Question

A sphere of mass $M$ and radius $R_2$ has a concentric cavity of radius $R_1$ as shown in figure. The force $F$ exerted by the sphere on a particle of mass $m$ located at a distance $r$ from the centre of sphere varies as $(0\le r\le \infty )$

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Region 1 $=0<r<R_1$
In this region, because of cavity, the gravitational force will be zero.
Hence, the graph of this section should be like
For region 2:
$R_1<r<R_2$

Let the mass density be ρ,
then $F=\frac{Gm}{r^2}\times \rho \times \frac{4}{3}\pi (r^3-R_1^3)$

$F=\frac{4}{3}Gmr\rho \pi [1-\frac{R_1^3}{r^3}]$
here, $r>R_1$
so,
Region 3 $=R_2<r<\infty$

$F=\frac{4}{3}Gm\rho \pi \left[\frac{R_2^3-R_1^3}{r^2}\right]$


Hence, option B is correct.