calculate the redshift gh/c^2 if h is the distance from the ground to a satellite in low-Earth orbit, 300km. Suppose the "light" is actually a radio wave with a frequency of 10^11 Hz. How many cycles would the transmitter emit if it ran for one day? How many fewer would be received in one day by the satellite? How long did it take the transmitter to generate these "extra" cycles?
9.8 m/s^2*300000m/(9*10^16) m^2/s^2 = 3.27*10^-11.
The red-shifted frequency will be that times 10^11 Hz, or 3.27Hz. I don't know how to use this to find the answers to the questions being asked.
To find how many cycles the transmitter would emit in one day, we need to know the frequency and the duration. In this case, the frequency is given as 10^11 Hz, and we can convert one day into seconds by multiplying it by 24 hours, 60 minutes, and 60 seconds:
1 day = 24 hours/day * 60 minutes/hour * 60 seconds/minute = 86400 seconds.
So, the number of cycles emitted by the transmitter in one day would be:
(10^11 Hz) * (86400 seconds) = 8.64 * 10^15 cycles.
Now, to determine how many fewer cycles would be received by the satellite in one day, we need to consider the redshift. The formula for calculating the redshift, z, is:
z = Δf / f,
where Δf is the difference in frequency and f is the original frequency. In this case, the redshift is given as 3.27 * 10^-11 and the original frequency is 10^11 Hz. Substituting these values, we can solve for Δf:
3.27 * 10^-11 = Δf / (10^11 Hz).
Rearranging the equation, we find:
Δf = (3.27 * 10^-11) * (10^11 Hz) = 3.27 Hz.
So, the frequency received by the satellite would be 10^11 Hz - 3.27 Hz = 9.99999673 * 10^10 Hz.
To determine the number of cycles received by the satellite in one day, we multiply the frequency received by the duration (in seconds):
(9.99999673 * 10^10 Hz) * (86400 seconds) = 8.64 * 10^15 cycles.
Comparing this with the number of cycles emitted, we can see that the number of cycles received is the same as the number of cycles emitted, as there is no significant difference due to the redshift.
Lastly, to find the time it took the transmitter to generate these "extra" cycles, we simply divide the number of cycles emitted (8.64 * 10^15 cycles) by the original frequency (10^11 Hz):
(8.64 * 10^15 cycles) / (10^11 Hz) = 8.64 * 10^4 seconds.
Therefore, it took the transmitter approximately 8.64 * 10^4 seconds to generate these "extra" cycles.