Physics Problems: Orbital period, Speed, Energy of a Satellite...?

Question

A satellite of mass 200 kg is launched from a site on the Equator into an orbit at 200 km above Earth’s surface. (a) If the orbit is circular, what is the orbital period of this satellite? (b) What is the satellite’s speed in orbit? (c) What is the minimum energy necessary to place this satellite in orbit, assuming no air friction?

I don’t understand this problem in my book. Please help me by explaining how to solve and what equations used to solve them. Thank You.

Answer 1

mass of the Earth=M, radius of the Earth=R~6367 km

mass of the satellite=m=200kg, satellite’s altitude above the Earth’s surface=r=200km

At the Earth's surface mg=γmM/R² ==> γM=gR²

b) Centripetal force=Gravitational force

mv²/(R+r)=γmM/(R+r)²

v²=gR²/(R+r)

v=R√[g/(R+r)]

v=6367e03√[9.8/(6367e03+200e03)]

v=7778 m/s

a) T=2π/ω=2π(R+r)/v=2π(6367+200)*1000/7778

T=5305s=1h 28m

c) U=Gravitational potential energy

U= -γMm/(R+r)= -gmR²/(R+r)

Emin=U(in orbit)-U(on Earth's surface)+Ekin=

-gmR²/(R+r)-(-gmR²/R)+0.5mv²=

gmR²[1/R-1/(R+r)]+0.5mv²=

gmRr/(R+r)+0.5mgR²/(R+r)=

[gmR/(R+r)]*(r+R/2)=6.43e09 J