Circular Platform?

Question

A large horizontal circular platform (M=77.1 kg, r=3.27 m) rotates about a frictionless vertical axle. A student (m=60.3 kg) walks slowly from the rim of the platform toward the center. The angular velocity ω of the system is 3.90 rad/s when the student is at the rim. Find ω (in rad/s) when the student is 1.63 m from the center.

I have no idea where to start on this problem, so any help is appreciated. Thanks.

Answer 1

There are two approaches: conservation of angular momentum or the conservation of (linear) momentum. They both deal with the conservation of momentum. I like the linear momentum approach because I keep forgetting the angular momentum equations for the various shapes of rotating bodies.

So, the linear momentum of your body on the rim of the wheel is P = mV; where m is the body's mass and V is the tangential velocity of the wheel at the rim, R radius = 3.27 m. V = w R; where w = the angular velocity while on the rim.

Similarly P = mv; where v is the tangential velocity at radius r when the student walks toward the center. v = W r; where W is the angular velocity at r = 1.63 m.

Since linear momentum is conserved P = mV = mwR = mWr = mv = P Thus, wR = Wr so that W = w (R/r). Roughly, given the radii of the problem, the angular velocity at r will be twice what it was when the student was at R, the rim.

Lesson learned: The mass falls out; so the angular velocities would have doubled close in at r no matter what the mass was. The angular momentum of the wheel is a constant because its mass distribution and shape were not altered. These are the factors that influence angular momentum and velocity. So it has no influence on the change in angular velocity.