Newton's Second Law of Rotation

[Momentum09]

Homework Statement

We have a rigid bar (length L) with an axis of rotation through the center of the bar. The bar is attached to a pulley with radius Rp at the axis of rotation. Two lead weights (mass M) can be screwed on the bar at equal distances R from the axis of the pulley. In each trial, the distance R is increased.

1. If we were to do an experiment with a bar that uniformly lost mass through the course of a single trial, how would the angular acceleration and the moment of inertia of the system change?
2. If were were to repeat the experiment with a bar that lost mass from the ends through the course of a single trial of the experiment, how would the angular acceleration and the moment of inertia of the system change?
3. If were were to repeat the experiment with a pulley that's radius decreased during the course of a trial, how would the angular acceleration and the moment of inertia of the system change?

Homework Equations

angular acceleration = angular momentum/moment of inertia
moment of inertia = mr^2

The Attempt at a Solution

1. The angular acceleration would increase because the moment of the inertia decreases as the mass of the bar decreases.
2. I'm not sure...
3. The angular acceleration would increase because the moment of inertia decreases since the radius of the pulley decreases.

 

[anathema]

However, the change in the moment of inertia may be less significant compared to the change in the angular acceleration.

 

[ogb p]

However, the decrease in radius would also decrease the lever arm, which could potentially offset the increase in angular acceleration. The moment of inertia would also decrease.

I would like to clarify and expand upon the above responses.

1. If the bar uniformly loses mass, the moment of inertia would decrease since it is directly proportional to the mass and the distance from the axis of rotation. This decrease in moment of inertia would result in an increase in angular acceleration, as stated in the attempt at a solution. Additionally, the angular momentum would also decrease as the mass decreases, but this would not affect the angular acceleration unless there are external forces acting on the system. Overall, the decrease in mass would result in a faster rotation of the bar.

2. If the mass is lost from the ends of the bar, the moment of inertia would decrease at a faster rate compared to the uniform loss of mass. This is because the mass is being removed from a larger distance from the axis of rotation, resulting in a larger decrease in moment of inertia. This would also result in a faster increase in angular acceleration compared to the uniform loss of mass.

3. If the pulley's radius decreases, the moment of inertia would decrease since it is directly proportional to the radius squared. This would result in an increase in angular acceleration, as stated in the attempt at a solution. However, the decrease in radius would also decrease the lever arm, making it harder for the weights to rotate the bar. This could potentially offset the increase in angular acceleration. Additionally, the decrease in radius would also decrease the distance the weights are from the axis of rotation, resulting in a decrease in moment of inertia.