Question

A uniform solid disk has a radius of R = 0.45 m and mass of M = 4.35 kg and is free to rotate about a fixed axis through its center of mass as shown below. It is initially at rest, and at t = 0 a constant torque is applied such that the disk starts to turn with angular acceleration a = 8*pi rad/s2.


How many revolutions has the disk made at t = 10 seconds?

What is the kinetic energy in the disk at t = 10 seconds

Answer 1

A = 8pi
integrate acceleration to get angular velocity:
V = 8pi*t +C
This C is a constant of integration, and to find it, we need to know the velocity at some time. Luckily they tell us, "The disk is initially at rest." This means that
V(0) = 8pi*0 + C = 0
so C = 0 and V = 8pi*t
Integrate Velocity to get position, or in this case, number of revolutions
P = (8pi/2)t^2 + C
again, this constant comes up, but since we know that it has not made any revolutions at time t=0, C = 0 again. So we get
P = (8pi/2)t^2
Evaluate at time t=10 to get angular displacement. Divide by 2pi to find out how many revolutions that is.

Answer 2

Now to find the kinetic energy at time t, you can use the formula:
(1/2)I*w^2
where I is the moment of inertia and w is the angular velocity.

Answer 3

Thank You!!

Answer 4

My pleasure =D