A test particle is moving in circular orbit in the gravitational field

Question:

A test particle is moving in circular orbit in the gravitational field produced by a mass density $r(r)=\frac{\mathrm{K}}{r^{2}}$. Identify the

correct relation between the radius $R$ of the particle's orbit and its period $\mathrm{T}$ :

  1. (1) $\mathrm{T} / \mathrm{R}$ is a constant

  2. (2) $\mathrm{T}^{2} / \mathrm{R}^{3}$ is a constant

  3. (3) $\mathrm{T} / \mathrm{R}^{2}$ is a constant

  4. (4) TR is a constant


Correct Option: 1

Solution:

(1) $F=\frac{G M m}{r}=\int a \frac{\rho(d V) m}{r^{2}}$

$=m G \int_{0}^{R} \frac{k}{r^{2}} \frac{4 \pi r^{2} d r}{r^{2}}$

$=-4 \pi k G m\left(\frac{1}{r}\right)_{0}^{R}$

$=-\frac{4 \pi k G m}{R}$

Using Newton's second law, we have

$\frac{m v_{0}^{2}}{R}=\frac{4 \pi k G m}{R}$

or $v_{0}=\mathrm{C}$ (const.)

Time period, $T=\frac{2 \pi R}{v_{0}}=\frac{2 \pi R}{C}$

or $\frac{T}{R}=$ cons

$\rightarrow \quad \rightarrow \quad \rightarrow \quad \rightarrow$

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