A 2300-kg satellite is orbiting the earth at an altitude equal to twice the eart

Question

A 2300-kg satellite is orbiting the earth at an altitude equal to twice the earth's radius. Find the satellite's weight at that altitude. Find the gravitational force exerted by the earth on the satellite at that altitude. What is the local g value at an earth altitude of 250 km (about where the Space Shuttle often orbits)? What is the speed of the Space Shuttle's orbit at an altitude of 250 km? At what earth altitude is the local g value 90% of earth's surface g value? What is the speed of a satellite orbiting earth where local g is 10% of earth's surface g? A certain planet (not Earth) has a radius of 5.59 times 106 m. A satellite that orbits this planet at an altitude of 490 km will complete a revolution every 170 minutes. What is the value of g on the surface of this planet?

Explanation / Answer

(a)

(i)

g = gravitational acceleration = gravity = at surface of Earth is = GMe/Re2

Now at R=2Re : new gravity = g' = GMe/(2Re)2 = g/4

Mass being 2300 kg, the weight will be = 2300 x (g/4) = 5635 Newtons

(ii)

The force exerted by Earth on this satellite will be same as given earlier in (i)

F=ma=mg'=mg/4=5635 Newtons

(b)

(i)

gravity value at 250 km will be given by

g'' = GMe/(Re+250000)2 =(6.67x10-11) ( 5.972 x 1024 )/(6371000 + 250000)2 = 9.086 SI units

(ii)

Centripetal force = mv2/R = F = 5635 Newtons

Thus v = ((6371000 + 250000)(5635)/(2300))1/2

=> v = 4027 SI units

(iii)

g' = 90 % of g = 0.9g = GMe/(Re+R')2 =(6.67x10-11) ( 5.972 x 1024 )/(6371000 + R')2

(6371000 + R')2 = (6.67x10-11) ( 5.972 x 1024 )/(0.9 x 9.8)

(6371000 + R')2 = 4.516 x 1013

R' = 349119 metres = 350 kms

(iv)

g' = 10 % of g = 0.1g = GMe/(Re+R')2 =(6.67x10-11) ( 5.972 x 1024 )/(6371000 + R')2

(6371000 + R')2 = (6.67x10-11) ( 5.972 x 1024 )/(0.1 x 9.8)

(6371000 + R')2 = 4.01 x 1014

R' = 13789358 m = 1379 kms

So speed will be given by :

mv2/r = mg'

v=(g'r)1/2

v=((0.98)(20160358))1/2

v=4445 SI units

Following same argument (c) can be solved.

Hope this helps. Kindly rate well.
Cheers.

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