Question

Assume a planet (Earth for example) is an uniform sphere of radius R and it has a narrow radial tunnel through its center. Assume that we can position an object anywhere along this tunnel or outside the planet (again Earth as an example). Let Fr equal magnitude of the gravitational force on the object that is traveling through the center of the planet. When the object is found at the planet’s surface, how far from the surface is there a point where the magnitude is 15 Fr if we move the object

One away from the planet

Two into the tunnel?

Answer 1

Let mass of the apple be "m". Force on it at surface of the planet can be expressed as follows:

Where M is mass of the planet, m is mass of the object and r is the radius of planet.

Let at height h the gravitational force on object be 15 Fr

Dividing the equation (1) and (2):

If we go deep inside the planet to get force on object equal to 15 Fr

where d is density of planet.

dividing two equations (3) and (4)