A uniform board of length and weight is balanced on a fixed semicircular cylinder of radius as shown in the figure. If the plank is tilted slightly from its equilibrium position, then show that time period of oscillations is given by .

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- May 29th 2009, 09:39 AMfardeen_gen[SOLVED] Time period of oscillations?
A uniform board of length and weight is balanced on a fixed semicircular cylinder of radius as shown in the figure. If the plank is tilted slightly from its equilibrium position, then show that time period of oscillations is given by .

- May 30th 2009, 01:48 AMOpalg
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 .