Question

Prove that the weight of an object on the moon is one-sixth of its weight on the earth.

Answer 1

Step 1: Values of mass and radius of earth and moon.

Mass of earth is ${\mathbf{m}}_{\mathbf{e}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{24}}\mathit{k}\mathit{g}$

Mass of moon is ${\mathbf{m}}_{\mathbf{m}\mathbf{}}\mathbf{=}\mathbf{}\mathbf{7}\mathbf{.}\mathbf{36}\mathbf{\times }{\mathbf{10}}^{\mathbf{22}}\mathit{k}\mathit{g}$

Radius of earth is ${\mathbf{R}}_{\mathbf{e}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{6}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{m}$

Radius of moon is ${\mathbf{R}}_{\mathbf{m}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{74}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{m}$

Step 2: Acceleration due to gravity on earth

Given by,

${\mathbf{g}}_{\mathbf{e}}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{G}{\mathbf{m}}_{\mathbf{e}}}{{{\mathbf{R}}^{\mathbf{2}}}_{\mathbf{e}}}$

$\mathbf{=}\mathbf{}\frac{\mathbf{6}\mathbf{.}\mathbf{67}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\mathbf{\times }\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{24}}}{{\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{)}}^{\mathbf{2}}}$

$\mathbf{=}\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{}\mathit{m}{\mathit{s}}^{\mathbf{-}\mathbf{2}}$

Step 3: Acceleration due to gravity on moon

Given by,

${\mathbf{g}}_{\mathbf{m}\mathbf{}}\mathbf{=}\mathbf{}\frac{\mathbf{G}{\mathbf{m}}_{\mathbf{m}}}{{{\mathbf{R}}^{\mathbf{2}}}_{\mathbf{m}}}$

$\mathbf{=}\mathbf{}\frac{\mathbf{6}\mathbf{.}\mathbf{67}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{11}}\mathbf{\times }\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{24}}}{{\mathbf{(}\mathbf{6}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{)}}^{\mathbf{2}}}$

$\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{}\mathit{m}{\mathit{s}}^{\mathbf{-}\mathbf{2}}$

Step 4: Assume the mass of the body be m.

Then,

The weight of the body on earth is = $\mathit{m}{\mathit{g}}_{\mathbf{e}}$

The weight of the body on the moon is = $\mathit{m}{\mathit{g}}_{\mathbf{m}}$

Therefore,

The ratio of the weight of the body on the moon and earth is $\mathbf{=}\mathbf{}\frac{\mathbf{m}{\mathbf{g}}_{\mathbf{m}}}{\mathbf{m}{\mathbf{g}}_{\mathbf{e}}}$

$\mathbf{=}\mathbf{}\frac{\mathbf{1}\mathbf{.}\mathbf{6}\mathbf{}\mathbf{m}{\mathbf{s}}^{\mathbf{-}\mathbf{2}}}{\mathbf{9}\mathbf{.}\mathbf{8}\mathbf{}\mathbf{m}{\mathbf{s}}^{\mathbf{-}\mathbf{2}}}$

$\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{6}}$

Hence, Weight on the moon is $\frac{\mathbf{1}}{\mathbf{6}}$weight on earth.