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7.Gravitation
normal
Suppose the gravitational force varies inversely as the $n^{th}$ power of the distance. Then, the time period of a planet in circular orbit of radius $R$ around the sun will be proportional to
A
$R^n$
B
${R^{\left( {n + 1} \right)/2}}$
C
${R^{\left( {n - 1} \right)/2}}$
D
$R^{-n}$
Solution
$\mathrm{T}=\frac{2 \pi \mathrm{r}}{\mathrm{v}}$
and $\quad \frac{\mathrm{mv}^{2}}{\mathrm{r}}=\frac{\mathrm{k}}{\mathrm{r}^{\mathrm{n}}} \Rightarrow \mathrm{v}=\frac{\mathrm{A}}{\mathrm{r}^{\frac{\mathrm{n}-1}{2}}}$
$\therefore \mathrm{T}=\frac{2 \pi \mathrm{r}}{\mathrm{A}} \mathrm{r}^{\frac{\mathrm{n}-1}{2}}=\frac{2 \pi}{\mathrm{A}} \mathrm{r}^{\frac{\mathrm{n}+1}{2}} \Rightarrow \mathrm{T} \propto \mathrm{r}^{\frac{\mathrm{n}+1}{2}}$
Standard 11
Physics
