A satellite of mass m is orbiting Earth in a stable circular orbit of radius R. The mass and radius of Earth are ME and RE , respectively. Express your answers to parts (a), (b), and (c) the following in terms of m, R, ME , RE , and physical constants, as appropriate. a. Derive an expression for the speed of the satellite in its orbit.
b. Derive an expression for the total mechanical energy of the satellite-Earth system in its orbit.
c. Derive an expression for the period of the satellite’s orbit.

a)   v = ,   b)  Em = - ½ m , c)  T = 2π

Explanation:

a) For this exercise we must use Newton's second law with the gravitational force

          F = ma

          = m a

the acceleration of the satellite is centripetal

          a =

we substitute

           = v²

          v =

the distance is from the center of the earth

b) mechanical energy is the sum of kinetic energy plus potential energy

         Em = K + U

         Em = ½ m v² - G m M / (R + R_e)

we substitute the expression for the velocity

         Em = ½ m    -  

         Em = - ½ m  

c) as the orbit is circulating, the velocity modulus is constant

         v = d / t

in a complete orbit the distance traveled of the circle is

        d = 2π (R + R_e)

where time is called period

         v = 2π (R + R_e)

         T = 2π (R + R_e) / v

we substitute the speed value

        T = 2π (R + R_e)

        T = 2π

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