9.3. Circular Orbits http://www.ck12.org
FIGURE 9.11
A satellite in orbit about the Earth.a. What is the orbiting satellite’s acceleration due to gravitygsat this location?
Answer:Using a familiar formula forgswe have,gs=Gmr 2 e.
But what value should we use forr? The distance from the center of the Earth to the satellite is:r=Re+h=
6370km+250km=6620km.
We can express this in terms of a ratio to the radius of the Earth, Rre=6620 km6370 km→r= 1. 039 Re. Substituting into the
equation forgswe have:
a=
Gme
( 1. 039 Re)^2=
1
( 1. 039 )^2
Gme
Re^2=
1
( 1. 039 )^2
g= 9 .08m/s^2b. What is the speed of the satellite?
Answer: We know thata=v
2
r and we have just found thata=gs=^9 .08m/s
(^2). We also knowr=6630 km=
6. 63 × 106 m, thus
v=
√
ar=√
( 9 .08m/s^2 )( 6. 63 × 106 m) = 7. 76 × 103 m/sThis is about 17,353 miles per hour, a typical “near-Earth” orbital velocity.
c. We often see pictures of astronauts floating within their spacecraft. It is not unusual for their craft to have a
similar orbit about the Earth as the satellite in this example. Sometimes you’ll hear someone claim that the reason
the astronaut is floating is because “there is no gravity up there,” or because the astronaut is “weightless.” Is it true
that the astronaut no longer experiences a gravitational force and is, by extension, weightless?
Answer:No, it is not true. As we can see from part A, the gravitational acceleration is close to 93% as much as it is
on the surface of the Earth→^99 ..^0881 × 100 = 92 .6%. If we were on a planet with a gravitational acceleration of 9. (^08) sm 2 ,
an astronaut who weighs 600 N on Earth would weigh 555 N on this planet. She would hardly be weightless. The
same 600 N astronaut does indeed have a 555 N force acting on her within the spacecraft. The gravitational force
from the Earth is what maintains her orbit about the Earth. If gravity were somehow to “turn-off” while she was in
orbit, the astronaut would move off tangentially to the orbit that she had been following- note the dashed red line
inFigure9.12. The fact that she does not fly off tangentially reinforces the notion that a force must be present to
maintain her circular motion path (recall Newton’s First Law). In fact, the distancehin Figure9.12 represents the
distance she effectively has fallen in her orbit about the Earth. Astronauts are in a perpetual state of free fall about
the Earth and this is why they “float.” They have an apparent weight of zero, only in that a scale reading of their
weight, while in orbit, would read zero.
You can see an example of apparent weightlessness in free fall by trying a simple experiment. Get an empty soda