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Two particles of identical mass are moving in circular orbits under a potential given by $\mathrm{V}(\mathrm{r})=\mathrm{Kr}^{\mathrm{n}}$, where $\mathrm{K}$ is a constant. If the radii of their orbits are $r_{1}, r_{2}$ and their speeds are $v_{1}, v_{2}$, respectively, then
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The correct answer is:
$v_{1}^{2} r_{1}{ }^{\mathrm{n}}=v_{2}{ }^{2} \mathrm{r}_{2}{ }^{\mathrm{n}}$
$\begin{array}{l}
\mathrm{V}(\mathrm{r})=\mathrm{Kr}^{-\mathrm{n}} \\
\text { gravitational field }=\mathrm{E}=-\frac{\mathrm{d} \mathrm{V}}{\mathrm{dr}} \\
=(-\mathrm{K}) \frac{\mathrm{d}}{\mathrm{dr}}\left(\mathrm{r}^{-\mathrm{n}}\right) \\
=(-\mathrm{K})(-\mathrm{n}) \mathrm{r}^{-\mathrm{n}-1} \\
=\frac{\mathrm{Kn}}{\mathrm{r}^{\mathrm{n}+1}}
\end{array}$
force on mass $=\mathrm{E} \times \mathrm{M}$, where $\mathrm{M}=$ mass of body
$\begin{array}{l}
\therefore \quad M E_{1}=\frac{M V_{1}^{2}}{r_{1}} \quad M E_{2}=\frac{M V_{2}^{2}}{r_{2}} \\
\therefore \frac{V_{1}^{2}}{V_{2}^{2}}=\frac{r_{1} E_{1}}{r_{2} E_{2}} \\
\Rightarrow \frac{V_{1}^{2}}{V_{2}^{2}}=\frac{r_{1}}{r_{2}} \frac{K n}{r_{1}^{n+1}} \frac{r_{2}^{n+1}}{K n} \\
\Rightarrow \frac{V_{1}^{2}}{V_{2}^{2}}=\frac{r_{2}^{n}}{r_{1}^{n}} \\
\Rightarrow V_{1}^{2} r_{1}^{n}=V_{2}^{2} r_{2}^{n}
\end{array}$
\mathrm{V}(\mathrm{r})=\mathrm{Kr}^{-\mathrm{n}} \\
\text { gravitational field }=\mathrm{E}=-\frac{\mathrm{d} \mathrm{V}}{\mathrm{dr}} \\
=(-\mathrm{K}) \frac{\mathrm{d}}{\mathrm{dr}}\left(\mathrm{r}^{-\mathrm{n}}\right) \\
=(-\mathrm{K})(-\mathrm{n}) \mathrm{r}^{-\mathrm{n}-1} \\
=\frac{\mathrm{Kn}}{\mathrm{r}^{\mathrm{n}+1}}
\end{array}$
force on mass $=\mathrm{E} \times \mathrm{M}$, where $\mathrm{M}=$ mass of body
$\begin{array}{l}
\therefore \quad M E_{1}=\frac{M V_{1}^{2}}{r_{1}} \quad M E_{2}=\frac{M V_{2}^{2}}{r_{2}} \\
\therefore \frac{V_{1}^{2}}{V_{2}^{2}}=\frac{r_{1} E_{1}}{r_{2} E_{2}} \\
\Rightarrow \frac{V_{1}^{2}}{V_{2}^{2}}=\frac{r_{1}}{r_{2}} \frac{K n}{r_{1}^{n+1}} \frac{r_{2}^{n+1}}{K n} \\
\Rightarrow \frac{V_{1}^{2}}{V_{2}^{2}}=\frac{r_{2}^{n}}{r_{1}^{n}} \\
\Rightarrow V_{1}^{2} r_{1}^{n}=V_{2}^{2} r_{2}^{n}
\end{array}$
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