Question

What happens to the force between two objects if : (a) The mass of one object is doubled? (b) The mass of both the objects are doubled? (c) The distance between the objects is doubled and tripled?

Answer 1

Universal law of gravitation

According to the universal law of gravitation, Every body on earth attracts every other body with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

It is given by:

$\mathit{F}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}}$

$F=$Force of the gravitation

$G=$Gravitational constant

$M=$mass of object $1$

$m=$mass of object $2$

$d=$distance between object $1$and object $2$

(ii) The mass of one object is doubled

$\mathit{F}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}}$

$\mathit{m}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathit{m}$

$\mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{\text{'}}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}}$

Substituting the value in the above formula

$\mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{2}\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}}$

$\mathit{F}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathit{F}$

If the mass of one object is doubled then force is also doubled.

(ii) The masses of both objects are doubled

$$\begin{gathered} F=\frac{GmM}{d^2} \\ m\text{'}=2m \\ M\text{'}=2M \end{gathered}$$

$\mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{\text{'}}\mathbf{M}\mathbf{\text{'}}}{{\mathbf{d}}^{\mathbf{2}}}$

Substituting the value in the above formula

$$\begin{gathered} \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{G}\mathbf{\left(}\mathbf{2}\mathbf{m}\mathbf{\right)}\mathbf{\left(}\mathbf{2}\mathbf{M}\mathbf{\right)}}{{\mathbf{d}}^{\mathbf{2}}} \\ \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{4}\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}} \\ \mathit{F}\mathbf{\text{'}}\mathbf{=}\mathbf{4}\mathit{F} \end{gathered}$$

If the masses of both objects are doubled, then the force will be four times.

(iii) The distance between the objects is doubled:

$\mathit{F}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}}$

if the distance is doubled then

$\mathit{d}\mathbf{\text{'}}\mathbf{=}\mathbf{2}\mathit{d}$

$\mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{\mathbf{d}{\mathbf{\text{'}}}^{\mathbf{2}}}$

$$\begin{gathered} \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{\left(}\mathbf{2}\mathbf{d}\mathbf{\right)}}^{\mathbf{2}}} \\ \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}} \\ \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}}\mathit{F} \end{gathered}$$

If the distance is doubled then force becomes one-fourth.

If the distance is tripled then:

$$\begin{gathered} d\text{'}=3d \\ \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{\text{'}}\mathbf{2}}} \end{gathered}$$

Substituting the values in the above formula

$$\begin{gathered} \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{\left(}\mathbf{3}\mathbf{d}\mathbf{\right)}}^{\mathbf{2}}} \\ \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{9}}\frac{\mathbf{G}\mathbf{m}\mathbf{M}}{{\mathbf{d}}^{\mathbf{2}}} \\ \mathit{F}\mathbf{\text{'}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{9}}\mathit{F} \end{gathered}$$

If the distance is tripled force thus becomes one-ninth.