Question

The gravitational force between two objects is $\mathrm{F}$. If the masses of both objects are halved without changing the distance between them, then the gravitational force would become?

• A. $\frac{\mathrm{F}}{4}$
• B. $\frac{\mathrm{F}}{2}$
• C. $\mathrm{F}$
• D. $2\mathrm{F}$

Answer 1

The correct option is A

$\frac{\mathrm{F}}{4}$

The explanation for the correct answer:

Option (A): $\frac{\mathrm{F}}{4}$

1. The force of gravitation between two objects is given as : ${\mathrm{F}}_{\mathrm{G}}=\mathrm{G}\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{{\mathrm{r}}^2}$, where $F_G$ is the gravitational force, $m_1$ and $m_2$ are two masses, $r$ is the distance between two masses and $G$ is the gravitational constant.
2. Now, According to the question masses are halved, i.e ${\mathrm{m}}_1^{\text{'}}=\frac{{\mathrm{m}}_1}{2}$ and ${\mathrm{m}}_2^{\text{'}}=\frac{{\mathrm{m}}_2}{2}$ and other parameters remains same.
3. The equation for the force of gravitation can be re-written as: ${\mathrm{F}}_{{\mathrm{G}}^{\text{'}}}=\mathrm{G}\frac{{\mathrm{m}}_1^{\text{'}}{\mathrm{m}}_2^{\text{'}}}{{\mathrm{r}}^2}$; substituting the value of ${\mathrm{m}}_1^{\text{'}}\mathrm{and}{\mathrm{m}}_2^{\text{'}}$

$$\begin{gathered} \Rightarrow {\mathrm{F}}_{{\mathrm{G}}^{\text{'}}}=\mathrm{G}\frac{{\displaystyle \frac{{\mathrm{m}}_1}{2}}\times {\displaystyle \frac{{\mathrm{m}}_2}{2}}}{{\mathrm{r}}^2} \\ \Rightarrow {\mathrm{F}}_{{\mathrm{G}}^{\text{'}}}=\mathrm{G}\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{4{\mathrm{r}}^2}=\frac{1}{4}\mathrm{G}\frac{{\mathrm{m}}_1{\mathrm{m}}_2}{{\mathrm{r}}^2} \\ \Rightarrow {\mathrm{F}}_{{\mathrm{G}}^{\text{'}}}=\frac{1}{4}{\mathrm{F}}_{\mathrm{G}} \end{gathered}$$

Thus when masses are halved keeping other parameters the same the force of gravitation will be $\frac{1}{4}$ times the original value.

Hence, Option (A) is correct.