Question

A spherical cavity of radius r is curved out of a uniform solid sphere of radius $R$ as shown in the figure below.

The distance of the centre of mass of the resulting body from that of the solid sphere is given by

• A. $\frac{R-r}{2}$
• B. $\frac{R+r}{2}$
• C. $D$
• D. $\frac{r^{3}}{R^{2}+Rr+r^{2}}$

Answer 1

Answer: (D)

Centre of mass of solid sphere is at origin (centre of sphere).
Now, centre of mass of sphere with cavity is calculated as follows.

$x_{CM}=\frac{m_{1}x_{1}-m_{1}x_{2}}{m_{1}-m_{2}}$
$=\frac{\left(\frac{4}{3}\pi R^{3}.\rho\right)\left(o\right)-\left(\frac{4}{3}\pi r^{3}\right).\rho\left(R-r\right)}{\frac{4}{3}\pi R^{3}\rho-\frac{4}{3}\pi r^{3}\rho}$
where, $p =$ density of material of sphere.
$\Rightarrow x_{CM}=\frac{-r^{3}\left(R-r\right)}{\left(R^{3}-r^{3}\right)}$
$=\frac{-r^{3} \left(R-r\right)}{\left(R-r\right)\left(R^{2}+Rr+r^{2}\right)} $
$d =\frac{r^{3}}{R^{2}+Rr+r^{2}}$

So, distance between two mass centres is
$d=\frac{r^{3}}{R^{2}+Rr+r^{2}}$