Question

Show that angular momentum of a satellite of mass '
$m _{s}$
' revolving around earth of mass '
$m _{e}$
' in an orbit of radius '
$r$
' is equal to '
$\sqrt{gm_{e}{m_{s}}^{2}r}$
'.

​

Answer 1

Given :

Mass of satellite = $\sf{m_s}$

Mass of earth = $\sf{m_e}$

Radius of orbit = r

To Prove :

Angular momentum of satellite is

• $\sf L=\sqrt{Gm_em_s^2r}$

Solution :

❖ Orbital velocity is the velocity at which object moves around celestial body.

Orbital velocity of satellite of mass $\sf{m_s}$ is given by

$\dag\:\underline{\boxed{\bf{\gray{v_o=\sqrt{\dfrac{Gm_e}{r}}}}}}$

• G denotes gravitational constant
• $\sf{m_e}$ denotes mass of earth
• r denotes radius of orbit

♦ Angular momentum of satellite is given by

$\dag\:\underline{\boxed{\bf{\orange{L=m_sv_or}}}}$

By substituting the given values;

$\sf:\implies\:L=m_s\times\sqrt{\dfrac{Gm_e}{r}}\times r$

$\sf:\implies\:L=\sqrt{\dfrac{Gm_e}{r}\times m_s^2\times r^2}$

$:\implies\:\underline{\boxed{\bf{\red{L=\sqrt{Gm_em_s^2r}}}}}$

Hence proved!