Question

A clock is placed in a satellite that orbits Earth with an orbital period of 92 min. By what time interval will this clock differ from an identical clock on Earth after 1 week? (Assume that special relativity applies and neglect general relativity.)

The answer should be in seconds

Answer 1

let r is the radius of orbit of the satellite.

we know, mass of the earth, Me = 5.97*10624 kg

T = 92 in

= 92*60

= 5520 s

we know, time period of a satellite,

T = 2*pi*r^(3/2)/sqrt(G*Me)

T^2 = 4*pi^2*r^3/(G*Me)

r^3 = G*Me*T^2/(4*pi^2)

r = ( G*Me*T^2/(4*pi^2))^(1/3)

= (6.67*10^-11*5.97*10^24*5520^2/(4*pi^2))^(1/3)

= 6.748*10^6 m

orbital speed of the satellite, v = 2*pi*r/T

= 2*pi*6.748*10^6/5250

= 8076 m/s

Time measured by a clock on the earth,

T' = T/sqrt(1 - (v/c)^2)

= 5250/sqrt(1 - (8076/(3*10^8))^2)

= 5250.00000190 s

so, T' - T = 5250.00000190 - 5250

= 1.90*10^-6 s <<<<<<<<------------------------------Answer