A uniform rod of length L is used as a physical pendulum. Its pivot can be chosen at any point on the rod at distance a from the centre. For small oscillations, at what value of a is the time period minimum?
Options

Question

A uniform rod of length L is used as a physical pendulum. Its pivot can be chosen at any point on the rod at distance a from the centre. For small oscillations, at what value of a is the time period minimum?

Options

G1 Physics · Accepted Answer

For a physical pendulum, T = 2π√[I/(Mga)]. For a uniform rod pivoted at distance a from its centre, I = M(L²/12+a²). Thus the equivalent simple-pendulum length is lₑ = I/(Ma) = a+L²/(12a). The period is minimum when this expression is minimum. Differentiate with respect to a: d/da [a+L²/(12a)] = 1−L²/(12a²). Set it equal to zero: 1 = L²/(12a²). Therefore, a² = L²/12, so a = L/√12. Correct option: B.