(a)
The radius of the orbit of the GPS satellite.
(a)
Answer to Problem 41P
The radius of the orbit of the GPS satellite in which ir revolves around the earth is
Explanation of Solution
Given info: The time period of the satellite moving around earth in the circular orbit is
The value of force of gravitational constant
The mass of earth is
From Newton’s second law, the nature of the force between the earth and satellite system is the gravitational force between the earth and the satellite and the
The formula to calculate gravitational force is
Here,
The speed of the satellite is
Here,
The time period of the satellite is,
The formula to calculate the centripetal force is
Here,
Substitute
Equate equation (1) and (2)
Substitute
Thus the radius of the orbit of the satellite is
Conclusion:
Therefore, the radius of the orbit of the GPS satellite in which ir revolves around the earth is
(b)
The speed of the orbiting satellite.
(b)
Answer to Problem 41P
The speed of the satellite is
Explanation of Solution
The formula to calculate the speed of the satellite revolving around the earth in a circular orbit is,
Substitute
Thus the speed of the satellite is
Conclusion:
Therefore, speed of the satellite revolving around the earth in a circular orbit is
(c)
The fractional change in the frequency due the time dilation.
(c)
Answer to Problem 41P
The fractional change in the received signal frequency is
Explanation of Solution
Given info: The broadcast signal frequency of the GPS satellite is
The formula to calculate the frequency of any signal is,
Here,
Differentiate the above equation.
Thus, the fractional change in the frequency is the equal to fractional change in the time period.
The formula to calculate the fractional increase in time period is,
Here,
The formula to calculate the relativistic factor is,
Here,
Substitute
Substitute
Take the Binomial expansion series expansion of the term
Substitute
Thus the fractional change in frequency is
Conclusion:
Therefore, fractional change in the received frequency is
(d)
The magnitude of the fractional change in frequency in terms of due to gravitational blue shift.
(d)
Answer to Problem 41P
The fractional change in frequency due to the fractional blue shift is
Explanation of Solution
The formula to calculate the gravitational blue shift is,
Here,
The formula to calculate the gravitational potential energy between the earth’s surface and the satellite orbit is
Here,
Substitute
Substitute
Substitute
Thus the fractional change in frequency due to the gravitational blue shift is
Conclusion:
Therefore, fractional change in frequency due to the gravitational blue shift is
(e)
The overall fractional change in the frequency due to both time dilation and gravitational blue shift.
(e)
Answer to Problem 41P
The overall fractional change in the frequency is
Explanation of Solution
The overall fractional change in the frequency is the sum of the both the fractional changes.
Thus the overall fractional change is the sum of the fractional change in the frequency due to the time dilation and fractional change in the frequency due to the gravitational blue shift.
The formula to calculate the overall fractional change is,
Overall fractional change =
Substitute
Conclusion:
Therefore, the overall fractional change in the frequency is
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Chapter 38 Solutions
Physics for Scientists and Engineers with Modern Physics
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