Disks A and B are mounted on shaft S S and may be connected or disconnected by c

Question

Disks A and B are mounted on shaft SS and may be connected or disconnected by clutch C. (See the figure below (Figure 1).) Disk A is made of a lighter material than disk B, so the moment of inertia of disk A about the shaft is one-third that of disk B. The moments of inertia of the shaft and clutch are negligible. With the clutch disconnected, A is brought up to an angular speed 0. The accelerating torque is then removed from A, and A is coupled to disk B by the clutch. (You can ignore bearing friction.) It is found that 2700 J of thermal energy is developed in the clutch when the connection is made.

What was the original kinetic energy of disk A?

Explanation / Answer

There are no external torques to the system, so angular momentum is conserved, so the final angular speed satisfies:

( I_A + I_B ) = I_A 0

It is given that I_A = 1/3 I_B, therefore

= 1/4 0.

Now consider the kinetic energies before and after coupling.

K_after = 1/2 (I_A + I_B) ^2 = I_A 0^2 / 8

K_before = 1/2 I_A 0^2

Therefore the loss in kinetic energy is 3/8 I_A 0^2.

This went into heat, and since we want that maximized to Qmax (2500J in this problem) we have

3/8 I_A 0^2 < Qmax

So

1/2 I_A 0^2 < 4/3 Qmax

So the maximum initial kinetic energy of A is 4*2700J/3 = 3,600J

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