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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.
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Solution
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.
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