The magnitude of gravitational field at distance ' $r_1$ ' and ' $r_2$ ' from the centre of a uniform sphere…

The magnitude of gravitational field at distance ' $r_1$ ' and ' $r_2$ ' from the centre of a uniform sphere of radius ' $R$ ' and mass ' $M$ ' are ' $\mathrm{F}_1$ ' and ' $\mathrm{F}_2$ ' respectively. The ratio ' $\left(F_1 / F_2\right)$ ' will be (if $r_1 \gt R$ and $r_2 \lt R$)
  1. $\frac{\mathrm{R}^2}{\mathrm{r}_1 \mathrm{r}_2}$
  2. $\frac{\mathrm{R}^3}{\mathrm{r}_1 \mathrm{r}_2^2}$
  3. $\frac{\mathrm{R}^3}{\mathrm{r}_1^2 \mathrm{r}_2}$
  4. $\frac{R^4}{r_1^2 r_2^2}$

Solution

For $r \gt R, F_1=\frac{G M}{r_1^2}$ $\begin{aligned} & \text {For } r \lt R, F_2=\frac{\mathrm{GMr}_2}{\mathrm{R}^3} \\ \therefore \quad & \frac{\mathrm{~F}_1}{\mathrm{~F}_2}=\frac{\mathrm{GM}}{\mathrm{r}_1^2} \times \frac{\mathrm{R}^3}{\mathrm{GMr}_2}=\frac{\mathrm{R}^3}{\mathrm{r}_1^2 \mathrm{r}_2}\end{aligned}$

Asked in: MHT CET 2024 (02 May Shift 1)

Practice more Gravitation questions on Aicharya