Angular Momentum: Inelastic Collision of Disk-Ball System

[maxhersch]

Homework Statement

A uniform disk of radius R=1m rotates counterclockwise with angular velocity ω=2rads/s about a fixed perpendicular axle passing through its center. The rotational inertia of the disk relative to this axis is I=9kg⋅m². A small ball of mass m=1 is launched with speed v=4m/s in the plane of the disk and remains stuck to the disk after collision and the trajectory of the ball is d=0.5m above the horizontal axis. Neglect gravity and assume the axle is frictionless.

a) Find the angular velocity after the collision .
b) Compute the loss of kinetic energy during the collision.

Homework Equations

L = Iω = r × p = m(r × v)
K = ½Iω²
K = mv²

The Attempt at a Solution

I have done a similar problem in which a ball is dropped on a disk rotating on a vertical axis in which case the ball does not cause any net torque. In this case the problem is simply L = (I+md²)ω which you can solve for ω. However, in this problem, the ball does cause a torque I attempted to use the same equation but I know this isn't the right way to do it. I can't find an example of this kind of problem anywhere online so hopefully someone here can help me.

 

[Bystander]

The axle is horizontal? The ball sticks to the disc where? Edge? 0.5 meter from axle? Ball is launched with, or counter to the direction of disc's rotation?

 

[Narwhalest]

So... as hella confusing as that question is worded, I'm going to assume a thing or two...

Since the ball is in the plane of the disk, I'm going to assume that it is shot in the direction (or counter to it -) of the spin. My response to this is to calculate the angular momentum of the ball at R = 0.5m away from center. Just imagine that both the ball and the tip of the disk travel in a circular fashion, just that before the ball sticks, it begins to rotate separate of the disk. And then they combine. After that, since you'll now have the Angular Momentum of Both, you can apply the conservation of angular momentum (and parallel axis theorem).

As to the Kinetic Energy, calculate both the Rotational Kinetic Energy of the disk and also the Linear Kinetic Energy of the ball (before colliding). Then compare to the Energy calculated afterwards.

(Also, your equation for the Linear Kinetic Energy is incorrect, try adding a 1/2 in front -like the rotational one)