What determines the speed of two balls on a rotating disk?

[november1992]

Homework Statement

Homework Equations

I=m$r^{2}$
L=ωI
ω=$\frac{L}{I}$

The Attempt at a Solution

I thought that since the moment of inertia was larger for the ball on the outside its angular speed would be slower. So then it would take longer to hit the wall.

 

[Alucinor]

You're missing something important in this question which is the difference between angular velocity (rad/s) and linear velocity (m/s). You don't need to be using angular momentum/moment of inertia.

Using ω is right while the girls are holding the ball, but when it is let go when it hits the wall depends on v, which is related to ω by some geometric considerations. If you notice, the two girls are sitting in a straight line with each other, one further out than the other, and they are always in that straight line, meaning that the girl further out has to travel further than the girl on the inside in the same amount of time. What can you figure from that?

 

[november1992]

I was thinking about that before, but I thought it would be strange if the ball on the outside traveled faster than the ball on the inside. So, I assumed they were traveling at the same speed.

 

[Alucinor]

Their ω is no doubt equal, they're both going through the same amount of radians as the other in the same time.

The problem is with arc lengths though--which is equal to θr (make sure θ is in radians). The girl on the outside moves further in the same amount of time, so the ball is actually moving faster.

Physics gets weird when things start rotating.

 

[asteriatic]

Your reasoning is correct. The moment of inertia is a measure of an object's resistance to rotational motion. The farther away an object is from the axis of rotation, the greater its moment of inertia. Therefore, the ball on the outside of the rotating disk will have a larger moment of inertia and will rotate at a slower angular speed compared to the ball on the inside. This means that it will take longer for the ball on the outside to make a full rotation and hit the wall. This can be explained by the equation ω=\frac{L}{I}, where ω is the angular speed, L is the angular momentum, and I is the moment of inertia. Since the ball on the outside has a larger moment of inertia, its angular speed will be smaller for a given angular momentum.