How Does Changing Arm Position Affect Angular Velocity and Work Done?

[PeterPoPS]

Homework Statement

A man is standing on a rotating plate (rotates without friction). In case A he is hold 1weight in each hand (straight out) with the mass m and he is rotatin with angular velocity omega_0.
Then he moves his arms down so they are parallell to the body (case B) and increases his angular velocity to omega_1.
Calculate the new angular velocity omega_1 and the work U which the man must do to change his body configuration to case B.
You can assume that the moment of inertia on z-axis for the man is I_0 in case A and I_0 / 2 in case B.
The length of the arms of the man is l, so each weight are l units out in case A and lowered l units (and ofcourse moved l units toward the body) in case B.

the answers are given
omega_1 = 2 omega_0
and
U = 0.5*I_0*(omega_0)^2 - 2mgl


My Problem is i get how to calculate the angular velocity omega_1
But I don't understand how to even start on calculating the work done.

Best regards /Peter

Homework Equations

The Attempt at a Solution

 

[Nate810]

For omega_1 I used the formulaI = I_0 + 2ml^2so I_1 / I_0 = omega_1 / omega_0and then got that omega_1 = 2 omega_0

 

[Moetasim]

To calculate the work done, we need to use the formula W = F*d, where W is the work done, F is the force applied, and d is the distance over which the force is applied. In this case, the force is the weight of the objects in the man's hands, and the distance is the length of his arms. So for case A, we have W = 2mg*l, and for case B, we have W = 2mg*2l.

To find the difference in work done, we subtract the work done in case A from case B, giving us a final answer of U = 2mg*l. However, this only accounts for the work done by the man's arms, and not the work done by his body as a whole.

To account for the work done by his body, we need to consider the change in moment of inertia. In case A, the moment of inertia is I_0, and in case B, it is I_0/2. Therefore, the work done by his body is given by the formula U = 0.5*I_0*(omega_0)^2 - 0.5*I_0/2*(omega_1)^2.

Combining this with the work done by his arms, we get the final answer of U = 0.5*I_0*(omega_0)^2 - 2mgl. This accounts for both the change in angular velocity and the change in moment of inertia, giving us the total work done by the man to change his body configuration from case A to case B.