Question

Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through its center. Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let FR be the magnitude of the gravitational force on the apple when it is located at the planet's surface.
a. How far from the surface is there a point where the magnitude is 1/2FR if we move the apple away from the planet?
b. How far from the surface is there a point where the magnitude is 1/2FR if we move the apple into the tunnel?

Answer 1

Answer:

Explanation:

a )

Force at the surface = FR

Value of g at height h = g( 1 - 2 h / R )

force at height h = F_R ( 1 - 2 h / R)

F_R / 2 = F_R ( 1 - 2 h / R)

=  F_R   - 2  F_R xh / R

F_R / 2 = 2  F_R x h / R

1 / 2 = 2h / R

4h = R

h = R / 4 . Ans

b )

Value of g at depth  d = g( 1 -  d / R )

force at depth d  = F_R ( 1 - d / R)

F_R / 2 = F_R ( 1 - d / R)

=  F_R   -   F_R xd / R

F_R / 2 =   F_R x d / R

1 / 2 = d / R

2d  = R

d = R / 2