Question

A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let r be the distance of the body from the centre of the star and let its linear velocity be v, angular velocity ω, kinetic energy K, gravitational potential energy U, total energy E and angular momentum l. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease.

Answer 1

The situation is shown in the diagram, where a body of mass m is revolving around a star of mass M.

Linear velocity of the body `v = sqrt((GM)/r)`

⇒ `v ∝ 1/sqrt(r)`

Therefore, when r increases, v decreases.

Angular velocity of the body `ω = (2π)/T`

According to Kepler's law of the period,

`T^2 ∝ r^3` ⇒ `T = kr^(3/2)`

Where k is a constant

∴  `ω = (2π)/(kr^(3/2))` ⇒ `ω ∝ 1/r^(3/2)`  .....`(∵ ω = (2π)/T)`

 Therefore, when r increases, ω decreases.

The kinetic energy of the body,

K = `1/2 mv^2`

= `1/2 m xx (GM)/r`

= `(GMm)/(2r)`

∴ `K ∝ 1/r`

Therefore, when r increases, KE decreases.

The gravitational potential energy of the body,

`U = - (GMm)/r` ⇒ `U ∝ 1/r`

Therefore, when r increases, PE becomes less negative i.e., increases.

Total energy of the body,

`E = KE + PE`

= `(GMm)/(2r) + (- (GMm)/r)` 

= `- (GMm)/(2r)`

Therefore, when r increases, total energy becomes less negative, i.e., increases.

Angular momentum of the body,

`L = mvr`

= `mr sqrt((GM)/r)`

= `msqrt(GMr)` 

∴ `L ∝ sqrt(r)`

Therefore, when r increases, angular momentum L increases.