The gravitational force F on a particle at position r within a spherical object

Question

The gravitational force F on a particle at position r within a spherical object
of uniform density ? has the form, F = - 4 G p ? m r rˆ / 3

where m is the mass of the particle, rˆ = r/ | r | and G is the universal
gravitational constant. The direction of the force is always towards the centre of the object. A cosmonaut travels to a solid spherical asteroid to test this formula by drilling a straight shaft through the centre of the asteroid with the shaft being open at both ends. To help her to test whether the form of the force is correct, she chooses a Cartesian coordinate system with the x-axis lying along the shaft, and the zero of the x-axis at the centre of the asteroid.
Assuming that the shaft is sufficiently narrow that it has no effect on the force, write down the x-component of the force, Fx, for a particle in the shaft.

Her aim is to drop a small ball from one end of the shaft at x = R (where R is the radius of the asteroid) and to compare the subsequent motion with theory. In the following, assume that the ball is sufficiently small to fit down the shaft, and that it does not make contact with the sides of the shaft.
Explain why the ball is expected to undergo simple harmonic motion.

Determine an expression for the period of motion that the cosmonaut expects. (Hint: you may find that Equation 1.53 in PM is a good starting point.)

The cosmonaut performs an experiment by dropping a small ball down the shaft, and uses a laser to accurately determine the ball’s position as a function of time. The ball is released from x = R at time t = 0. At small times, she finds that the x-component of the position of the ball has the following form:
x(t) = A + Bt + Ct 2 . (i) Determine the constant A.
Given that the ball is released from rest, derive an expression for the velocity, and hence determine the value of constant B.
The ball has a mass m = 0.10 kg; it weighs 1.2 N at the surface of the asteroid. Determine the acceleration due to gravity at the surface of the asteroid. Hence use differentiation to determine the value of constant C.
Briefly give a reason why the position of the ball follows this equation close to the surface of the asteroid.

The experiment continues and the motion is found to deviate slightly from the prediction, with the following function found to be a good approximation to the position of the ball:
x(t) = 0.90 R cos(? t) + 0.10 R cos(3? t).
Determine an expression for the x-components of the velocity and acceleration.

Explanation / Answer

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