Preparation problems
9-12. At what altitude will a satellite complete a
circular orbit of the Earth in 2.0 hours?
Solution:
We need to find a relationship between the period of a
circular orbit and its altitude. We can start by considering the
relationship between the gravitational force and the radial force
keeping the satellite in uniform circular motion
However, we also know that the velocity of an object in uniform circular
motion is given by
Substituting this in, we get
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é
ê
ë
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GM
4 p2
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T2 |
ù
ú
û
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(1/3)
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This will give us the radius of the orbit. If we recall that the
altitude of the orbit is the height above the Earth's surface, we see
that we want to know
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h = r - RE = |
é
ê
ë
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GM
4 p2
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T2 |
ù
ú
û
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(1/3)
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- RE |
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where RE is the radius of the Earth. Plugging in our known values, we get an answer of
9-21. Where should a satellite be placed to orbit the
Sun with a period of 100 days?
Solution:
We need to find a relationship between the radius of a
circular orbit and its period. We can start by considering the
relationship between the gravitational force and the radial force
keeping the satellite in uniform circular motion
where M is the mass of the sun and r is the distance at which the
satellite orbits. However, we also know that the velocity of an object
in uniform circular motion is given by
Substituting this in, we get
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é
ê
ë
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GM
4 p2
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T2 |
ù
ú
û
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(1/3)
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Recalling that T = 8.64 ×106 s, we get an answer of
9-22. Determine the orbital period of the Hubble Space
Telescope, which orbits Earth at an altitude of 610 km.
Solution:
We need to find a relationship between the altitude of a
circular orbit and its period. We can start by considering the
relationship between the gravitational force and the radial force
keeping the satellite in uniform circular motion
where M is the mass of the sun and r is the distance at which the
satellite orbits. Additionally, we also know that the velocity of an
object in uniform circular motion is given by
Substituting this in, we get
For the last step, we have used the fact that we can write the
radius as r = RE + h, where RE is the radius of the Earth and h
is the altitude of the orbit. Putting in our known numbers, we get
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T = 5806 s = 1 hour 37 minutes |
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Follow-up problems
9-9. A sensitive gravitometer is carried to the top of
Chicago's Sears Tower, where its reading for the acceleration of gravity
is 0.00136 m/s2 lower than at street level. Find the height of the
building.
Solution:
We start by writing an equation to represent the information
we are given in the problem:
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Dg = gtop - gbottom = -0.00136 m/s2 |
| (1) |
Now we can find the acceleration due to gravity at any distance from the
center of the earth by recalling
Knowing this, we can rewrite Eq. (1) as
Here we have assumed that the bottom of the Sears Tower is at a distance
RE (mean radius of Earth) from the center of Earth. While this is
not exactly true, it will be close enough for our problem.
Now we simply must solve for h, the height of the building.
Doing this we get
Plugging in our known values, we get an answer of
Solutions translated from TEX by TTH, version 1.57.
