# Conservation of Energy vs Conservation of Momentum in Rotational Dynamics

It is clear to me why angular momentum is always conserved, and how in some cases energy is not necessarily conserved within the system (in those cases where bodies deform, or friction is involved).

However, I am unable to see why energy is not conserved in the following situation:

You have a spinning disk with some angular velocity, and then you place a new on top of that disk, and then both bodies began rotating with some new angular velocity.

There is no deformation of bodies like in an inelastic collision, and there is no friction involved.

So why doesn't conservation of energy hold in this system?

• Would you be able to see why energy isn't conserved if you put a initially stationary block on a large moving surface and they both move together at a new velocity? – Brian Moths Aug 12 '15 at 0:51
• There is no friction involved -- Of course there is. If no friction was involved the two objects could not come to some common angular velocity. – David Hammen Aug 12 '15 at 0:52
• @DavidHammen Ahh... there was what I was missing. Post that as an answer and I'll accept it. – 1110101001 Aug 12 '15 at 1:39

Energy is conserved, but if you ignore some kinds of energy then it will look like it isn't conserved.

You can imagine a really big disk with some radial pointing two by fours attached at the one o'clock, two o'clock etcetera positions then attach springs to each two by four with the spring pointing in the clockwise/counter-clockwise directions. Add a nice level surface to the end of the spring. Now you can make each surface very smooth and we can have no friction yet motion can be transferred by the springs.

Now if you have two disks like that and put the parts with the two by fours facing each other and rotate them so they are half an hour out of phase then the two disks don't touch (for now).

You can quickly torque one of them and it will deform. It has to deform because it has to get a bit bigger radially until the radially stretching gives each portion/of the disk enough radial acceleration from the pressure imbalance from the parts radially farther out and the parts radially farther in give it enough radial acceleration to go in a circle at a steady speed.

So we now have one disk with a fixed angular speed. And another disk with no angular speed. Eventually when the moving disk has rotated about 1/24 of a revolution (a bit less, how much less depending on the thickness of those two by fours and the size of those springs) then the two disks will start to make contact.

As they make contact the springs will compress and the moving disk will slow down a bit and the stationary one will start to rotate. Eventually the springs will be compressed as much as they are going to be when the two disks are moving at the same speed.

At this point you could have some anchors that are just the right size to go through some holes in the two by four to keep the springs from expanding once they got to that magic compression distance. Or you can let the springs expand. If you let them expand the formerly stationary disk will eventually have all the rotation and continue that way for about 1/12 of a revolution then the process repeats.

Now if you want friction just imagine more two by fours with springs and each one thinner and smaller so at first one has all the kinetic energy then they share it and the springs have some energy and then the other one has the energy but the energy passes back and forth so quickly that it just looks like the whole thing moves at that average speed.

The friction is like that, different parts move a bit so the real motion of real parts only has the right average speed.

But whether you constrained those springs when the disks moved at equal speed or let the energy slosh back and forth energy is conserved. In a realistic disk the energy gets distributed as heat.

And a real surface has uneven parts that do press and stick and such similar to the two by fours in this example.

• Could you please add a sketch for the 2 by 4 with springs setup? I don't quite visualize it. – 1110101001 Aug 12 '15 at 7:31

As one of the comments mentions, it is simpler to consider a linear case. Dropping a body of mass $m$ on one moving with mass $M$ and velocity $v$ is essentially considered the instantaneous transformation $M \to M + m$. Momentum must be conserved in the collision, but the mass of the system effectively increases, producing a smaller kinetic energy:

$$KE_\text{new} = \frac{p^2}{2(M+m)} = \frac{M}{M+m}KE_\text{old}$$

The momentum simply cannot increase because the force is directed towards the vertical, being absorbed entirely by the ground (assuming complete rigidity).

• Assuming complete rigidity is frame dependent. But you can have a simple head on collision in space. You could even have a head on collision of two springs. – Timaeus Aug 12 '15 at 1:49