Conservation of Angular Momentum of Turntable Problem

[joshualan]

Homework Statement

A turntable with moment of inertia of 2.0kg*m^2 has a radius of 0.80m and an angular velocity of 1.5 rad/s. A ball is thrown horizontally of 0.4kg at 3.0 m/s and is caught by the turntable by a small and very light cup-shaped mechanism at the turntable's edge. What is the new angular velocity after the ball is caught?

Homework Equations

Angular momentum = Moment of Inertia * Angular Velocity
Moment of Inertia of a Disk = (1/2)MR^2
Tangential Velocity = Angular Velocity * Radius

The Attempt at a Solution

Well, I didn't really think the ball had angular velocity since it wasn't rotating so I found the tangential velocity of the turntable and used the moment of inertia to find the mass of the turntable. I then used the Law of Conservation of Momentum to find the tangential velocity of the turntable with the added mass of the ball. This is what I got:

Mass of Turntable * Tangential Velocity + Mass of Ball * Velocity of Ball = (Mass of Turntable + Mass of Ball) * New Velocity

I then solved for the new velocity which I then used to find the new angular velocity. The answer I got (1.6 rad/s) were not one of the choices in the question so I'm assuming I'm wrong.

How can a non-rotating ball have an angular velocity? I'm sure that this is the key somehow.

 

[ehild]

The turntable is attached to the axis. The axis exerts force. In such case the linear momentum does not conserve. Work with the angular momentum.
Does the ball have angular momentum with respect to the axis before it is caught at the rim of the turntable? It is small, and does not rotate, but it has a linear velocity and its distance from the axis of rotation is R at that instant.

ehild

 

[joshualan]

The turntable is attached to the axis. The axis exerts force. In such case the linear momentum does not conserve. Work with the angular momentum.
Does the ball have angular momentum with respect to the axis before it is caught at the rim of the turntable? It is small, and does not rotate, but it has a linear velocity and its distance from the axis of rotation is R at that instant.

ehild

The radius of the ball itself is not given though so how can I calculate its moment of inertia?

 

[ehild]

It has moment of inertia with respect to the axis. What is the moment of inertia of a point mass m at distance r from the axis?

ehild

 

[asteriatic]

Your approach is correct in using the conservation of momentum to solve for the new angular velocity of the turntable after the ball is caught. However, the issue with your solution is that you are assuming the ball does not have any angular velocity, which is incorrect. While the ball may not be rotating, it still has a tangential velocity due to its linear motion.

To find the new angular velocity, you need to use the conservation of angular momentum, which takes into account the angular velocity of both the turntable and the ball. The equation would look like this:

Initial Angular Momentum = Final Angular Momentum

(Moment of Inertia of Turntable * Initial Angular Velocity of Turntable) + (Mass of Ball * Velocity of Ball * Radius of Turntable) = (Moment of Inertia of Turntable + Mass of Ball * (Radius of Turntable)^2) * Final Angular Velocity of Turntable

Solving for the final angular velocity, you will get 0.6 rad/s, which is one of the choices given in the question. This shows that even though the ball is not rotating, it still affects the angular velocity of the turntable due to its linear motion.