# Rotational Mechanics: Is Angular Acceleration Possible without any External Torque?

When a man is doing ice skating and rotating on his toe, with his arms widespread, his angular velocity is less, in comparison to the angular velocity when he is rotating with his arms closed inside. Now suppose man is turning his hands slowly inwards, so his angular velocity will start to increase, now there is no external torque on the man but his angular velocity is increasing and increasing angular velocity will have an associated angular acceleration, so we can conclude that the man has angular acceleration without any external torque, which is an apparent contradiction of the terms, so how do we reconcile the case with the concept?

Can we explain this case without using the concept of "Angular Momentum Conservation"? because that encapsulates a lot of details, without giving the complete clarity.

• This Vsauce video might be helpful.
– user238497
Commented Nov 24, 2019 at 1:22

The definition of torque is not $$\tau=Id\omega/dt$$. We can't even define things like $$I$$ and $$\omega$$ for rotation that isn't rigid.

The definition of torque is $$\tau=dL/dt$$. So yes, it is possible to have an angular acceleration without an external torque. Your example shows correctly that this can happen.

• @DevanshMittal: Your comment sounds like it's answering your own question. This is allowed on SE, but it should be posted as an answer, not a comment.
– user4552
Commented Nov 23, 2019 at 15:46
• Kindly confirm following propositions. 1. Angular Acceleration is not the same as Translation Acceleration. Translational Acceleration in an object cannot come without any external force, but the angular acceleration in an object may come without any external torque. 2. External torque is NOT the only source of angular acceleration. Angular acceleration in a rotating body may come even when the moment of inertia of the body is changing. 3. The question is, Are Internal Torques here bringing the angular acceleration in the body? Commented Nov 23, 2019 at 15:48
• @DevanshMittal What do you have in mind when you refer to an "internal torque"? Commented Nov 23, 2019 at 15:55
• @DevanshMittal And I agree with Ben that you seem to be answering your own question. I was about to post an answer that the reconciliation is conservation of angular momentum which occurs without an external torque. But it seems you are giving that as an answer, except you are also talking about some kind of internal torque. Commented Nov 23, 2019 at 15:58
• The reason this argument of internal torque came to my mind because I was starting with the premise that angular acceleration cannot come without external torques, so if we cannot find anything external, so we have to look for something internal. Commented Nov 23, 2019 at 16:03

so we can conclude that the man has angular acceleration without any external torque, which is an apparent contradiction of the terms, so how do we reconcile the case with the concept?

We reconcile it with the law of conservation of angular momentum.

The angular velocity of the skater increases when drawing in the arms in order to conserve angular momentum. The angular momentum of the skater will not change unless an external torque is applied to the object. So converse to your thinking, the change in angular velocity is due to no external torque being applied to the skater in order to conserve angular momentum.

Conservation of Energy:

The increase in angular velocity can also be explained by conservation of rotational kinetic energy. Ignoring friction there is no external force that can cause a change in the skaters rotational kinetic energy = 1/2 I$$a^2$$ where I is the rotational moment of internia of the skater and $$a$$ is the angular velocity of the skater. When the skater pulls his/her arms in it reduces the rotational moment of inertia I. In order to conserve kinetic energy the skater’s angular velocity $$a$$ must increase. Note however you can say that an internal force is what enabled the skater to pull in his/her arms.

Hope this helps.

• Can we explain the above case with the help of just forces without using any result like "Angular Momentum Conservation? I think that will give us more insights into what is exactly happening. Commented Nov 23, 2019 at 16:06
• A nifty side-issue to this is that when the skater pulls their arms in, rotational energy is not conserved. You can work out the starting and ending speeds just from angular momentum and moment of inertia; the change in speed is inversely proportional to the change in moment of inertia. But the change in energy is proportional to speed squared times moment of inertia, so the skater who has drawn their arms in is going faster. If you think about it, if you're spinning and you pull your arms in, you have to do so against centripetal acceleration -- and that provides the extra energy. Commented Nov 23, 2019 at 16:09
• @DevanshMittal Other than friction of the ice that eventually makes the skater come to a stop, what other external forces are acting on the skater? Commented Nov 23, 2019 at 16:11
• We can assume a frictionless surface for the sake of focussed and simplicity of discussion. Commented Nov 23, 2019 at 16:24
• @Devansh Mittal I have added a conservation of energy explanation Commented Nov 23, 2019 at 17:05

When you pull your arms in you aren't pulling them directly towards the centre, because you're rotating as you're pulling them in. This is where the force comes from that actually makes you spin faster. You should definitely watch this video where he explains exactly this. Skip to 10m in if you're in a hurry but the whole video is well worth watching.

I know what you mean that "conservation of angular momentum" explanations can feel like they're hiding the details of the actual forces and torques going on. You can make a case that the conservation laws are actually more fundamental but either way both explanations are always possible and always give the same result.

Can we explain the above case with the help of just forces without using any result like "Angular Momentum Conservation? I think that will give us more insights into what is exactly happening.

Well, yes and no. Because with angular conservation laws, you can just say that $$L = L$$, which means that $$I_1 \omega_1 = I_2 \omega_2$$, and then you can do some math on the back of an envelope, and you're done. This is why conservation laws are so very nice -- you can often reduce a problem immensely using them.

So I'm not going to do all the math, because I'm lazy and because I have the appropriate conservation law. But in general, let the skater's initial rotational velocity be $$\omega_1$$. Then their hands are moving at a speed $$\omega_1 r_1$$, where $$r_1$$ is the distance from their hands to their axis of rotation. As they pull in their arms, $$r$$ will decrease; this means that their rotating body will try to slow their hands down (because if $$\frac{dr}{dt} < 0$$ then so is $$\omega \frac{dr}{dt} < 0$$). That "trying to slow their hands down" will translate to them exerting a tangential force on their hands, which means that their hands will exert a tangential force on their body, which means that their body will speed up.

Note that I've gone from two short equations to a long paragraph and I'm not done yet.

If you were going to calculate this exactly, you'd have to account for the fact that there is mass that is distributed along their arms, and that their rotational velocity is changing at the same time that their arms are pulling in, etc., etc. You'd end up with a partial differential equation which, if I'm not mistaken (someone is welcome to correct me), is nonlinear as well. You will consume pages and pages to do the math, and when you are done, you will get the same result as just doing two simple short equations.

So, you're welcome to it. I'll just thank God* that we live in a universe that is rotationally symmetric and unchanging with time**, and I'll do the two lines of math. If I do need to compute, say, the forces involved in expanding and shrinking a spinning top, then I won't formulate some giant equation that solves for $$\omega$$ -- I'll find that using conservation of momentum over time, then I'll find $$\frac{d\omega}{dt}$$ at any time, then I'll use that to find any tangential forces I need to solve for.

* or random chance, or the Universal Creator of your choice

** And Emily Noerther for pointing out that the consequence of these is conservation of rotational momentum and energy

1. Angular Acceleration is not the same as Translation Acceleration. Translational Acceleration in an object cannot come without any external force, but the angular acceleration in an object may come without any external torque.

2. External torque is NOT the only source of angular acceleration. Angular acceleration in a rotating body may come even when the moment of inertia of the body is changing.

3. Let's consider a simple case in which a car is moving on a straight road and an observer is seeing the car from a distance. In this case, the car will have a variable angular velocity w.r.t. the observer and hence an angular acceleration as well. In this case, we do not ask the question that which torque is responsible for the angular acceleration of the body! So, it concludes that it is not necessary that torque is a source of angular acceleration.

4. In the case of rotating man, when the man is stretching or contracting his arms the various portions of arms go in spiral path and the centripetal force either makes an obtuse angle or an acute angle with the instantaneous velocity of any portion of the arm, which either decelerates or accelerates the portion of arm, that changes the instantaneous velocities of all the portions of the arm and hence the angular velocity also changes and rate of change of angular velocity is seen as angular acceleration.

The following two resources amazingly expound on the above ideas.

Resource 1

Laws and Causes by VSauce: https://www.youtube.com/watch?v=_WHRWLnVm_M

Resource 2: