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We have a balancer initially at the end of a plank that spins around an axis at its center. The initial rotation period is T[itex]_{0}[/itex]. The balancer starts to walk with constant speed along the plank and we are asked to answer some questions about it.

a) What is the balancer's torque?

Since the force over the balancer is central it is consequently parallel to the radius and the torque is, therefore, zero. Right?

b) What are the angular momentum of the system (the balancer is to be trated as a point particle and the plank has momentum of inertia I=[itex]\frac{1}{12}[/itex]MD[itex]^{2}[/itex]) when the balancer reaches the center of the plank?

The angular momentum is given by [itex]\stackrel{\rightarrow}{L}[/itex]=[itex]\stackrel{\rightarrow}{r}[/itex]×[itex]\stackrel{\rightarrow}{p}[/itex]

where [itex]\stackrel{\rightarrow}{r}[/itex] is the vector connecting the position of the balancer to the axis of rotation. At the center of the plank r=0 so the total angular momentum is due exclusively to the plank. It is

L=I[itex]\omega[/itex]=[itex]\frac{1}{12}[/itex]MD[itex]^{2}[/itex][itex]\omega[/itex]

where [itex]\omega[/itex] is the angular velocity.

Right?

c) What is the rotation period in the iten b) situation?

We have that [itex]\omega[/itex]=2[itex]\pi[/itex]/T so

T=[itex]\frac{2\pi{I}}{L}[/itex]

I would like to express this independently of L but L is a function of T which cancels the other T. How can I determine it?

Thank you.