When the point of contact is at an angle with the vertical, the forces on the plank are its weight W (at the centre of the plank) and an upwards force N at the point of contact. The angular equation of motion (about the point of contact) is , where I is the moment of inertia about the point of contact, and d is the horizontal distance from the centre of the plank to the point of contact.
Assuming that is small, we can use the approximate value . Also, we can assume that I is the same as the moment of inertia about the centre of the plank. The formula for the moment of inertia of a rod of mass m and length l about its midpoint is . So we take .
Then the (approximate) angular equation of motion is , or . That is an SHM equation for a motion with period .