Question

The moment of inertia of the uniform semicircular disc of mass $M$ and radius $R$ about a line perpendicular to the plane of the disc through the center is

• A. $M{R}^2$
• B. $\frac{1}{2}M{R}^2$
• C. $\frac{1}{4}M{R}^2$
• D. $\frac{2}{5}M{R}^2$

Answer 1

The correct option is B

$\frac{1}{2}M{R}^2$

Step 1: Given data

Uniform semicircular disc with mass $M$ and radius $R$.

We have to find the moment of inertia about a line perpendicular to the plane of the disc through the center.

Step 2: Calculation

Since the mass of a semicircular disc is $M$, we can consider its moment of inertia of it as half of the moment of inertia of a circular disc of mass $2M$

We know, a moment of inertia of a disc of mass $2M$ is

$I_{circulardisc}=\frac{1}{2}\left(2M\right)R^2=M{R}^2$

Therefore, moment of inertia of semicircular disc with mass $M$,

$$\begin{gathered} I_{semicirculardisc}=\frac{1}{2}{I}_{circulardisc} \\ \Rightarrow I_{semicirculardisc}=\frac{1}{2}M{R}^2 \end{gathered}$$

The moment of inertia of the semicircular disc with mass $M$ and radius $R$ about a line perpendicular to the plane of the disc through the center is $\frac{1}{2}M{R}^2$

Hence, option B is correct