Question

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.

Answer 1

Answer:

a)   v = $\sqrt{G \frac{M_e}{(R+R_e)} }$,   b)  Em = - ½ m $G \frac{M_e}{(R+R_e)}$, c)  T = 2π $\sqrt{\frac{ (R+R_e)^3 }{G M_e } }$

Explanation:

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

          F = ma

          $G \frac{m M_e}{(R+R_e)^2 }$ = m a

the acceleration of the satellite is centripetal

          a = $\frac{v^2}{(R+R_e)}$

we substitute

            $G \frac{m M_e}{(R+R_e)^2 } = m \frac{v^2}{ (R+R_e)}$

          $G \frac{M_e}{(R+R_e)}$  = v²

          v = $\sqrt{G \frac{M_e}{(R+R_e)} }$

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  $G \frac{M_e}{(R+R_e)}$  - $G \frac{M_e}{(R+R_e)}$  

         Em = - ½ m $G \frac{M_e}{(R+R_e)}$  

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) $\sqrt{\frac{(R+R_e) }{G M_e } }$

        T = 2π $\sqrt{\frac{ (R+R_e)^3 }{G M_e } }$

Answer 2

(a) An expression for the speed of the satellite in its orbit.

$V=\sqrt{G\dfrac{M_e}{R+R_e}$

(b) An expression for the total mechanical energy of the satellite-Earth system in its orbit.

$E_m =\dfrac{1}{2}mG\dfrac{M_e}{(R+R_e)}$

(c) An expression for the period of the satellite’s orbit.

$T=2\pi\sqrt\dfrac{(R+R_e)^3}{GM_e}$

What are satellites?

A satellite is a moon, planet or machine that orbits a planet or star. For example, Earth is a satellite because it orbits the sun

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

F = ma

$ma =G\sqrt{\dfrac{mM_e}{(R+R_e)}$

the acceleration of the satellite is centripetal

$a=\dfrac{v^2}{R+R_e}$

we substitute

$G\dfrac{mM_e}{(R+R_e)}=m\dfrac{v^2}{(R+R_e)}$

$G\dfrac{M_e}{(R+R_e)}=v^2$

$v=\sqrt{G\dfrac{M_e}{(R+R_e)}$

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

        Em = K + U

$Em =\dfrac{1}{2}m v^2 - \dfrac{G m M} {(R + R_e)}$

we substitute the expression for the velocity

$E_m=\dfrac{1}{2}mG\dfrac{M_e}{(R+R_e)}-G\dfrac{M_e}{(R+R_e)}$

$E_m=-\dfrac{1}{2}G\dfrac{M_e}{(R+R_e)}$

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

$v=\dfrac{d}{t}$  

in a complete orbit the distance traveled of the circle is

$d = 2\pi (R + R_e)$

where time is called period

$v = 2\pi (R + R_e)$

$T = \dfrac{2\pi (R + R_e)} { v}$

we substitute the speed value

$T = 2\pi (R + R_e) . \sqrt{\dfrac{(R+R_e)}{GM_e}$

$T=2\pi\sqrt{\dfrac{(R+R_e)}{GM_e}$

(a) An expression for the speed of the satellite in its orbit.

$V=\sqrt{G\dfrac{M_e}{R+R_e}$

(b) An expression for the total mechanical energy of the satellite-Earth system in its orbit.

$E_m =\dfrac{1}{2}mG\dfrac{M_e}{(R+R_e)}$

(c) An expression for the period of the satellite’s orbit.

$T=2\pi\sqrt\dfrac{(R+R_e)^3}{GM_e}$

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