A clock is placed on a satellite that orbits Earth with a period of 90.0 minutes

Question

A clock is placed on a satellite that orbits Earth with a period of 90.0 minutes at an altitude of 3.00 x102 km.

a) Considering only special relativistic effects, by what time interval will this clock differ from an identical clock on Earth after exactly one year?

b) Considering only general relativistic effects, by what time interval will this clock differ from an identical clock on Earth after exactly one year?

c) Taking into account both special and general relativistic effects, by what time interval will this clock differ from an identical clock on Earth after exactly one year?

Explanation / Answer

period T = 90 min

orbit radius r = 3.0e+4 + 6.4 e+6 m   , add earth radius

                      = 6.43e+6 m

speed of the settelite v = 2*pi*r/T   = 2*3.14*6.4e+6/90*60 = 7.45e+3 m/s

with special releativistic effect

to = tf (1-v2/c2)1/2 = tf ( 1- 7.45e+3/3.0e+8)2)1/2 = tf (1- 0.5 * 6.1669e-10)

The difference in clocks after 1 yr   = 365*24*3600* (0.5 * 6.1669e-10) = 9.72 e-3 s

with general relativity there is gravitational time dialtion

to = tf (1- 2GM/rc2)1/2

     = to = tf (1 - 2*6.67e-11*6.0e+24/6.43e+6*9.0e+16)1/2    = tf (1- 6.9e-10)

time difference = 365*24*3600*6.9e-10 = 2.18e-2 s

c) time difference considering both effects = 2.18e-2 , when we are condiering general releativistic effects it is about non-inertial frames. it also includes the effects of special relativity where we are considering non-accelerating frames.

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