Question

Your task is to place your spacecraft in circular orbit about Mars with an orbital period of 8 hours and 40 minutes. The mass of Mars is 6.45 x 1023 kg, and the radius of Mars is 3394 km. What will be the radius of your circular orbit?

Answer 1

Gravitational forces between 2 objects is given by: F=GMm/r2, where G is universal gravitational constant=6.67*10-11 Nm2/kg2,M is mass of first object,m is mass of second object, r is spearation between the masses.

Let the mass of moon be M and that of spacecraft be m.

Since the spacecraft performs uniform circular motion, it experiences a force given by: F=mv2/r, where v is velocity and r is the radius of circular trajectory.

So,GMm/r2=mv2/r

=>v2=GM/r

=>v=(GM/r)1/2

Also,period of revoultion=distance/speed=2r/v, where r is radius and v is velocity.

So,period of revolution T= 2r/(GM/r)1/2=2r3/2/(GM)1/2

Here,T=8 hours 40 mins= 8*60*60+40*60 seconds= 31200 seconds.

M=6.45*1023 kg.

So,31200=2r3/2/(6.67*10-11*6.45*1023)1/2

=>2r3/2=31200*(6.67*10-11*6.45*1023)1/2=2.046*1011

=>r3/2=2.046*1011/(2)=3.256*1010

=>r=(3.256*1010)2/3=10197268.17 m= 10197.27 km