# Conservation of momentum, spinning disk

1. Mar 9, 2010

### bookman

Assumptions: 2D problem only, no gravity, no friction, no external torque.

Flat circular disk spinning at a constant angular velocity w.

Disk is initially at (0,0), but is NOT restrained in translation. Disk has some finite mass.

A small ball (finite mass) is tethered to a string which is fixed to the center of the disk. The ball sits in a radial groove at some intermediate radius on the disk. The groove extends straight to the outer edge. The disk and ball system is initially balanced so that the disk spins about its physical center. The disk spins at a constant angular velocity (w). There is tension in the string. For some non-external reason, the string breaks, and the ball is “free”, but restrained tangentially by the groove.

So, the action of the ball (in my mind):
Since it’s restrained by the groove to circular motion, there will be a centripetal (radial) acceleration. The acceleration will be r*w^2, where r is increasing over time. Knowing the initial radial velocity (0) and initial radial position, I can calculate the outward motion of the ball. Once the ball leaves the groove at the outer perimeter, there will no longer be an acceleration component, but there will be both radial and tangential velocity components to the ball motion.

But, what happens to the unrestrained spinning disk? When the string broke, and the ball began moving radially outward, was momentum being conserved with the disk moving in the opposite direction? Was there a linear component added to the disk’s rotational motion? (Since the centripetal acceleration is directed towards the disk center). Are the disk and ball considered a “system” in which linear and angular momentum are conserved while the ball is in the groove? Did the disk move the instant the string broke? Does the fact that the mass center is slightly changing have an appreciable effect on the ball or disk? I'm trying to think through these items and am only confusing myself...