega}{dt}$. If the displacement ($r$) of the center of mass from the reference axis is non-zero, this matters.

One can have a non-zero net torque without an associated non-zero rotational acceleration.

 

[A.T.]

Does this imply that it is impossible to achieve rotational equilibrium (not just about certain point(s) but for a body) without the net force acting on the body being zero?

Based on the definition your book apparently uses, yes. Because of what @jbriggs444 wrote above.

 

[rmln]

If there are only those two forces then there is a non-zero net linear force. That means that there is a non-zero rate of linear momentum change. ($\sum F = ma = \frac{dp}{dt}$).

That, in turn, means that if we take our rotational axis somewhere other than on the line traversed by the center of mass that the non-zero rate of change of linear momentum results in a non-zero rate of change of angular momentum. $L = r \times p + I \omega$ and $\frac{dL}{dt} = r \times \frac{dp}{dt} + I \frac{d \omega}{dt}$. If the displacement ($r$) of the center of mass from the reference axis is non-zero, this matters.

One can have a non-zero net torque without an associated non-zero rotational acceleration.

Thank you very much! Actually, the reason I asked this question was that in our textbook (Serway-Jewett), there is a question:

, for which the textbook's solution is option b) (no force equilibrium but torque equilibrium). In light of what you wrote, this seems wrong to me. Can someone please enlighten me why b) is the correct answer?

 

[Herman Trivilino]

TL;DR Summary: torque equilibrium clarification

Hi, I have the following question:
Suppose you have an ideal rod and two forces of equal magnitude are applied to its ends, in such a way:

View attachment 361558


Now it is obvious that the rod is in rotational equilibrium but my textbook says that for rotational equilibrium, the net torque must be 0 about any axis, however in this case it is only 0 about its midpoint. So what am I missing here?

Thanks in advance!

My guess is that what you are missing is that you are not mentioning what is likely stated in the texbook. Is there really no mention in the textbook of the support force at the center of the rod? That force, by the way, is pushing upward on the rod.

If indeed the texbook makes no mention of the rod being supported at its center, then you are indeed correct and you are missing the nothing. The texbook, on the other hand, is missing the very important reference to the support force at the center of the rod. Without that support force the net torque on the rod is not not zero about every point, and you are very clever for realizing that.

The oft-mentioned notion that if something is in rotational equilibrium about one point, then the net torque about every point is zero, depends on the object also being in translational equilibrium.

 

[A.T.]

Thank you very much! Actually, the reason I asked this question was that in our textbook (Serway-Jewett), there is a question:

View attachment 361593
View attachment 361594
, for which the textbook's solution is option b) (no force equilibrium but torque equilibrium). In light of what you wrote, this seems wrong to me. Can someone please enlighten me why b) is the correct answer?

Because 'torque equilibrium' in this case has apparently a different meaning than 'rotational equilibrium' as per your first post.

I was never a fan such questions, which don't test actual understanding, just memorization of often non standardized terminology.

 

[rmln]

My guess is that what you are missing is that you are not mentioning what is likely stated in the texbook. Is there really no mention in the textbook of the support force at the center of the rod? That force, by the way, is pushing upward on the rod.

If indeed the texbook makes no mention of the rod being supported at its center, then you are indeed correct and you are missing the nothing. The texbook, on the other hand, is missing the very important reference to the support force at the center of the rod. Without that support force the net torque on the rod is not not zero about every point, and you are very clever for realizing that.

The oft-mentioned notion that if something is in rotational equilibrium about one point, then the net torque about every point is zero, depends on the object also being in translational equilibrium.

Thanks for the answer! This wasn't a textbook exercise, I made it up.

Because 'torque equilibrium' in this case has apparently a different meaning than 'rotational equilibrium' as per your first post.

I was never a fan such questions, which don't test actual understanding, just memorization of often non standardized terminology.

I may have missed it, but interestingly, the book doesn't really explain the difference between rotational and torque equilibrium (if there's any). It defines rotational equilibrium of a rigid body as it having no angular acceleration about any axis, which if I understand correctly is just an alternative way of saying that the net force acting on it is zero(?). Funnily enough, I searched the textbook for any mentions of 'torque equilibrium' and the only results were in the quick quiz you see in this thread, and in another quick quiz just below it, so I assumed these were interchangeable.

I also wonder what would any of you answer to the quiz above?

 

[A.T.]

I may have missed it, but interestingly, the book doesn't really explain the difference between rotational and torque equilibrium (if there's any).

If both is used in the same book with different meanings, then that's really confusing.

It defines rotational equilibrium of a rigid body as it having no angular acceleration about any axis, which if I understand correctly is just an alternative way of saying that the net force acting on it is zero(?).

Not quite. Zero net force is a requirement for zero net torque about any axis, but it doesn't by itself imply it.

I also wonder what would any of you answer to the quiz above?

Without any further information from the book, and forced to choose one of the 4 alternatives I would choose d). To me, 'torque equilibrium' without specification of the axis means about any axis. If they meant 'torque equilibrium around the intersection of the lines of action' (leading to answer b), they should have stated this explicitly.

 

[Lnewqban]

Can someone please enlighten me why b) is the correct answer?

What makes you believe that option b) could not be the correct answer?

Please, see:
https://engcourses-uofa.ca/books/statics/forces-and-moments/couple-moment/

https://en.wikipedia.org/wiki/Couple_(mechanics)

 

[A.T.]

What makes you believe that option b) could not be the correct answer?

Please, see:
https://engcourses-uofa.ca/books/statics/forces-and-moments/couple-moment/

https://en.wikipedia.org/wiki/Couple_(mechanics)

View attachment 361638

View attachment 361639

View attachment 361640

Your examples match answer a) not b). But what do they have to do with the quiz question?

 

[Herman Trivilino]

TL;DR Summary: torque equilibrium clarification

Now it is obvious that the rod is in rotational equilibrium but my textbook says that for rotational equilibrium, the net torque must be 0 about any axis, however in this case it is only 0 about its midpoint. So what am I missing here?

So, I hope it's now clear to you that you've either misread your textbook or your textbook has made an error. The first condition (translational equilibrium) must be satisfied for that claim to be true. In other words, for the object to be in rotational equilibrium about any arbitrary point.