A uniform disc of radius r is suspended through a small hole drilled in it. Determine the minimum possible time period of the disc for small oscillations, along with the distance of the hole from the centre required to achieve this minimum time period.

Step-by-step solution

The time period of a physical pendulum is given by T = 2 I mgd . For a uniform disc of radius r and mass m , the moment of inertia about its center of mass is I_ cm = 1 2 mr^2 . By the parallel axis theorem, the moment of inertia about a pivot at a distance d from the center is I = I_ cm + md^2 = 1 2 mr^2 + md^2 . Substituting I into the time period formula, we get T = 2 1 2 mr^2 + md^2 mgd = 2 r^2 2d + d g . To find the minimum time period, the expression ( r^2 2d + d ) must be minimized. Differentiating with resp