Review. A global positioning system (GPS) satellite moves in a circular orbit with period 11 h 58 min. (a) Determine the radius of its orbit. (b) Determine its speed. (c) The nonmilitary GPS signal is broadcast at a frequency of 1 575.42 MHz in the reference frame of the satellite. When it is received on the Earth’s surface by a GPS receiver (Fig. P38.41), what is the fractional change in this frequency due to time dilation as described by
where Ug is the change in gravitational potential energy of an object–Earth system when the object of mass m is moved between the two points where the signal is observed. Calculate this fractional change in frequency due to the change in position of the satellite from the Earth’s surface to its orbital position. (e) What is the overall fractional change in frequency due to both time dilation and gravitational blueshift?
Figure P38.41
(a)
Answer to Problem 39.65P
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
Explanation:
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 centripetal force between them. Both the forces should be equal so that the satellite can revolve in an orbit.
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)
Answer to Problem 39.65P
Explanation of Solution
Explanation
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)
Answer to Problem 39.65P
Explanation of Solution
Explanation
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)
Answer to Problem 39.65P
Explanation of Solution
Explanation
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)
Answer to Problem 39.65P
Explanation of Solution
Explanation
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
Want to see more full solutions like this?
Chapter 39 Solutions
Physics for Scientists and Engineers, Technology Update, Hybrid Edition (with Enhanced WebAssign Multi-Term LOE Printed Access Card for Physics)
- 2. Consider the situation described in problem 1 where light emerges horizontally from ground level. Take k = 0.0020 m' and no = 1.0001 and find at which horizontal distance, x, the ray reaches a height of y = 1.5 m.arrow_forward2-3. Consider the situation of the reflection of a pulse at the interface of two string described in the previous problem. In addition to the net disturbances being equal at the junction, the slope of the net disturbances must also be equal at the junction at all times. Given that p1 = 4.0 g/m, H2 = 9.0 g/m and Aj = 0.50 cm find 2. A, (Answer: -0.10 cm) and 3. Ay. (Answer: 0.40 cm)please I need to show all work step by step problems 2 and 3arrow_forwardFrom number 2 and 3 I just want to show all problems step by step please do not short cut look for formulaarrow_forward
- Look at the answer and please show all work step by steparrow_forward3. As a woman, who's eyes are h = 1.5 m above the ground, looks down the road sees a tree with height H = 9.0 m. Below the tree is what appears to be a reflection of the tree. The observation of this apparent reflection gives the illusion of water on the roadway. This effect is commonly called a mirage. Use the results of questions 1 and 2 and the principle of ray reversibility to analyze the diagram below. Assume that light leaving the top of the tree bends toward the horizontal until it just grazes ground level. After that, the ray bends upward eventually reaching the woman's eyes. The woman interprets this incoming light as if it came from an image of the tree. Determine the size, H', of the image. (Answer 8.8 m) please show all work step by steparrow_forwardNo chatgpt pls will upvotearrow_forward
- Physics for Scientists and Engineers with Modern ...PhysicsISBN:9781337553292Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPhysics for Scientists and EngineersPhysicsISBN:9781337553278Author:Raymond A. Serway, John W. JewettPublisher:Cengage LearningPrinciples of Physics: A Calculus-Based TextPhysicsISBN:9781133104261Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning
- Physics for Scientists and Engineers: Foundations...PhysicsISBN:9781133939146Author:Katz, Debora M.Publisher:Cengage LearningModern PhysicsPhysicsISBN:9781111794378Author:Raymond A. Serway, Clement J. Moses, Curt A. MoyerPublisher:Cengage LearningPhysics for Scientists and Engineers, Technology ...PhysicsISBN:9781305116399Author:Raymond A. Serway, John W. JewettPublisher:Cengage Learning