(a)
The radius of the orbit of GPS satellite.
(a)
Answer to Problem 65P
The radius of the orbit of the GPS satellite is
Explanation of Solution
Write the expression for the gravitational force acting on the satellite.
Here,
Write the expression for the centripetal force acting on the satellite.
Here,
Write the expression for the velocity of the satellite.
Here,
Use expression (III) in (II).
Equate expressions (IV) and (I) and solve for
Conclusion:
Substitute
Therefore, the radius of the orbit of the GPS satellite is
(b)
The speed of the satellite.
(b)
Answer to Problem 65P
Speed of the satellite is
Explanation of Solution
Equation (III) gives the orbital speed of the satellite.
Conclusion:
Substitute
Therefore, speed of the satellite is
(c)
The fractional change in the frequency due to time dilation.
(c)
Answer to Problem 65P
The fractional change in the frequency due to time dilation is
Explanation of Solution
Write the expression for the frequency.
Here,
Differentiate expression.
The small fractional decrease in the frequency received from the satellite is equal to the fractional increase in period of oscillator due to the time dilation.
Write the expression for Lorentz factor.
Use expression (IX) in (VIII).
Write the binomial expansion for
Substitute expression (XI) in (X).
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 from Earth’s surface to its orbital position.
(d)
Answer to Problem 65P
The fractional change in frequency due to the change in position of the satellite from Earth’s surface to its orbital position is
Explanation of Solution
Write the expression for the change in gravitational potential energy.
Here,
Write the given expression for the fractional change in frequency.
Conclusion:
Substitute
Substitute
Therefore, the fractional change in frequency due to the change in position of the satellite from Earth’s surface to its orbital position is
(e)
The overall fractional change in frequency due to both time dilation and gravitational blue shift.
(e)
Answer to Problem 65P
The overall fractional change in frequency is
Explanation of Solution
Write the expression for the overall fractional change in frequency.
Here,
Conclusion:
Substitute
Therefore, the overall fractional change in frequency is
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Chapter 39 Solutions
Physics for Scientists and Engineers with Modern Physics, Technology Update
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