Page 122 - Schaum's Outline of Theory and Problems of Applied Physics

CHAP. 9] CIRCULAR MOTION AND GRAVITATION 107

2
At the earth’s surface, where g = g 0 = 9.8 m/s and r = r e = 6400 km,
Gm e
g 0 =
r 2
e
From these formulas for g and g 0 we find that, at the distance r from the earth’s center, the acceleration of gravity is
2

r e
g = g 0
r
When r = r e + h = 6400 km + 1000 km = 7400 km,
2

6400 km 2 2
g = (9.8 m/s ) = 7.3 m/s
7400 km
SATELLITE MOTION

Gravitation provides the centripetal forces that keep the planets in their orbits around the sun and the moon in
its orbit around the earth. The same is true for artificial satellites put into orbit around the earth.

SOLVED PROBLEM 9.15
Find the velocity an artificial satellite must have to pursue a circular orbit around the earth just above the
surface.
2
In a stable orbit, the gravitational force mg on the satellite must be equal to the centripetal force mv /r required.
Hence
mv 2
= mg
r
2
v = rg
√
v = rg
The mass of the satellite is irrelevant. To find v, we use the radius of the earth r e for r and the acceleration of gravity
at the earth’s surface for g. This gives

√ 2 3
6
v = r e g = (6.4 × 10 m)(9.8 m/s ) = 7.9 × 10 m/s
which is about 18,000 mi/h. With a smaller velocity than this, a space vehicle projected horizontally above the earth
will fall to the surface; with a larger velocity, it will have an elliptical rather than a circular orbit.

SOLVED PROBLEM 9.16

A satellite in a geostationary orbit circles the earth above the equator with a period of exactly 1 day, so it
stays above a particular place all the time. Most of the satellites in such orbits act as relays for telephone
calls and television programs. Find the altitude of a geostationary orbit.
A satellite that moves in a circular orbit of radius r covers a distance 2πr in period T. Hence its velocity is
√
v = 2πr/T. From the solution to Prob. 9.15, another formula for the satellite speed is v = rg. Setting these
formulas equal allows us to eliminate v:
2πr √
= rg
T
2
From the solution to Prob. 9.14, g = (r e /r) g 0 , so

2
2πr r g 0
e
=
T r