Physics 41 HW Set 1 Chapter 15

74. Review problem. Imagine that a hole is drilled through the center of the Earth to the other
side. An object of mass m at a distance r from the center of the
Earth is pulled toward the center of the Earth only by the mass
within the sphere of radius r. (a) Write Newton's law of
gravitation for an object at the distance r from the center of the
Earth, and show that the force on it is of Hooke's law form, F = –
kr, where the effective force constant is k = (4/3) Gm. Here
is the density of the Earth, assumed uniform, and G is the
gravitational constant. (b) Show that a sack of mail dropped
into the hole will execute simple harmonic motion if it moves
without friction. When will it arrive at the other side of the
Earth?
(a)
Newton’s law of universal gravitation is
Thus,
Which is of Hooke’s law form with
GM m Gm 4
3
F r
2 2
r r
3
4
F Gm r
3
4
k
Gm
3
(b)
The sack of mail moves without friction according to
4
Gm rm a
3
4
2
a
Gr
r
3
Since acceleration is a negative constant times excursion from equilibrium, it executes SHM
with
4G
and period
3
2
3
T
G
The time for a one-way trip through the earth is
We have also
T 3
2 4G
3
e e
2 2
e 3Re
GM G 4 R 4
g GR
R
3
e
so
4G
g
and
3 Re
R e
6
T
6.37 10 m
3
2.53 10 s 42.2 m in .
2
2 g 9.8 m s
Recall from physics 40 Chapter 13 that the period of an object in orbit around the Earth at the
surface is twice this 84 minutes and the orbital velocity is 7.9 km/s.