Question

A spherically-symmetric planet is made of an empty inner core of radius a surrounded by a thick shell of radius 2a, which is surrounded by another thick shell of radius 3a. The inner shell has a volume density that only depends on radius and is given by s1(r) = br^3, where b is a known positive constant. The outer shell also has a volume density that only depends on radius and is given by s2(r) = y/(r^2), where y is a known positive constant.

[Planet 2]

b=alpha

s= rho

y=gamma

Find the gravitational field in the following regions in terms of G, b, y, a, and r.

r < a:

a < r < 2a:

2a < r < 3a:

r > 3a:

Answer 1

we will use gauss theorem for gravitational field :

where M is the total mass contained inside the volume V which has surface S. and is the gravitational field.

for r<a

Since there is no mass inside inner core i.e r<a (as given by the question) .So right hand side is zero.

The only way Left hand side can be zero is if we have is zero... So the gravitational field is zero.

a<r<2a

Let us find the total mass inside a spherical thin shell of radius r

Now since the density depends only on the radius we have the gravitational field directed radially and constant at a particular radius 'r'. This means the equipotential surfaces are spheres in space. So:

   .............so we have found for a<r<2a

Now for 2a<r<3a

Like above case .... consider a spherical Gaussian surface if radius r such that 2a<r<3a

We need to find the total Mass inside this surface:

Again the gravitational field only depends on "r". i.e equipotential surfaces are spheres. So 'g' is constant on the gaussian surface :

Thus we have found the field..

Now for r>3a

Similarly a spherical gaussian surface of radius r such that r>3a is considered and since density depends on r only we have the equipotential surface as spheres. So the field is constant on the gaussian surface. We only need to calculate the total mass :

So:

Thus we have found the fields in every region of space .