Question

A spherical cavity is made in a lead sphere of radius $R$ such that its surface touches the outside surface of lead sphere and passes through the centre. The shift in the centre of mass of the lead sphere as a result of this hollowing, is

• A. $\frac{\text{R}}{7}$
• B. $\frac{\text{R}}{14}$
• C. $\frac{\text{R}}{2}$
• D. R

Answer 1

Answer: (B)

Let centre of mass of lead sphere after hollowing be at point $O_2$ , where $O{O}_2=x$
Mass of spherical hollow $m=\frac{\frac{4}{3}\pi {\left(\frac{\text{R}}{2}\right)}^2\text{M}}{\left(\frac{4}{2}\pi {\left(\text{R}\right)}^3\right)}=\frac{\text{M}}{8}$ and
$\text{x}={\text{OO}}_1=\frac{\text{R}}{2}$

$\therefore \text{x}=\frac{\text{M}\times 0-\left(\frac{\text{M}}{8}\right)\times \frac{\text{R}}{2}}{\text{M}-\frac{\text{M}}{8}}=\frac{\frac{\text{MR}}{16}}{\frac{7\text{M}}{8}}=-\frac{\text{R}}{14}$
$\therefore \text{shift}=\frac{\text{R}}{14}$