# How equivalent are heat energy and work energy in connection with a spinning flywheel?

Let's say we have two identical spinning flywheels, that have arbitrary geometry, and are made of copper.

Now we apply some heat energy at the center point of flywheel A, causing it to slow down a bit, because some mass was added into it.

And we apply a force, but no torque, at the center of flywheel B, causing it to slow down a bit, because some kinetic mass-energy was added into it. (the flywheel starts to move linearly)

If the energies added are the same, are the new spinning rates the same?

Have we explained the mechanism behind time dilation of moving flywheels?

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What do you mean for flywheel B by "apply a force at the center", do you slow it down by friction or do you mean some kind of acceleration? – Alexander Feb 7 '12 at 10:24
I think he's saying that he is giving the flywheel a $v_cm$, thus increasing its energy. As energy $\propto$ mass, the mass has increased, and thus the moment of inertia has increased. So $\omega$ decreases. That's the OP's logic (I think) The main issue here is, how does one deal with rotational kinetic energy in relativity? I'm not too sure of this, so I'm not answering this one. – Manishearth Feb 7 '12 at 10:55
I think he's saying the force is applied along the axis of rotation, so that the wheel starts moving linearly. Then its rate of rotation appears to slow down to an observer at rest due to time dilation. But I think we need some clarification from the OP before we can really answer this one. – Nathaniel Feb 8 '12 at 9:27
Kartsa, if you're asking what I think you're asking then I think the answer is no. The amount of slow-down caused by adding heat will depend on how the extra energy ends up distributed, which depends on the flywheel's geometry and its size in particular, because the further the extra mass is from the centre, the more it will affect the moment of inertia. The slow-down due to time-dilation, on the other hand, will be the same for all flywheels, so the two numbers won't in general be the same. If you can clarify your question I'll post this as an answer. – Nathaniel Feb 8 '12 at 9:31
Clarified. But you are mistaken: Heating an object that is made of one material kind of makes the object denser, same way as adding kinetic energy. – kartsa Feb 9 '12 at 11:14