Question

Find the moment about the diameter of a disc when the moment of inertia of the disc about its geometrical axis is $I$.
(A) $I$
(B) $2I$
(C) $\dfrac{I}{2}$
(D) $\dfrac{I}{4}$

Answer 1

Hint
The geometrical axis of the disc passes through the centre perpendicularly to the diameter. Thus, the moment of inertia of the disc along the diameter will be calculated by using the perpendicular axis theorem on the moment of inertia.

Complete step by step answer
Given, the moment of inertia of the disc about its geometrical axis i.e. ${{\rm{I}}_{\rm{z}}}$ is ${\rm{I}}$.
According to the perpendicular axis theorem, the moment of inertia of a planar body about an axis must be equal to the sum of the moment of inertia of any two mutually perpendicular axes.
i.e. ${{\rm{I}}_{\rm{z}}} = {{\rm{I}}_{\rm{x}}} + {{\rm{I}}_{\rm{y}}}$
Now, on considering the symmetry of the disc, we have-
${{\rm{I}}_{\rm{x}}} = {{\rm{I}}_{\rm{y}}}$
Because the diameter of the disc will be uniform in all directions. Also, ${{\rm{I}}_{\rm{x}}}$ and ${{\rm{I}}_{\rm{y}}}$ will be the moment of inertia about the diameter of the disc.
Since ${{\rm{I}}_{\rm{z}}} = {\rm{I}}$
Thus, ${\rm{I}} = {{\rm{I}}_{\rm{x}}} + {{\rm{I}}_{\rm{x}}} = 2{{\rm{I}}_{\rm{x}}}$
${{\rm{I}}_{\rm{x}}} = \dfrac{{\rm{I}}}{2}$
i.e. the moment of inertia about the diameter of the disc will be $\dfrac{{\rm{I}}}{2}$ where ‘$I$’ is the moment of inertia along the geometrical axis of the disc.
Therefore, (C) $\dfrac{{\rm{I}}}{2}$ is the required solution.

Note
We consider the symmetry of the disc because it is uniform in the planar of the body. In the given question, the geometrical axis is considered on the z-axis; therefore, x and y axes will lie on the plane of the disc.