1. Mail Chute Through the Earth (40 points total)
Assume the Earth is sphere of radius R, has uniform density, massM,
and tunnel dug through the Earth from pole to pole. A box of massm
(very well wrapped!) is held just above the North Pole openingand
dropped from rest (see diagram). Assume:
· You can ignore friction and all effects due to the various
motions of the Earth
· The tunnel is so thin it has no measurable effect on the
gravitational field of the Earth.
a) Use Gaussâ law to find an expression for F(r), the magnitude ofthe gravitational force felt by mass
m at distance r from the center of the Earth, where r < R. (10points)
b) Show that m will move through the Earth in simple harmonicmotion. Find T, the period of that
motion. Use: MEarth = 5.98x1024 kg, REarth = 6.37x106 m, and G =6.67x10-11 N-m2/kg2 to get a
numerical answer. (8 points)
c) Use a computer program to draw a graph of a(r), (magnitude only,so it is positive at all points),
showing points from r = 0 to r = 2R outside the Earth). (10points)
NOTE: You do not have to show a derivation for F(r) for r >R.
d) Find a numerical value for vmax, the maximum speed of m, usingthe values for the constants given in
part b. (Initial conditions: ro = R, vo = 0.) (12 points)
HINT: One method is to use conservation of energy.
1. Mail Chute Through the Earth (40 points total)
Assume the Earth is sphere of radius R, has uniform density, massM,
and tunnel dug through the Earth from pole to pole. A box of massm
(very well wrapped!) is held just above the North Pole openingand
dropped from rest (see diagram). Assume:
· You can ignore friction and all effects due to the various
motions of the Earth
· The tunnel is so thin it has no measurable effect on the
gravitational field of the Earth.
a) Use Gaussâ law to find an expression for F(r), the magnitude ofthe gravitational force felt by mass
m at distance r from the center of the Earth, where r < R. (10points)
b) Show that m will move through the Earth in simple harmonicmotion. Find T, the period of that
motion. Use: MEarth = 5.98x1024 kg, REarth = 6.37x106 m, and G =6.67x10-11 N-m2/kg2 to get a
numerical answer. (8 points)
c) Use a computer program to draw a graph of a(r), (magnitude only,so it is positive at all points),
showing points from r = 0 to r = 2R outside the Earth). (10points)
NOTE: You do not have to show a derivation for F(r) for r >R.
d) Find a numerical value for vmax, the maximum speed of m, usingthe values for the constants given in
part b. (Initial conditions: ro = R, vo = 0.) (12 points)
HINT: One method is to use conservation of energy.
