(a)
The radius of the orbit.
(a)
Answer to Problem 43P
The radius of the orbit is
Explanation of Solution
The gravitational force acting on the satellite is equal
Write the expression for the force acting on the satellite.
Here,
Equate the gravitational force, and centripetal force.
Here,
Substitute,
Here,
Conclusion
Substitute,
Therefore, the radius of the orbit is
(b)
The speed of the satellite.
(b)
Answer to Problem 43P
The speed of the satellite is
Explanation of Solution
Write the expression for speed in terms of period.
Conclusion:
Substitute,
Therefore, the speed of the satellite is
(c)
The fractional change in the frequency due to time dilation.
(c)
Answer to Problem 43P
The fractional change in the frequency due to time dilation is
Explanation of Solution
Write the expression for the frequency.
Here,
Take the derivative of equation (V) on both sides.
Substitute,
The fractional change in the frequency is equal to the fractional change in time period according to equation (VII).
Substitute,
Substitute,
Here,
Conclusion:
Substitute,
Therefore, the fractional change in the frequency due to time dilation is
(d)
The fractional change in frequency due to the change in position of the satellite.
(d)
Answer to Problem 43P
The fractional change in frequency due to the change in position of the satellite is
Explanation of Solution
Write the expression for the gravitational potential.
The fractional change in frequency due to the change in position of the satellite is.
Conclusion:
Substitute,
Substitute,
Therefore, the fractional change in frequency due to the change in position of the satellite is
(e)
The overall fractional change in the frequency.
(e)
Answer to Problem 43P
The overall fractional change in the frequency is
Explanation of Solution
The fractional change in frequency due to the change in position of the satellite is
Hence the overall change in the frequency is.
Conclusion:
Therefore, the overall fractional change in the frequency is
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Chapter 9 Solutions
Principles of Physics: A Calculus-Based Text
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