Physics - Gravitation Question with Solution | TestHub

For the planet to orbit around the star, the centripetal force must be provided by gravitational force. Hence, ${\mathrm{F}}_{\mathrm{d}}={\mathrm{F}}_{\mathrm{a}}$. ${\mathrm{F}}_{\mathrm{a}}\propto -{\mathrm{r}}^{-7/2}$ ...(Given) (-ve sign indicates force is towards the centre of orbit) $$\begin{array}{ll}& \text{ Hence, }\mathrm{a}\propto -{\mathrm{r}}^{-7/2}\\ \therefore & -{\omega }^2\mathrm{r}\propto -{\mathrm{r}}^{-7/2}\\ \therefore & {\omega }^2\propto {\mathrm{r}}^{-9/2}\\ \therefore & \frac{4{\pi }^2}{{\mathrm{T}}^2}\propto {\mathrm{r}}^{-9/2}\\ & \Rightarrow {\mathrm{T}}^2\propto {\mathrm{r}}^{9/2}\end{array}$$