If the period of a satellite is T=24 h = 86400 s that means it is in geostationary orbit around Earth. That means that the force of gravity FgĀ and the centripetal force Fcp are equal:
Fg=Fcp
m*g=m*(vĀ²/R),
Ā
where m is mass, v is the velocity of the satelite and R is the height of the satellite and g=G*(M/rĀ²), where G=6.67*10^-11 mĀ³ kgā»Ā¹ sā»Ā², M is the mass of the Earth and r is the distance from the satellite.Ā
Masses cancel out and we have:
G*(M/rĀ²)=vĀ²/R, R=r so:
G*(M/r)=vĀ²
r=G*(M/vĀ²), since v=Ļr it means vĀ²=ĻĀ²rĀ² and we plug it in,
r=G*(M/ĻĀ²rĀ²),
rĀ³=G*(M/ĻĀ²),Ā Ļ=2Ļ/T, it meansĀ ĻĀ²=4ĻĀ²/TĀ² and we plug that in:
rĀ³=G*(M/(4ĻĀ²/TĀ²)), and finally we take the third root to get r:
r=ā{(G*M*TĀ²)/(4ĻĀ²)}=4.226*10^7 m= 42 260 km which is the height of a geostationary satellite.Ā
