Question

Hey ppl!

Give me the expression for total energy of a orbiting satellite ​

Answer 1

$\sf{\huge{\underline {GRAVITATION}}}$ :

$\sf{\large {\underline {Total\:Energy \:of \:An \:Orbiting \: Satellite}}}$ :

Consider a satellite of mass $\tt{m}$ , moving with velocity $\tt{v₀}$ in an orbit of radius $\tt{r}$. Because of gravitational pull of the earth, the satellite has $\tt{Potential\:Energy}$ which is given by

U = - $\frac{GMm} {r}$

The $\tt{Kinetic \:Energy}$ of a satellite due to its orbital motion

K = $\frac{1}{2}$mv₀^2

K = $\frac{1}{2}$m$\frac{(GM)}{r}$

Total energy of the satellite is

E = U + K

E = - $\frac{GMm} {r}$ + $\frac{1}{2}$m$\frac{(GM)}{r}$

E = - $\frac{GMm} {2r}$

Answer 2

The kinetic energy of an object in orbit can easily be found from the following equations:

Centripetal force on a satellite of mass m moving at velocity v in an orbit of radius r = mv2/r  

But this is equal to the gravitational force (F) between the planet (mass M) and the satellite:  

F =GMm/r2 and so mv2 = GMm/r

But kinetic energy = ½mv2 and so:

kinetic energy of the satellite = ½ GMm/r

Kinetic energy in orbit = ½ mv2 = + ½GMm/r

All satellites have to be given a tangential velocity (v) to maintain their orbit position and this process is called orbit injection.  

Total energy of satellite in orbit = -GMm/2r

However the total energy INPUT required to put a satellite into an orbit of radius r around a planet of mass M and radius R is therefore the sum of the gravitational potential energy (GMm[1/R-1/r]) and the kinetic energy of the satellite ( ½GMm/r).