Where does the torque come from?

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In summary, the question being discussed is why does a hammer, or any other object, twist when tossed in the air without any external force acting on it. The answer is that the center of mass in the head of the hammer is located some small distance from the axis of the handle, which causes a slight torque on the head when the handle is flipped in a vertical arc. This is known as the intermediate axis theorem and can also be seen in objects with three unique moments of inertia, such as a cell phone or a gyroscope. The Tennis Racket Theorem is a qualitative explanation for this phenomenon, but the exact reason for it is not fully understood.
  • #1
Old Geezer
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Here's the question that I mentioned in my introduction:

If anyone can give me the answer, please keep it very simple - I was never any good at physics or higher math.

It's very easy and only requires a claw-hammer to do. Very low tech.

Hold the hammer at the end of the handle with the claw pointing to the right. Now flip it into the air having it do only one 360 degree turn. Catch it and notice where the claw is now. It's pointing to the left. Flip it once more and the claw points to the right. Why? Why does it twist when tossed? I'm not giving it any kind of twist when I toss it.

That's my question, and it's been bothering me for years.

Thanks for any help you can give me.
 
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  • #2
Old Geezer said:
Hold the hammer at the end of the handle with the claw pointing to the right. Now flip it into the air having it do only one 360 degree turn. Catch it and notice where the claw is now. It's pointing to the left. Flip it once more and the claw points to the right. Why? Why does it twist when tossed? I'm not giving it any kind of twist when I toss it.
The center of mass in the head of the hammer is located some small distance from the axis of the handle, so when the handle is flipped in a vertical arc with the handle sideways, there will be a slight torque on the head with respect to the axis of the handle.
 
  • #3
Astronuc said:
The center of mass in the head of the hammer is located some small distance from the axis of the handle, so when the handle is flipped in a vertical arc with the handle sideways, there will be a slight torque on the head with respect to the axis of the handle.
Thanks for the reply, but it seems to me that I've tried it using a 'T' shaped piece of wood (or something similar) and had the same result. The center of gravity would then be in the center of the cross piece at the end of the longer section.
 
  • #4
The human body is not an ideal machine.
 
  • #5
True, but I don't think that's the reason.
 
  • #6
I just tried this with two separate objects: a back scratcher, and a 12" length of 1 x 2 wood. In both cases, they flipped from one side to the other, like the claw hammer in the original example.

In other words, it seems the hammer does it because anything will do it. People should try this and see what result they get.
 
  • #7
Old Geezer said:
Hold the hammer at the end of the handle with the claw pointing to the right. Now flip it into the air having it do only one 360 degree turn. Catch it and notice where the claw is now. It's pointing to the left. Flip it once more and the claw points to the right. Why? Why does it twist when tossed? I'm not giving it any kind of twist when I toss it.

https://en.wikipedia.org/wiki/Tennis_racket_theorem



 
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  • #8
I guess that's the answer, then. Thanks for your help. Wikipedia seem to have all the answers to almost anything provided you know where to look.

Euler's therom is far beyond me, though.Paul N.
 
  • #9
Old Geezer said:
I guess that's the answer, then. Thanks for your help.
You're welcome.

Old Geezer said:
Euler's therom is far beyond me, though.
The plots based on conservation laws in the videos above are nice, but still quite abstract. Terry Tao offers an more concrete explanation based on the movement of rigidly connected point masses, which might be the basis for a more intuitive visualization:

http://mathoverflow.net/a/82020
 
  • #10
Please read my new member introduction, then you'll understand why the explanation is, unfortunately, beyond me.

If anyone wants to know where to find the cracks on a 'Tilt-A-Whirl' carnival ride, I can tell them. I spent 23 years as a Safety & Health inspector for the NYS Dept. of Labor and know how to inspect rides, ski lifts, elevators and a lot of other things. That stuff I understand; physics, no.
 
  • #11
CWatters said:
The human body is not an ideal machine.

If it was then the axis of rotation would be exactly the same as the object's second principal axis and the effect would not occur.
 
  • #12
Old Geezer said:
Please read my new member introduction, then you'll understand why the explanation is, unfortunately, beyond me.

Best (plain English) explanation I can manage is that...

You will have seen that a spinning gyroscope is stable and doesn't fall over. However not all spinning objects are stable. Some are unstable and very small errors in the way you spin the object become exaggerated.
 
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  • #13
CWatters said:
You will have seen that a spinning gyroscope is stable and doesn't fall over. However not all spinning objects are stable.

In the case of the gyroscope there is a qualitative explanation based on linear dynamics of point masses:



I wonder if there is something similar for the intermediate axis theorem.
 
  • #14
The Tennis Racket Theorem is the right answer.
 
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  • #16
Spinning objects with three unique moments of inertia are unstable about their intermediate axis. Try spinning a cell phone about all three axes (separately), about the axes with max and min moments, the rotation is stable, but about the intermediate axis the rotation is unstable. It's true that if you were top spin it perfectly about the intermediate principle axis, it would be stable, but this is the same kind of stability as balancing a ball on top of another ball.

The second video posted by A.T. shows a very nice illustration of the torque free rotation of objects with three unique moments of inertia.

But if you are truly asking "why?" I don't know. I studied the topic for over a year and was not able to figure it out. There are lots of solutions; you can evaluate Euler's equations numerically easily and see the instability, you could probably evaluate them analytically too and see the solution. Poinsot created something called the Poinsot construction that is also a really nice qualitative description of the motion (involving polhods and herpolhodes...).

It was a machinist who first showed me the hammer instability; I think the only satisfying answer to me would be to explain the behavior from the point of view of shrinking me down to a very small size (aka ant man) and riding on the hammer and asking myself, what forces do I feel?

If you would like, I can provide references to the sources I read on torque free rotation.

Chris
 
  • #17
CWatters said:
Yeah, I linked that already. It's an explanation using angular dynamics quantities, just like the standard angular momentum vector explanation of a gyro. What I haven't found is a good explanation based on point masses, similar to the satellite analogy in the gyro video above.
 
  • #18
A.T. said:
Yeah, I linked that already. It's an explanation using angular dynamics quantities, just like the standard angular momentum vector explanation of a gyro. What I haven't found is a good explanation based on point masses, similar to the satellite analogy in the gyro video above.
Have you ever heard of D'Alembert's principle? I was thinking it might be a useful technique to try to understand the rotation about the intermediate axis, but I just have not have had time to do it.

Chris
 
  • #19
Did anyone catch that The title of the post "Where does Torque come from?" was posted a "moment" ago?
 
  • #20
from what i have read, the torque comes from the centrifugal force acting on the T tip if the tip is displaced from center... this force gets bigger as the displacement from aligned gets larger... until you have a huge momentum that carries the tip all the way to the opposite direction... but i always wonder why it can stay at the extreme for more than a bounce... like, to me, the thing should bounce from side to side, with the instability growing each and every flip/bounce... or maybe if there is drag, then it should decay to one of the other states [ min Momentum or max Momentum??]..and just wondering, if the flip force hits 0 at the apex?

any thoughts
 
  • #21
There is no torque. When you write out Euler's equations in this case (force-free), your source terms to the three non-linear coupled differential equations = 0.
 
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  • #22
It's amazing that the question would seem to have an easy answer, and yet, one professor recently posted (and he attributed Ed Purcell (a prominent physicist who may have said it )), rigid body dynamics, is one of the most difficult subjects for undergraduates in physics to understand. (Not quantum mechanics, not relativity ) This is particularly true of the Tennis Racket Theorem.

A complete treatment of the problem requires Jacobi Elliptic, and Theta functions.

I congratulate the poster for the curiosity over the years of this unusual phenomena. Most observers would overlook it or regard it as unimportant. I understand when putting satellites in orbit, spin rates around an intermediate axis are a real concern.
 
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  • #24
Isn't this the intermediate axis theorem? If so there's a good intuitive explanation here...

 
  • #25
bland said:
Isn't this the intermediate axis theorem? If so there's a good intuitive explanation here...


I wish there was a good intuitive explanation along these lines (linear dynamics). Unfortunately the explanation in this Veritasium video (based on a post by Terry Tao) doesn't really work, as explained in the StandupMaths video and many comments before. Tao since then updated his post, to include Coriolis forces in order to fix the problem, but those of course aren't very intuitive.

The explanation with the ellipsoids, as nice as it looks visualized, is not very intuitive either. The ellipsoids represent quite abstract quantities, which are not easy to interpret in terms of linear dynamics (which are intuitive for more people).
 
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  • #26
A.T. said:
I wish there was a good intuitive explanation along these lines (linear dynamics). Unfortunately the explanation in this Veritasium video (based on a post by Terry Tao) doesn't really work, as explained in the StandupMaths video and many comments before. Tao since then updated his post, to include Coriolis forces in order to fix the problem, but those of course aren't very intuitive.

I also found it a bit confusing why he refers to the internal tension forces so much. I think it's easier to say that, e.g. in the case where the ##M## masses are rotating about the ##\hat{y}## axis, then in the rotating frame if we subject the ##m## masses to a small disturbance, the torques about the ##\hat{y}## axis of the Coriolis forces acting on the ##m## masses are small whilst the moment of inertia about the ##\hat{y}## axis is large (##\sim 2MR^2##). And the result is no deviation of the ##M## masses into the ##xz## plane in this regime. And the more regular motion he described in the first part takes place.

Whilst if we set it up so that the ##m## masses are now rotating about the ##\hat{y}## axis, then in the rotating frame if we subject the ##M## masses to a small disturbance, the torques about the ##\hat{y}## axis of the Coriolis forces acting on the ##M## masses are large whilst the moment of inertia about the ##\hat{y}## axis is small (##\sim 2mR^2##). And the result is significant, un-ignorable, deviation of the ##m## masses into the ##xz## plane. And the motion goes all funny.

I think it's easier conceptually to deal with rigid bodies with only external forces and torques, and not worry about internal stresses!
 
  • #27
etotheipi said:
I think it's easier conceptually to deal with rigid bodies with only external forces and torques, and not worry about internal stresses!
Yes, but if you want to explain rotational phenomena in terms of linear dynamics, you have to model the rigid body as a set of point masses. Then the forces internal to the whole rigid body are external forces to the point masses.
 
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1. What is torque?

Torque is a measure of the turning force applied to an object. It is a rotational equivalent of linear force and is typically measured in Newton-meters (Nm) or pound-feet (lb-ft).

2. Where does torque come from?

Torque comes from the application of a force at a distance from the axis of rotation. This force creates a moment, which causes an object to rotate around its axis.

3. What factors affect the amount of torque generated?

The amount of torque generated is dependent on the magnitude of the applied force, the distance from the axis of rotation, and the angle between the force and the lever arm.

4. How is torque related to power?

Torque and power are closely related, as power is defined as the rate at which work is done, or the rate at which torque is applied. The more torque an object has, the more power it can produce.

5. Can torque be negative?

Yes, torque can be negative. A negative torque occurs when the direction of the force is opposite to the direction of rotation, causing the object to slow down or rotate in the opposite direction.

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