Conservation of Angular Momentum Question

[timtng]

An ice skater doing a toe spin with outstretched arms has an angular velocity of 4 rad/s. She then tucks in her arms, decreasing her moment of inertia by 7.5%

a. What is the resulting angular velocity?
b. By what factor does the skater's kinetic energy change?

For a, I use IW = I'W' >> 1(4rad/s) = (1-.075)W', then solve for W'
I got 4.32 rad/s for W'. I don't know if I'm doing it correctly. Also, I need help on part b.

Thx

 

[NateTG]

Part a looks ok.

IIRC the formula for rotational energy is
1/2Iω²

The ratio is the old kinetic energy divided by the new kinetic energy.

 

[M98Ranger]

.

a. The resulting angular velocity can be calculated using the conservation of angular momentum equation, which states that the initial angular momentum is equal to the final angular momentum. In this case, the initial angular momentum is given by I1W1, where I1 is the initial moment of inertia and W1 is the initial angular velocity. The final angular momentum is given by I2W2, where I2 is the final moment of inertia and W2 is the final angular velocity. Since the angular momentum is conserved, we can set these two equations equal to each other:

I1W1 = I2W2

We are given that the initial angular velocity is 4 rad/s and the moment of inertia decreases by 7.5% when the skater tucks in her arms. This means that the final moment of inertia is 0.925 times the initial moment of inertia. Therefore, we can rewrite the conservation of angular momentum equation as:

I1(4 rad/s) = (0.925I1)W2

Solving for W2, we get:

W2 = (4 rad/s)(I1/0.925I1) = 4.32 rad/s

Therefore, the resulting angular velocity is 4.32 rad/s.

b. To calculate the change in kinetic energy, we can use the equation for kinetic energy in terms of angular momentum, which is given by:

K = 1/2I1W1^2

The initial kinetic energy is given by:

K1 = 1/2I1(4 rad/s)^2 = 8I1

Similarly, the final kinetic energy is given by:

K2 = 1/2(0.925I1)(4.32 rad/s)^2 = 8I1

Therefore, the change in kinetic energy is:

ΔK = K2 - K1 = 8I1 - 8I1 = 0

This means that the skater's kinetic energy does not change when she tucks in her arms. This is due to the conservation of angular momentum, which states that the total angular momentum of a system remains constant unless an external torque is applied. In this case, since no external torque is applied, the skater's angular momentum remains constant and therefore, her kinetic energy also remains constant.