Tangent force on a rotating rod

[abomination5]

Homework Statement

I've been working on this problem for several hours with no luck. Any help would be appreciated. (It's not homework)

A thin rod of mass M and length L is hinged at the bottom, and almost balanced vertically. It starts to fall. Find the tangential component of the force that the part of the rod below r exerts on the part of the rod above r.

(Theta is the angle between the rod and the vertical, r is the distance on the rod from the hinge)

Homework Equations

SOLN: F = Mg(L-r)(3r-L)/(4 L^2) * Sin(theta)

d x F = I *(angular acceleration)

The Attempt at a Solution

I've already determined the angular acceleration to be

3/2 g/L Sin(theta)

using the Lagrangian.

I've calculated the moment of inertia to be:

1/3 M (L-r)/L [(L-r)^2 + 3*r^2]

using the parallel axis theorem.

I've tried for several hours to get the final equation for the force but I haven't been able to do so yet... maybe there is a mistake in the moment of interia?

 

[Doc Al]

The Attempt at a Solution

I've already determined the angular acceleration to be

3/2 g/L Sin(theta)

Good.

using the Lagrangian.

I've calculated the moment of inertia to be:

1/3 M (L-r)/L [(L-r)^2 + 3*r^2]

Rather than go this route, focus on the center of mass of the piece in question. What is its acceleration? What must be the net force on it?

 

[abomination5]

Doc Al,

Thanks for your help. I was able to get the solution this morning. I won't spoil the rest of the details for anyone who wants to try it.