Question

The gravitational force between two bodies is $\text{'}\mathrm{F}\text{'}$. If the mass of each body is doubled and the distance between them is halved then find the new force between them.

Answer 1

Step-1: The Given data

The masses of the object are doubled. i.e. ${\mathrm{m}}_1^{\text{'}}=2{\mathrm{m}}_1$ and ${\mathrm{m}}_2^{\text{'}}=2{\mathrm{m}}_2$. and the distance between them is halved, i.e. ${\mathrm{r}}^{\text{'}}=\frac{\mathrm{r}}{2}$

Step 2: Formula used

The force of gravitation acting between two objects is given as: ${\mathrm{F}}_{\mathrm{G}}=\mathrm{G}\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{{\mathrm{r}}^2}$; here ${\mathrm{m}}_1\mathrm{and}{\mathrm{m}}_2$ are the masses of both objects and $\mathrm{r}$ is the distance between them

Step-3: Calculate the new force of gravitation $\left({\mathrm{F}}_{\mathrm{G}}^{\text{'}}\right)$.

The equation of gravitation can be rewritten as: ${\mathrm{F}}_{\mathrm{G}}^{\text{'}}=\mathrm{G}\frac{{\mathrm{m}}_1^{\text{'}}{\mathrm{m}}_2^{\text{'}}}{{\left({\mathrm{r}}^{\text{'}}\right)}^2}$

Substituting the value of ${\mathrm{m}}_1^{\text{'}},{\mathrm{m}}_2^{\text{'}}\mathrm{and}{\mathrm{r}}^{\text{'}}$

$$\begin{gathered} \Rightarrow {\mathrm{F}}_{\mathrm{G}}^{\text{'}}=\mathrm{G}\frac{2{\mathrm{m}}_1\times 2{\mathrm{m}}_2}{{\left({\displaystyle \frac{\mathrm{r}}{2}}\right)}^2} \\ \Rightarrow {\mathrm{F}}_{\mathrm{G}}^{\text{'}}=\mathrm{G}\frac{4{\mathrm{m}}_1{\mathrm{m}}_2}{{\displaystyle \frac{{\mathrm{r}}^2}{4}}} \\ \Rightarrow {\mathrm{F}}_{\mathrm{G}}^{\text{'}}=\mathrm{G}\frac{16{\mathrm{m}}_1{\mathrm{m}}_2}{{\mathrm{r}}^2} \end{gathered}$$

$$\begin{gathered} \Rightarrow {\mathrm{F}}_{\mathrm{G}}^{\text{'}}=16\mathrm{G}\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{{\mathrm{r}}^2} \\ \Rightarrow {\mathrm{F}}_{\mathrm{G}}^{\text{'}}=16{\mathrm{F}}_{\mathrm{G}} \end{gathered}$$

Hence, the force of gravitation will becomes $16$ times of the original force when the mass of each body is doubled and the distance between them is halved.