Question

$I_A,I_B$ and $I_C$ are the moments of inertia of a ring, semicircular ring and an arc of the same mass about the axes as shown below. Which of the following options is correct?

• A. $I_A>I_B>I_C$
• B. $I_A<I_B<I_C$
• C. $I_A=I_B=I_C$
• D. $I_A>I_B=I_C$

Answer 1

The correct option is C $I_A=I_B=I_C$

Given that $m_A=m_B=m_C=m\text{kg}$
For any arc of radius $r$ and angle $0<\theta ,2\pi$,
$I=\int dm{r}^2=m{r}^2$
Circular and semicircular ring are just special cases when $\theta =2\pi$ or $\pi$
Therefore, about the given axes,
Moment of inertia of a ring $\left(I_A\right)=m{r}^2$
Moment of inertia of a semicircular ring $\left(I_B\right)=m{r}^2$
Moment of inertia of an arc $\left(I_C\right)=m{r}^2$
$I_A=I_B=I_C$