Question

Assuming the expression for the moment of inertia of a thin uniform disc about a transverse axis through its centre, obtain an expression for its moment of inertia about any diameter. Hence, write the expression for the corresponding radius of gyration.

Answer 1

Consider a thin uniform disc of mass M and radius R in the xy plane, as shown in below figure. Let I_x , I_y and I be the moments of inertia of the disc about the x, y and z axes respectively. But, I_x = I_y , since each

represents the moment of inertia (MI) of the disc about its diameter and, by symmetry, the MI of the disc about any diameter is the same. As I_z is the MI of the disc about the z-axis through its centre and perpendicular to its plane,

I_Z = \(\frac12\) MR² ....(1)

According to the theorem of perpendicular axes,

Radius of gyration : The radius of gyration of the disc for rotation about its diameter is