homework and exercises - Equation of motion for a falling rod (with one end touching a frictionless surface)

I have a quick question about the equation of motion for a falling rod (with one end touching a frictionless surface). The end touching the surface is not fixed. I am given the moment of inertia about the center of mass. Only gravity is acting on the rod. The rod begins at an angle $\theta$ above the ground.

I know that the rod will rotate about the center of mass, and the point touching the frictionless surface will slide along the surface, but I am having trouble calculating the torque. For reference, the left end of the rod is touching the surface, and the right end is in air. I have calculated the torque, $\tau$, from the right moment arm as $\tau=\frac{mg\cos{\theta}}{4I_G}$ because the right half of the rod contains half of the mass and half of the length. I don't really understand how to calculate the total net torque, though.

Any hints would be appreciated.