Question

Show that the weight of an object on moon is $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6^{th}$}\right.$ of its weight on earth.

Answer 1

Step 1: Assumptions

Suppose an object of mass ‘m’

Let its weight on the moon be ‘$W_m$’

Let the mass of moon $=M$

The radius of moon $=R$

Gravitational constant $=G$

The weight of the same object on the earth $\mathbf{=}{\mathit{W}}_{\mathbf{e}}$

Mass of earth ${\mathit{M}}_{\mathbf{e}}\mathbf{=}\mathbf{100}\mathit{M}$ (as mass of earth is $100$ times than that of moon)

Radius of earth ${\mathit{R}}_{\mathbf{e}}\mathbf{=}\mathbf{4}\mathit{R}$ (as radius of earth is $4$times than that of moon)

Step 2: comparing weights

Applying the universal law of gravitation, the weight of the object on the moon will be

Weight on moon, ${\mathit{W}}_{\mathbf{m}}\mathbf{=}\frac{\mathbf{G}\mathbf{\times }\mathbf{M}\mathbf{m}}{{\mathbf{R}}^{\mathbf{2}}}$……..(i)

Applying the universal law of gravitation the weight of the object on the earth will be

Weight on earth, we$$\begin{gathered} {\mathit{W}}_{\mathbf{e}}\mathbf{=}\frac{\mathbf{G}\mathbf{\times }{\mathbf{M}}_{\mathbf{e}}\mathbf{\times }\mathbf{m}}{{\mathbf{\left(}{\mathbf{R}}_{\mathbf{e}}\mathbf{\right)}}^{\mathbf{2}}} \\ {\mathit{W}}_{\mathbf{e}}\mathbf{=}\frac{\mathbf{G}\mathbf{\times }\mathbf{100}\mathbf{M}\mathbf{\times }\mathbf{m}}{{\mathbf{\left(}\mathbf{4}\mathbf{R}\mathbf{\right)}}^{\mathbf{2}}} \end{gathered}$$

${\mathit{W}}_{\mathbf{e}}\mathbf{=}\frac{\mathbf{G}\mathbf{\times }\mathbf{100}\mathbf{M}\mathbf{\times }\mathbf{m}}{\mathbf{16}{\mathbf{R}}^{\mathbf{2}}}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{\left(}\mathit{i}\mathit{i}\mathbf{\right)}$

Dividing equation (i) by equation (ii), we get

$\frac{{\mathbf{W}}_{\mathbf{m}}}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{{\displaystyle \frac{G\times M\times m}{R^2}}}{{\displaystyle \frac{G\times 100M\times m}{16R^2}}}$

$$\begin{gathered} \frac{{\mathbf{W}}_{\mathbf{m}}}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{\mathbf{G}\mathbf{\times }\mathbf{M}\mathbf{\times }\mathbf{m}\mathbf{\times }\mathbf{16}{\mathbf{R}}^{\mathbf{2}}}{{\mathbf{R}}^{\mathbf{2}}\mathbf{\times }\mathbf{G}\mathbf{\times }\mathbf{100}\mathbf{M}\mathbf{\times }\mathbf{m}} \\ \frac{{\mathbf{W}}_{\mathbf{m}}}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{\mathbf{16}}{\mathbf{100}} \\ \frac{{\mathbf{W}}_{\mathbf{m}}}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{6}} \\ \mathit{o}\mathit{r}\mathbf{}\frac{\mathbf{W}\mathbf{e}\mathbf{i}\mathbf{g}\mathbf{h}\mathbf{t}\mathbf{}\mathbf{o}\mathbf{n}\mathbf{}\mathbf{m}\mathbf{o}\mathbf{o}\mathbf{n}}{\mathbf{W}\mathbf{e}\mathbf{i}\mathbf{g}\mathbf{h}\mathbf{t}\mathbf{}\mathbf{o}\mathbf{n}\mathbf{}\mathbf{e}\mathbf{a}\mathbf{r}\mathbf{t}\mathbf{h}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{6}} \end{gathered}$$

Weight on moon$\mathbf{=}\frac{\mathbf{1}}{\mathbf{6}}weightonearth$