A body of unit mass inside a uniform sphere at a distance r from its centre O experiences a gravitational attraction towards the centre ofwhere G is the gravitational constant and M_r is the mass of material inside the sphere of radius r.
(In other words it is as if the body were on the surface of a sphere of radius r, all the matter further from the centre than r will have no net gravitational effect on the body.)

Suppose a straight tube could be drilled right through the Earth, modelled as a uniform sphere of radius R and total mass M. A ball of unit mass is dropped into the tube at the surface of the Earth.
i) Work out the mass of a sphere of the Earth of radius r, in terms of R, M and r.
ii) Write down an expression for the gravitational force on the ball when at a distance r from the centre.
iii) Hence show that the ball will oscillate with simple harmonic motion with an amplitude equal to the radius of the Earth. Determine the period.