Question

In Figure a spherical part of radius $\frac{R}{2}$ is removed from a bigger solid sphere of radius $R$ . Assuming uniform mass distribution, a shift in the centre of mass will be:

• A. $\frac{-R}{6}$
• B. $-\frac{R}{14}$
• C. $\frac{R}{9}$
• D. $\frac{R}{3}$

Answer 1

The correct option is A $\frac{-R}{6}$

Let the mass of bigger sphere be $M$ with radius $R$. Then the mass of smaller sphere with radius $\left(\frac{R}{2}\right)$ is $\frac{M}{4}$

Centre of the big sphere which is $C.O.M$ be the origin $O.C.O.M$ of small sphere is at distance $\frac{R}{2}$

cut out $\to C.M$ of the remaining portion shift to $P$. mass of remaining portion $=\frac{3m}{4}$ from conservation of $C.O.M$

$C.M$ of big sphere $+C.M$ of small $\frac{3M}{4}\times (-OP)=M\times OO+\frac{M}{4}\times \frac{R}{2}$

$\Rightarrow OP=-\frac{R}{6}$