Physics Circulation Motion?

Question

Does anyone know how to derive expressions in terms of m, G, M, and x for the kinetic energy of a satellite and for the gravitational potential energy of a satellite?

Answer 1

If you want to keep a satellite in a circular orbit you must balance centripetal force with gravitational force.
Fg=Fc

Fc= m ac
ac=V^2/R ( hint kinetic energy Ke=0.5m V^2 )
R- radius of the orbit
Fg= G Mm/R^2

the potential energy of the satellite relative to Earth surface is
Pe= Fg(R-r)
r- radius of the Earth
Pe= G Mm (R-r) / R^2

Answer 2

let's assume the satellite is in circular orbit a distance r from the earth. to find its kinetic energy, we need to find its speed. the speed is found by equating centripetal to gravitational force:

GmM/r^2 = mv^2/r, where G is the newt. grav. cst, M is the mass of the earth, m is the mass of the satellite, and r is the radius of the orbit

KE is 1/2 mv^2, so the expression above shows that KE is GmM/2r

the potential energy in this case is -GmM/r, if you are taking calculus, you can derive this by integrating the expression for force.

Notice that 2KE+PE =0, an example of what is known as the virial theorem.