# A test particle is moving in a circular orbit in a gravitational field produced by a mass density $\rho (r)=\frac{K}{r^{2}}$ . Identity the correct relation between the radius $R$ of the particles orbit and its period $T$ : Option 1) $T/R\:\:is\:a \:constant$ Option 2) $T^{2}/R^{3}\:\:is\:a \:constant$ Option 3) $T/R^{2}\:\:is\:a \:constant$ Option 4) $TR\:\:is\:a \:constant$

Given

$\rho =\frac{K}{r^{2}}$

$E=\frac{G}{R^2}\int_{0}^{R}\frac{K}{r^{2}}.4\pi r^{2}dr=\frac{GK4\pi}{R}=\frac{C}{R}$

Now

$\frac{C}{R}m=mw^{2}R=m\left ( \frac{2\pi}{T} \right )^{2}R$

$\Rightarrow\frac{T}{R}=constant$

Option 1)

$T/R\:\:is\:a \:constant$

Option 2)

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

Option 3)

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

Option 4)

$TR\:\:is\:a \:constant$

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