Help: Hamster wheel rotational dynamics

[Ingenue1]

Homework Statement

A large hollow hamster ball is rolling along the ground at 5 miles per hour. A hamster is running inside, matching his pace at the bottom of the ball to stay by the ground. The hamster weighs .3 kg, and the ball has a mass of 1.5 kg with a radius of 0.4m.

Homework Equations

What is the angular velocity of the ball?
The hamster decides to do a loop the loop and grabs the inside of the ball with its claws. What is the new angular velocity of the ball?

What is the new velocity along the ground in miles per hour?

The Attempt at a Solution

 

[denverdoc]

Need at least some attempt at an answer. What relation is there between linear velocity and angular velocity?

How do we compute moment of inertia for a sphere?

What effect does the loop de loop have on the moment of inertia--it is now an additional point mass at radius, 0.4m.

For angular momentum to be conserved, what eqn can we use?

 

[kerimek]

To determine the angular velocity of the ball, we can use the equation ω = v/r, where ω is the angular velocity, v is the linear velocity, and r is the radius of the ball. Plugging in the given values, we get ω = (5 mph) / (0.4m) = 12.5 rad/s.

When the hamster grabs the inside of the ball and starts running in a circular motion, the angular velocity of the ball will change. This can be calculated using the conservation of angular momentum, where the initial angular momentum of the system (hamster + ball) is equal to the final angular momentum after the hamster starts running in a circular motion. We can express this as Iω = (I + mR^2) ω', where I is the moment of inertia of the ball, m is the mass of the hamster, R is the radius of the ball, and ω' is the new angular velocity of the ball. Solving for ω', we get ω' = Iω / (I + mR^2). Plugging in the given values, we get ω' = (0.5 kg m^2)(12.5 rad/s) / (0.5 kg m^2 + 0.3 kg)(0.4m)^2) = 8.33 rad/s.

To determine the new velocity of the ball along the ground, we can use the equation v = ωr, where v is the linear velocity, ω is the angular velocity, and r is the radius of the ball. Plugging in the new angular velocity of the ball, we get v = (8.33 rad/s) (0.4m) = 3.33 m/s. Converting this to miles per hour, we get v = (3.33 m/s)(2.237) = 7.46 mph.

Therefore, the new angular velocity of the ball is 8.33 rad/s, and the new velocity along the ground is 7.46 mph.