Question

A sphere of matter, of mass M and radius a, has a concentric cavity of radius b. Find the gravitational force F exerted by the sphere on a particle of mass m, located a distance rfrom the center of the sphere, as a function of r in the range 0 ? r ? ? . Consider points r = 0; b; a and ? in particular.

Answer 1

We can solve this problem by using Newton’s shell law, which states that “a uniform shell of matter attracts a particle that is outside the shell as if all the shell’s mass were concentrated at its centre; the shell exerts no net gravitational force on a particle that is located inside it”.

a) If the particle is at r that is outside the shell, then the particle is outside the sphere, and the whole mass of the sphere acts on the particle, as if it was all concentrated at its centre:

Now dividing the whole space into three regions :

1) 0<r<a

F(r)=0

2) a<r<b

That if F(r) is proportional to

3) b<r<?

That is F(r) is proportional to

In region 1 - F = 0F=0

In region 2 - F increase gradually

In region 3 - F decrease gradually

Deriviation :

Mass of the sphere = M

Particle is placed at a distance r from the center and there is a particle of mass m located there.

The mass of the spherical shell that will cause force will be mass located between the spheres of radius r and radius a. That is mass of volume

Mass is distributed uniformly in the shell. Thus density of the shell is

That is :

Thus force exerted on m is =

G.m.M' / d2

Substitue M' in the above equation to get the force.