AP Free Response Question
1999 C2
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A spherical, nonrotating planet has a radius
R
and a uniform density
throughout its volume. Suppose a narrow tunnel were drilled through the planet along one of its diameters, as shown in the figure above, in which a small ball of mass
m
could move freely under the influence of gravity. Let
r
be the distance of the ball from the center of the planet.
(a) Show that the magnitude of the force on the ball at a distance r < R from the center of the planet is given by F = —Cr, where
(b) On the axes below, sketch the force
F
on the ball as a function of distance
r
from the center of the planet.
The ball is dropped into the tunnel from rest at point P at the planet’s surface.
(c) Determine the work done by gravity as the ball moves from the surface to the center of the planet.
(d) Determine the speed of the ball when it reaches the center of the planet.
(e) Fully describe the subsequent motion of the ball from the time it reaches the center of the planet.
(f) Write an equation that could be used to calculate the time it takes the ball to move from point P to the center of the planet. It is not necessary to solve this equation.
Topic Formulas
Description
Published Formula
angular displacement
angular momentum
angular velocity
center of mass
centripetal acceleration
friction
gravitational force (vector)
gravitational potential energy
Hooke's Law
impulse
kinetic energy
linear momentum
linear velocity and angular velocity
moment of inertia
net torque
Newton's 2nd Law
Newton's Law of Universal Gravitation
period and frequency
period of a simple pendulum
period of a spring
potential elastic energy
potential energy
power (dot product)
rate of change of momentum
rate of change of work
rotational kinetic energy
torque
uniform acceleration  displacement and instantaneous velocity
uniform acceleration  instantaneous position
uniform acceleration  instantaneous velocity
work (dot product)
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