# Conservation of angular momentum or conservation of energy

I've read all the questions with similar titles but I couldn't find an answer.

Suppose I'm rotating with my arms extended on a frictionless surface. I have angular momentum and energy:

$$L_0=I_0\ \omega_0$$ $$E_0=\frac{1}{2}I_0\ \omega_0^{2}$$

Where $I_0$ is my moment of Inertia with my arms extended and $\omega_0$ is my initial angular velocity.

Suddenly, I decide to flex my arms to decrease my moment of Intertia. Then my angular momentum and energy are:

$$L_f=I_f\ \omega_f$$

$$E_f=\frac{1}{2}I_f\ \omega_F^{2}$$

If I use conservation of energy to calculate final angular velocity I get:

$$\omega_f = \sqrt{\frac{I_0}{I_f}}\omega_0$$

But if I use conservation of angular momentum:

$$\omega_f = \frac{I_0}{I_f}\omega_0$$

Both can't be right... Is energy not conserved in this problem? why?

Edit: Many anwers have pointed out that I'm actually doing work when I pull my arms back. Thanks for that clarification! What would happen if the system is a disk rotating with two persons in each side and they start walking towards the center? They walk using static frictional force which does not do work. Would energy be conserved then?

• Try including the work you did pulling in your arms... Commented Sep 10, 2017 at 21:36
• The key is in your question: the word flex. Commented Sep 10, 2017 at 21:41
• Why do you think that friction does not do work? Its a force like any other, and the objects in contact are moving...
– Eddy
Commented Sep 10, 2017 at 21:50
• Because static friction is applied on a point that doesn't move so there isn't any displacement along its direction and hence no work. Commented Sep 10, 2017 at 21:53
• Ah, so your confusion is that you think someone can stand on a spinning disk but there foot not move. How exactly?
– Eddy
Commented Sep 10, 2017 at 22:09

You do work on your arms as you pull them in, thus your energy has increased. The correct conserved quantity is angular momentum, as you deduce. The amount of work done on your arms can either be computed directly (force times distance type approach) or by using the solution from angular momentum (final energy minus initial energy type approach).

• Thanks for the answer! I edited the question in view of your answer. Commented Sep 10, 2017 at 21:46

The idea you sketch out in your edit will simply not work out like you imagine.

A static frictional force alone can't cause a massive object to start moving anywhere; on the contrary static friction is the force you have to overcome to set an object resting on a surface into motion. It's the static frictional force that saves you from having to do work to hold your position on the spinning disc and if you increase the disc's speed, at some point you'll start moving away from the discs center.

If you've ever done a ride on some kind of really fast spinning carousel, you should remember how ,much work you'd have to do to move towards the carousel's center.

• A static friction force can and does make things accelerate. The static part refers to the relative motion of the surfaces in contact.
– nasu
Commented Feb 21, 2022 at 1:17

]You stand in a bus or train. You Don't hold any handle or grab bar. The train starts smoothly and accelerates. You accelerate too. Your velocity increases and so does your KE. What force did the work responsible for the change in your KE? Don't bother to include muscle forces. Same thing happens to your suitcase sitting next to you. And it has no muscles or tendons. So, your assumption that static friction cannot do work is not correct. There is nothing unusual to explain. If you walk towards the center of the disk without sliding laterally you have to slow down your motion so there is a tangential acceleration and so, a tangential force who does work. This force is (or may be) static friction.