Assume the Earth is spherical (radius R) and has uniform density (mass M). A tun
ID: 2005197 • Letter: A
Question
Assume the Earth is spherical (radius R) and has uniformdensity (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 Gauss’ law to find an expression for F(r), the magnitude
of the gravitational force felt by mass at distance r from the
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 harmonic motion. 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.
Explanation / Answer
MEarth = 5.98x1024 kg, REarth = 6.37x106 m, G = 6.67x10-11 N-m2/kg2,
density = M/(4R3/3)
a) F*4r2 = 4GmMr/r2 where Mr = (4r3/3) = M(r/R)3
so F(r) = GmM/R3 * r
FT(r) = F(r) * x/r = GmM/R3 * x where x is the coordinate measured from the origin (the point closest to the center of the earth) along the tunnel.
b) note direction of FT is always opposite to x, so FT = -GmM/R3 * x = -kx
where k = GmM/R3 = g/R
period T = 2(m/k) = 2(R/g) = 2(6.37x106/9.81) = 5063 s