Question

A sphere of radius $4\text{cm}$ is removed from a sphere of radius $8\text{cm}$ as shown in figure. If $\text{O}$ is the $COM$ of the complete sphere, the position of $COM$ of the remaining sphere will shift towards left from $\text{O}$ by

• A. $-4/7\text{cm}$
• B. $4/7\text{cm}$
• C. $2/7\text{cm}$
• D. $-2/7\text{cm}$

Answer 1

The correct option is B $4/7\text{cm}$

Let point $\text{O}$ be the origin. Then, the COM of complete sphere will be at $(0,0)$.
Let mass of complete sphere be $M$, then mass of removed portion is
$M^{\prime }=\frac{M}{\frac{4\pi }{3}\times 8^3}\times \frac{4\pi }{3}\times 4^3=\frac{M}{8}$

Let us consider removed sphere as filled with negative mass, then mass of sphere will be $-M/8$ and its coordinates of $COM$ are $(4,0)$.

Then, $x$-coordinate of $COM$ of remaining part is
$X_{COM}^{\prime }=\frac{M_1{X}_1+M_2{X}_2}{M_1+M_2}=\frac{M\times 0-M/8\times 4}{M-M/8}$
$=\frac{-M/2}{7M/8}=\frac{-4}{7}\text{cm}$

Therefore, $COM$ shifts towards left by distance $\frac{4}{7}\text{cm}$