Question

A uniform sphere of mass $M$ and radius $R$ is surrounded by a concentric spherical shell of same mass but radius $2R.$ A point mass $m$ is kept at a distance $x(>R)$ in the region bounded by spheres as shown in the figure. The net gravitational force on the particle is

• A.
  $\frac{GMm}{x^2}$
• B.
  $\frac{GMmx}{R^3}$
• C.
  $\frac{G(M+m)}{x^2}$
• D. zero

Answer 1

The correct option is A

$\frac{GMm}{x^2}$
The gravitational force on the point mass $m$ due to uniform sphere is $$F=\frac{GMm}{x^2}$$ Where,
$x$ is the distance between centre of sphere and mass $m$.

$M$ is the mass of sphere.

The net gravitational force on the point mass due to the outer spherical shell will be zero because the gravitational force of attraction on a point mass due to various points of the spherical shell cancel each other completely as their vector sum is zero.

Thus, the net gravitational force on the particle will be $F=\frac{GMm}{x^2}$.

$\text{Key Concept:}$
Field due to uniform solid sphere of mass $M$ and radius $R$ at distance $r$ outside of sphere is $-GM/r^2$ and inside of the sphere is $-GMr/R^3$.

Field due to uniform hollow sphere of mass $M$ and radius $R$ at $r$ outside the sphere is $GM/r^2$ and inside is$=0$.

Why this question : It is absolutely important to be well versed with the field due to a solid sphere and a hollow sphere on both sides inside and outside. This concept is extremely important from a JEE standpoint.