Angular Momentum: what is conserved when bodies interact?

While we recognize Newton's second law: $LaTeX: \sum\vec{F}=m\vec{a},$ we also introduced force as a single interaction between two bodies whereby momentum is exchanged: $LaTeX: \sum\vec{F}=\frac{d\vec{p}}{dt},$ or a force causes the change in momentum of each object in opposite ways. Thus if $LaTeX: \sum\vec{F}_{system} = 0,$ then $LaTeX: \frac{d\vec{p}}{dt}=0,$ and we learned that momentum of a system is constant if the sum of external forces is zero. Now we take the same steps with rotational dynamics. We know that

$LaTeX: \sum\vec{\tau}=I\vec{\alpha},$ but also

$LaTeX: \sum\vec{\tau}=\frac{d\vec{L}}{dt},$ torque changes the angular momentum; equals angular momentum's rate of change.

If $LaTeX: \sum\vec{\tau}=0,$ then $LaTeX: \frac{d\vec{L}}{dt}=0,$ angular momentum doesn’t change… it is constant when there is no external torque.

Consider a collision we are familiar with from Chapter 3.1:

An inelastic collision between two masses, before and after.

A block runs into and sticks to a stationary block 4 times its mass. The resulting body, at LaTeX: 5M_o moves off at LaTeX: v_f = v_0/5, conserving momentum. In the collision, the smaller block receives an impulse to the left, and the larger object receives the same magnitude impulse to the right. Additionally, we can see that in this process we’ve lost 80% of the kinetic energy to thermal energy. It's easiest to see if we express kinetic energy as: $LaTeX: E_K=\frac{p^2}{2m},$ because we know that momentum is the same before and after, but the moving mass increased by a factor of 5 in the collision. Let’s do this for a rotational collision.

A turntable rotates freely on a low friction bearing. It would rotate for a long time like this because there's no torque acting on it. But then, we drop four identical non-rotating disks on it.

Exercise 4.7.1

A turntable rotating about its center with four disks about to be dropped onto it, exercise 4.7.1. The base of a turntable is rotating freely at rotational velocity $LaTeX: \omega_0$ on a very good bearing when four identical disks are dropped onto it. Please find:

The final rotational velocity. Please support your answer with the correct concepts and formulas and pay particular attention to which lens you use. Develop this carefully.
Is kinetic energy conserved? If not, how do you know? If so, how much energy is converted and what is it converted into?
What is the angular impulse $LaTeX: \Delta\vec{L},$ received by each body? Please include direction.
Consider the interaction whereby the stationary disks speed up and the rotating disk slows down. Can you identify an interaction that may generate heat?

Exercise 4.7.2

Two identical bodies tied together with a string are spinning in space about the center at angular speed, $LaTeX: \omega_i,$ when a motor at the center pulls them both inward such that the final diameter of their paths is 1/3 the original diameter, or, $LaTeX: d \Rightarrow \underline{\hspace{1mm}1/3\hspace{1mm}}d_i.$ If this is in outer space, we can be sure there are no outside forces. We might consider conserving kinetic energy and/or angular momentum.

Is it possible for the angular momentum to change? If so how?
Is it possible for energy to change? if so, where did the energy go or come from.

Is it possible to conserve both angular momentum and kinetic energy? Let’s find out!
What happens to the moment of inertia with this change?
If we conserve angular momentum what should be the new angular velocity? $LaTeX: \omega\Rightarrow\underline{\hspace{8mm}}\omega_i$
Would this change the kinetic energy? If so, by what factor: $LaTeX: KE\Rightarrow\underline{\hspace{8mm}}KE_{i}$
Can we conserve angular momentum and kinetic energy? If not, which one must have changed, and where did that change come from?