Mars rotates on its axis once every 24.8 hours. Part A

What is the speed of a geosynchronous satellite orbiting

Mars? Part B

What is the altitude of a geosynchronous satellite orbiting

Mars?

This question is difficult and kind of confusing. I will show you

the solutions:

a) Mass of mars: 6.4191×10^23 kg

F_centripetal = F_gravitational

m_s * a_g = m_s * a_c

Note: m_s is the mass of satellite, a_g is gravitational

acceleration, and a_c is centripetal acceleration

Mass of satellite cancels out from the above equation and why the

mass of satellite is not even given to you in the first

place.

Therefore, a_g=a_c

a_c = (omega)^2*r

a_g = GM/r^2 <-----M is the mass of Mars
(omega)^2*r = GM/r^2
r^3=GM/(omega)^2
r= cubic root {GM/(omega)^2}
speed of a geosynchronous satellite = (omega)*r = (omega)*cubic
root {GM/(omega)^2} = cubic root (GM*omega)
Note: simplified using the fact that omega can also be written as
cubic root (omega^3)
Also know that T=2pi/omega
or omega = 2pi/T
speed of a geosynchronous satellite =cubic root (GM*omega)= cubic
root (GM2pi/T)
= cubic root [(6.673 x 10^-11 x 6.4191x10^23 kg x 2 x pi) /
(24.8hours x 3600 seconds/hr)]
= 1440 m/s = 1.44km/s
b) v=omega*r
r=v/omega
omega = 2pi/T=2*3.14/(24.8hours x 3600 seconds/hr)]
v= 1440m/s from part a
r=2.05*10^7
But radius of mars is 3397000m
Altitude = 2.05*10^7-3397000=1.72*10^7m

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