Help with physics please?

Question

I have no idea what to do here, any explanation would be great.

In the figure the large mass disk has mass 2 kg radius 0.2m and initial angular velocity 50 rad/sec and the small disk has mass 4kg radius 0.1m and initial angular velocity 200 rad/sec (about 1900 rpm). A) find the common final angular velocity after the disks are pushed into contact. B) Is kinetic energy conserved during this process.

Answer 1

Use conservation of angular momentum. Angular momentum is I*w, where I = moment of inertia around the center of rotation, w = angular velocity. The sum of the two momentums prior to contact will equal the angular momentum of the combination

I1*w1 + I2*w2 = (I1 + I2)*w3

The angular velocity result is then

w3 = (I1*w1 + I2*w2)/(I1 + I2)

You do not have to actually compute the moments of inertia; since the shapes are the same, the moment will be directly proportional to R^2, so the equation simplifies to

w3 = (R1^2*w1 + R2^2*w2)/(R1^2 + R2^2)

Since you know the angular velocities involved, you can compare the kinetic energy before and after; rotational kinetic energy is .5*I*w^2; so compare

.5*I1*w1^2 + .5*I2*w2^2

with

,5*(I1 + I2)*w3^2.

I think you will find that KE is not conserved.

Answer 2

Ask youself what will the final state look like? Qualitatively, the two discs' edges will be touching (If I understand your description). Assuming no slipping, that means the point of contact on each disc is moving at the same speed. Since the radii are different their anguar velocity will be different. But the circumfrence of the smaller disc 2πR is also the "distance" a point on the circumfrence of the larger disc travels in the same time as it takes the small to complete one revolution. This means their speeds are correlated (but not the same). We also know that energy and angular momentum are conserved so if we know what the initial values are, we know what the final values are (the same). So all we have to do is solve for the speed (of one or the other) given that the total angular momentum is the same and is shared by both. In other words calculate the total initial angular momentum then use it to solve for the final speeds of both. (figure how the two speeds correlate with the total angular momentum). g'luck