Simple Harmonic Motion and angular frequency

[Abid Rizvi]

Homework Statement

A horizontal plank of mass m and length L is pivoted at one end. The plank's other end is supported by a spring of force constant k (see the figure below). The plank is displaced by a small angle θ from its horizontal equilibrium position and released. Find the angular frequency with which the plank moves with simple harmonic motion. (Use any variable or symbol stated above as necessary.)

The picture is uploaded

Homework Equations

##\omega## = ##\frac{\tau}{I}## (where omega is angular frequency)
F = kx

The Attempt at a Solution

So I said the force that puts the plank back to equilibrium is kx. Using the definition of arc length, I said kx = k##\theta##L. A force of Mg is also acting on the plank, so I had the total torque = K##\theta##L-Mg*##\frac{L}{2}## I know that the moment of inertia is ##\frac{1}{3}##M##L^2##. Using the formula for ##\omega##, I had ##\sqrt{\frac{k\theta L-Mg\frac{L}{2}}{3ML}}## But this is incorrect. What am I doing wrong?

 

[mfb]

kx = kθ\thetaL

This is a force, not a torque.

The units don't match afterwards due to this mistake.