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An Earth satellite used in the Global Positioning System moves in a circular orbit with period 11 h 58 min. (a) Determine the radius of its orbit. (b) Determine its speed. (c) The satellite contains an oscillator producing the principal nonmilitary GPS signal. Its frequency is 1 575.42 MHz in the reference frame of the satellite. When it is received on the Earth’s surface, what is the fractional change in this frequency due to time dilation, as described by
where ΔUg/m is the change in gravitational potential energy per unit mass between the two points at which the signal is observed. Calculate this fractional change in frequency. (e) What is the overall fractional change in frequency? Superposed on both of these relativistic effects is a Doppler shift that is generally much larger. It can be a redshift or a blueshift, depending on the motion of a particular satellite relative to a GPS receiver (Fig. P1.39).
(a)
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The radius of the orbit of the satellite.
Answer to Problem 39P
The radius of the orbit of the satellite is
Explanation of Solution
Write the expression of Newton’s gravitational law
Write the expression for centripetal force
Write the expression for the linear velocity of an object moving in circular path
Here,
Equating equation (I) and (II) and substitute equation (III)
Substitute
Conclusion:
Thus, the radius of the orbit of the satellite is
(b)
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The speed of the satellite.
Answer to Problem 39P
The speed of the satellite is
Explanation of Solution
Substitute
Conclusion:
Thus, the speed of the satellite is
(c)
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The fractional change in the frequency due to time dilation.
Answer to Problem 39P
The fractional change in the frequency due to time dilation is
Explanation of Solution
The fractional change in the frequency received on Earth is equal to the magnitude of the fractional increase in the period of moving oscillator due to time dilation.
Write the expression of fractional change in the frequency
Here,
Substitute
Conclusion:
Thus, the fractional change in the frequency due to time dilation is
(d)
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The fractional change in the frequency.
Answer to Problem 39P
The fractional change in the frequency is
Explanation of Solution
Write the expression for the gravitational potential energy
Here,
Substitute
The fractional change in the frequency is
Conclusion:
Thus, the fractional change in the frequency is
(e)
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The overall fractional change in the frequency.
Answer to Problem 39P
The overall fractional change in the frequency is
Explanation of Solution
The overall fractional change in the frequency is the sum of both the fractional changes.
Hence,
Conclusion:
Thus, the overall fractional change in the frequency is
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Chapter 1 Solutions
Modern Physics
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