Question

The moment of inertia of an annular disc of mass $M$, outer and inner radii $R$ and $r$ about its diameter is:
$\begin{aligned} & \text{A}\text{. }M\dfrac{\left( {{R}^{2}}\text{+}{{\text{r}}^{2}} \right)}{2}\text{ } \\\ & \text{B}\text{. }M{{R}^{2}}\text{ } \\\ & \text{C}\text{. }M{{r}^{2}} \\\ & \text{D}\text{. }M\dfrac{\left( {{R}^{2}}\text{+}{{\text{r}}^{2}} \right)}{4} \\\ \end{aligned}$

Answer 1

For finding the expression for the moment of inertia of an annular disc about its diameter, we need to know the expression for the moment of inertia of an annular disc about its central axis. Then we will apply a perpendicular axis theorem to determine the desired expression.

Formula used:
MOI of annular disc about its central axis, ${{I}_{C}}=M\dfrac{\left( {{R}^{2}}+{{r}^{2}} \right)}{2}$
Perpendicular axis theorem, ${{I}_{Z}}={{I}_{X}}+{{I}_{Y}}$

Complete step by step answer:
Moment of inertia, also known as mass moment of inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, similarly mass determines the force needed for a desired linear acceleration. Moment of inertia is the analogue of mass in rotational dynamics. Moment of inertia is also known as angular mass or rotational mass.
We are given an annular disc of mass $M$, with outer and inner radii $R$ and $r$ respectively.

The moment of inertia of an annular disc of mass $M$, outer and inner radii $R$ and $r$ about its central axis is given as,
${{I}_{C}}=M\dfrac{\left( {{R}^{2}}+{{r}^{2}} \right)}{2}$
Perpendicular axis theorem states that the moment of inertia of a planar lamina about an axis perpendicular to the plane of lamina is equal to the sum of the moments of inertia of the lamina about the two axes at right angles to each other at the point where the perpendicular axis passes through it.
${{I}_{Z}}={{I}_{X}}+{{I}_{Y}}$
Let the moment of inertia about its diameter is ${{I}_{d}}$
By perpendicular axis theorem,
$\begin{aligned} & {{I}_{d}}+{{I}_{d}}={{I}_{C}} \\\ & {{I}_{C}}=2{{I}_{d}} \\\ \end{aligned}$
${{I}_{d}}=\dfrac{{{I}_{C}}}{2}$
${{I}_{d}}=M\dfrac{\left( {{R}^{2}}+{{r}^{2}} \right)}{4}$
Moment of inertia of annular disc about its diameter is given as ${{I}_{d}}=M\dfrac{\left( {{R}^{2}}+{{r}^{2}} \right)}{4}$
Hence, the correct option is D.

Note: While calculating the moment of inertia of a body, the axis should be used accordingly to find the value. A body has different values of moment of inertia about different axes. Perpendicular axis theorem and Parallel axis theorem should be used carefully choosing the appropriate axes. Moment of inertia of an annular disc along one of its diameters is half of the moment of inertia along its central axis.