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Question

A test particle is moving in a circular orbit in the gravitational field produced by a mass density $ρ (r) = \frac{K}{r ^{2}}$ . Identify the correct relation between the radius R of the particle's orbit and its period T.

A

$T / R^{2}$ is a constant

B

$T R$ is a constant

C

$T^{2} / R^{3}$ is a constant

D

$T / R$ is a constant

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Solution

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Correct option is D)

$m = \int_{0}^{R} ρ 4 π r^{2} d r$
$m = 4 π k R$
$v \propto 4 π k$
$\frac{T}{R} = \frac{2 π}{4 π k}$
s0 $\frac{T}{R}$ is a constant.

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