Period of a circular arc shaped pendulum

The period T is equal to $$2\pi/w_n$$ where the natural frequency can be found from $$w_n^2{\theta}+\ddot{\theta} = 0$$. Since $$\tau = I\alpha$$, as there is no net torque about point P because gravity acts at the center of mass and equals the normal force, $$\ddot{\theta} = 0$$ so $$w_n = 0$$. Does that mean the period is infinite?

• Think about where the centre of mass of the bent rod is. Hint: it is not at P and if the rod is not at its central position then it is not directly below P either. Jan 1 at 15:30

No. Gravity does not act at point $$P$$ because your CoM is not at point $$P$$. Your CoM is lower than point $$P$$. Therefore, gravity produces a torque.
Your normal force will not produce a torque, since it's applied at point $$P$$, so the torque about point $$P$$ due to the normal force is zero.
• If the CoM is at the same position as the point of rotation, then gravity will provide no torque. If no other forces provide any torques, then you won't have any oscillations. The object will just continue doing what it was doing before -- if it was rotating with angular speed $\omega$ then it will keep rotating with that same angular speed, if it was not rotating then it will not rotate. Jan 1 at 16:01