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Question

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

A
$$T^2/R^3$$ is a constant
B
$$T/R$$ is a constant
C
$$T/R^2$$ is a constant
D
$$TR$$ is a constant
Solution
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Correct option is D. $$T/R$$ is a constant
$$m=\int^R_0\rho4\pi r^2dr$$
$$m=4\pi kR$$
$$v\propto \sqrt {4\pi k}$$
$$\dfrac{T}{R}=\dfrac{2\pi}{\sqrt{4\pi k}}$$
s0 $$\dfrac{T}{R}$$ is a constant.

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