A uniform rod of length l is suspended from one end and undergoes small oscillations. Determine the length of a simple pendulum that possesses a time period identical to that of the rod.

Step-by-step solution

The time period of a physical pendulum is given by T = 2 I mgd , where I is the moment of inertia about the point of suspension and d is the distance of the center of mass from the point of suspension. For a uniform rod of mass m and length l suspended from one end, the moment of inertia is I = ml^2 3 and the distance to the center of mass is d = l 2 . Substituting these values into the time period formula: T = 2 ml^2 3 mg ( l 2 ) = 2 2l 3g The time period of a simple pendulum of length L is given by T = 2 L g . Eq