Question

Consider a semi-circular disc of mass $M$ and radius $R$ . 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,

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

Answer 1

Answer: (B)

$I_{0}=\left(\frac{\left(mR\right)^{2}}{2}\right)$
$I_{0}=I_{c m}+md^{2}$
$\frac{m R^{2}}{2}=I_{c m}+m\left(\frac{4 R}{3 \pi }\right)^{2}$
$\Rightarrow I_{c m}=\frac{m R^{2}}{2}-m\left(\frac{4 R}{3 \pi }\right)^{2}$
$I_{c n}=\left[\frac{M R^{2}}{2} - M \left(\frac{4 R}{3 \pi }\right)^{2}\right]$