Question

a satellite of mass m is put in the circular orbit of height h from the surface of earth then the orbital velocity of satellite is

Answer 1

Step 1: Given that:

Mass of satellite = m

The height of the satellite from the surface of the earth = h

Step 2: Determination of the orbital velocity of satellite:

Let the orbital velocity of the satellite be $v_o$.

Now, if $M_e$ be the mass of earth and $R_e$ be the radius of earth then, the gravitational force between the earth and the satellite will be;

$F=G\frac{M_{e}m}{{(R_e+h)}^2}$ .........(1)

Where $R_e+h$ will be the distance between the centre of satellite and the centre of the earth.

Now, we know that for a body to move in a circular orbit, a centripetal force is required.

The centripetal force on the body is given by;

$F=\frac{m{v}^2}{r}$

Where; m is the mass of the body moving on circular path of radius r and v is the velocity of the body.

For satellite;

$F=\frac{m{v}_o^2}{(R_e+h)}$ ............(2)

From equations (1) and (2), we get;

$G\frac{M_{e}m}{{(R_e+h)}^2}=\frac{m{v}_o^2}{(R_e+h)}$

$\frac{m{v}_o^2}{(R_e+h)}=G\frac{M_{e}m}{{(R_e+h)}^2}$

$v_o^2=G\frac{M_e}{(R_e+h)}$

$v_o=\sqrt{\frac{G{M}_e}{(R_e+h)}}$

Thus,

The orbital velocity of the satellite will be $\sqrt{\frac{G{M}_e}{(R_e+h)}}$ .