Question

The moment of inertia of a uniform semi-circular disc about an axis passing through its centre of mass and perpendicular to its plane is (Mass of this disc is $M$ and radius is $R$ ).

• A. $\frac{\left(\text{MR}\right)^{2}}{2} - \text{M} \left(\frac{2 \text{R}}{\pi }\right)^{2}$
• B. $\frac{\left(\text{MR}\right)^{2}}{2} - \text{M} \left(\frac{4 \text{R}}{3 \pi }\right)^{2}$
• C. $\frac{\left(\text{MR}\right)^{2}}{2} + \text{M} \left(\frac{4 \text{R}}{3 \pi }\right)^{2}$
• D. $\frac{\left(\text{MR}\right)^{2}}{2} + \text{M} \left(\frac{2 \text{R}}{\pi }\right)^{2}$

Answer 1

Answer: (B)

$\left(\text{I}\right)_{0} = \left(\frac{\left(\text{MR}\right)^{2}}{2}\right)$
$\text{I}_{0} = \text{I}_{\text{cm}} + \text{Md}^{2}$
$\frac{\left(\text{MR}\right)^{2}}{2} = \left(\text{I}\right)_{\text{cm}} + \text{M} \left(\frac{4 \text{R}}{3 \pi }\right)^{2}$
$⇒ \, \, \left(\text{I}\right)_{\text{cm}} = \frac{\left(\text{MR}\right)^{2}}{2} - \text{M} \left(\frac{4 \text{R}}{3 \pi }\right)^{2}$
$\left(\text{I}\right)_{\text{cm}} = \left[\frac{\left(\text{MR}\right)^{2}}{2} - \text{M} \left(\frac{4 \text{R}}{3 \pi }\right)^{2}\right]$