Assume the Earth is spherical (radius R) and has uniform
density (mass M). A tunnel is now dug along any secant. A
box of mass m (very well wrapped!) is held just above one
opening and dropped from rest. 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), themagnitude
of the gravitational force felt by mass at distance r fromthe
center of the Earth, where r < R. Now, find FT(r), the(tangential) component of this force along
the direction of the tunnel.
b) Show that m will move through the Earth in simple harmonicmotion. Find T, the period of that
motion. Use: M Earth = 5.98x1024 kg, R Earth = 6.37x106 m, and G =6.67x10-11 N-m2/kg2 to get a
numerical answer.
Assume the Earth is spherical (radius R) and has uniform
density (mass M). A tunnel is now dug along any secant. A
box of mass m (very well wrapped!) is held just above one
opening and dropped from rest. 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), themagnitude
of the gravitational force felt by mass at distance r fromthe
center of the Earth, where r < R. Now, find FT(r), the(tangential) component of this force along
the direction of the tunnel.
b) Show that m will move through the Earth in simple harmonicmotion. Find T, the period of that
motion. Use: M Earth = 5.98x1024 kg, R Earth = 6.37x106 m, and G =6.67x10-11 N-m2/kg2 to get a
numerical answer.
