Ball dropped through a tunnel through the earth.

[PhysicsStudnt]

Homework Statement

A tunnel is drilled from the surface of the earth(mass assumed to be M and radius to be R) to its center . A body of mass m is dropped from the surface to the center through the tunnel. What will be the velocity with which the body of mass m will hit the tunnel.

Homework Equations

Loss in PE = Gain In KE

Loss in PE = PE at surface - PE at center of earth.

= |-GMm/R - 0|

1/2 mv[2] = |-GMm/R|

v = √(2GM/R)

The Attempt at a Solution

But in a textbook , the PE at the center of Earth is given to be -(3/2) (GMm/R)

what is the right way of approaching the problem.

 

[ShayanJ]

The point where the potential energy is zero, is arbitrary. You assumed it to be the centre of the Earth which is not in agreement with problem's assumption. Of course you can solve the problem with your current assumption too.

 

[I like Serena]

Let's assume that ##R## is the radius of the planet, and ##r## is the distance between the body and the center.

Then the attractive force is not just ##F=-\frac{GMm}{r^2}##.
The value of ##M## that you should use varies with ##r##, since we're inside the planet.

The ##M## that contributes is the mass in the sub sphere that is still between you and the center of the planet. To calculate it, we need to assume for instance that the mass density is constant.

What would the mass ##M(r)## of the sub sphere of radius ##r## be?