Question

A solid sphere of mass $m$ radius $r$ is placed inside a hollow thin spherical shell of mass $M$ and radius $R$ as shown in figure. A particle of mass $m^{\prime }$ is placed on the line joining the two centres at a distance $x$ from the point of 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 $2r<x<2R$.

• A. $\frac{Gm{m}^{\prime }(2r-x)}{2r^3}$
• B. $\frac{{Gmm}^{\prime }x}{2r^3}$
• C. $\frac{Gm{m}^{\prime }}{(x-r{)}^2}$
• D. $\frac{{Gmm}^{\prime }x}{2r^3}$

Answer 1

The correct option is C $\frac{Gm{m}^{\prime }}{(x-r{)}^2}$


The net gravitational field on a point mass $m$ inside a spherical shell of mass $M$ is zero, this is because of the symmetric mass distribution outside the position of the test mass $m$ .

So, the force from any spherically symmetric mass distribution on a mass inside its radius is zero.

Thus, the shell of mass $M$ exerts no force on mass $m^{\prime }$ as its gravitational field inside is zero.i.e., $F_{shell}=0$

So the force on the particle will come due to solid sphere of mass $m$ only.

The point mass $m^{\prime }$ is inside the shell is at a distance $x-r$ from the centre of the solid sphere of mass $m$ as shown in the figure.

Field due to the solid sphere at the point where $m^{\prime }$ is placed,

$E_{sphere}$=$\frac{Gm}{(x-r{)}^2}$

So, the net force on particle $m^{\prime }$ will be,

$F_{net}=m^{\prime }{E}_{sphere}+F_{shell}$

$\Rightarrow F_{net}=\frac{Gm{m}^{\prime }}{(x-r{)}^2}+0$

$\therefore F_{net}=\frac{Gm{m}^{\prime }}{(x-r{)}^2}$

Hence, option (c) is the correct answer.