Question

answer the question in the attachment​

Answer 1

Question :

A satellite of mass m is orbiting the earth in a circular orbit of radius R.

Statement I : Angular momentum of the satellite varies as 1/√R

Statement II : Linear momentum of the satellite varies as 1/R

Statement III : Kinetic energy varies as 1/R

To Find :

Which of the statements is/are correct.

Solution :

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

Orbital velocity of satellite is given by

$\bigstar\:\underline{\boxed{\bf{\gray{v_o=\sqrt{\dfrac{GM}{R}}}}}}$

• G denotes gravitational constant
• M denotes mass of earth
• R denotes radius of orbit

A] Angular momentum :

$\sf:\implies\:L=mv_oR$

• m denotes mass of satellite

$\sf:\implies\:L=m\sqrt{\dfrac{GM}{R}}R$

$\sf:\implies\:L=m\sqrt{GMR}$

$\bf:\implies\:L\propto\sqrt{R}$

∴ First statement is incorrect.

B] Linear momentum :

$\sf:\implies\:p=mv_o$

$\sf:\implies\:p=m\sqrt{\dfrac{GM}{R}}$

$\bf:\implies\:p\propto\dfrac{1}{\sqrt{R}}$

∴ Second statement is incorrect.

C] Kinetic energy :

$\sf:\implies\:K=\dfrac{1}{2}mv_o^2$

$\sf:\implies\:K=\dfrac{1}{2}m\left(\sqrt{\dfrac{GM}{R}}\right)^2$

$\sf:\implies\:K=\dfrac{GMm}{2R}$

$\bf:\implies\:K\propto\dfrac{1}{R}$

∴ Third statement is correct.

Option - D is the correct answer!