Dynamics homework, force rotating two masses on a plane

Question

I was working on the problem you can see here: http://d2vlcm61l7u1fs.cloudfront.net/media%2Fb33%2Fb33abad2-d802-4136-8987-7efd4032f03d%2FphpjIKozF.png

I've solved part a no problem. And, I have the solution, but I have no idea how it is derived from the equations I think I should be using.

I know $H_G=\Sigma\overline\rho_i \times m_i\dot{\overline r_i}$, and that $m_i = 2m$, $\overline\rho_i = L/2$, $\dot{\overline r_i}=\overline v_i = \dot\theta$. This would result in $2m(L/2)\dot\theta$, but the answer somehow has $(L/2)$ squared: $2m(L/2)^2\dot\theta$.

What am I missing that eliminates the extra multiple of $L/2$? (I know I need this answer to find $\dot H_G$.) I guess I'm using the wrong equation derivation somehow for $H_G$?

Answer 1

You need to transform the single force whose line of action does not go through the centre of mass into a force whose line of action goes though the centre of mass and a couple as shown here.

The couple will produce the angular acceleration of the system, $\tau = I_{\rm c}\ddot \theta$.