Question

A solid sphere of mass m and radius r is placed inside a hollow thin spherical shell of mass M and radius R as shown in figure (11-E1). A particle of mass m is placed on the line joining the two centres at a distance x from the point o contact of the sphere and the shell. Find the magnitude of the resultant gravitational force on this particle due to the sphere and the shell if (a) r< x< 2r,
(b) 2r < x < 2R and (c) x> 2R.

Answer 1

(a) $m_1$ is placed at a distance x from 'O' If r< x < 2r, Lets consider a thin shell of mass.
$dm=\frac{m}{\left(\frac{4}{3}\right)\pi r^3}\times \frac{4}{3}\pi x^3=\frac{m{x}^3}{r^3}$
Thus $\int dm=\frac{m{x}^3}{r^3}$
Then gravitational force
$F=\frac{Gmdm}{x^2}=\frac{8\frac{m{x}^3}{r^3}}{x^2}=\frac{Gmx}{r^3}$
(b) 2r < x < 2R, then F is due to only the sphere,
$\therefore F=\frac{Gm{m}^1}{(x-r{)}^2}$
(c) If x > 2R, then gravitational force is due to both sphere and shell, then
due to shell, $F=\frac{GM{m}^1}{(x-R{)}^2}$
Due to the sphere$=\frac{gm{m}^1}{(x-r{)}^2}$
So, resultant force $=\frac{gm{m}^1}{(x-r{)}^2}+\frac{GM{m}^1}{(x-R{)}^2}$