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
$d=\frac{r^3}{R^2+Rr+r^2}$