Question

Consider a heavily body which has a mass twice that of the earth and a radius thrice that of the earth. What will be the weight of the book on this heavily body, if its weight on earth is 900 N?

Answer 1

Step 1: Given data

Weight of book on earth,${\mathit{W}}_{\mathbf{e}}\mathbf{=}\mathbf{900}\mathit{N}$

Assuming Mass of heavily body = 2(Mass of earth)

${\mathit{M}}_{\mathbf{h}}\mathbf{=}\mathbf{2}{\mathit{M}}_{\mathbf{e}}$

Assuming Radius of heavily body = 3(Radius of the earth)

${\mathit{R}}_{\mathbf{h}}\mathbf{=}\mathbf{3}{\mathit{R}}_{\mathbf{e}}$

Step 2: Finding the weight of a heavy body

Weight of heavily body and weight of the earth is given by:

$$\begin{gathered} {\mathit{W}}_{\mathbf{e}}\mathbf{=}\frac{\mathbf{G}{\mathbf{M}}_{\mathbf{e}}\mathbf{m}}{{\mathbf{R}}_{\mathbf{e}}^{\mathbf{2}}} \\ {\mathit{W}}_{\mathbf{h}}\mathbf{=}\frac{\mathbf{G}{\mathbf{M}}_{\mathbf{h}}\mathbf{m}}{{\mathbf{R}}_{\mathbf{h}}^{\mathbf{2}}} \end{gathered}$$

$\frac{{\mathbf{W}}_h}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{{\displaystyle \frac{\mathbf{G}{\mathbf{M}}_{\mathbf{h}}\mathbf{m}}{{\mathbf{R}}_{\mathbf{h}}^{\mathbf{2}}}}}{\frac{{\mathbf{GM}}_{\mathbf{e}}\mathbf{m}}{{\mathbf{R}}_{\mathbf{e}}^{\mathbf{2}}}}$

$$\begin{gathered} \frac{{\mathbf{W}}_{\mathit{h}}}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{{\mathbf{M}}_{\mathbf{h}}}{{\mathbf{M}}_{\mathbf{e}}}\mathbf{\times }\frac{{\mathbf{R}}_{\mathbf{e}}^{\mathbf{2}}}{{\mathbf{R}}_{\mathbf{h}}^{\mathbf{2}}} \\ \frac{{\mathbf{W}}_{\mathbf{h}}}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{\mathbf{2}\mathbf{Me}}{{\mathbf{M}}_{\mathbf{e}}}\mathbf{\times }\frac{{\mathbf{R}}_{\mathbf{e}}^{\mathbf{2}}}{{\mathbf{\left(}\mathbf{3}{\mathbf{R}}_{\mathbf{e}}\mathbf{\right)}}^{\mathbf{2}}} \\ \frac{{\mathbf{W}}_{\mathbf{h}}}{{\mathbf{W}}_{\mathbf{e}}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{9}} \\ {\mathit{W}}_{\mathbf{h}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{9}}\mathbf{\times }{\mathit{W}}_{\mathbf{e}} \\ {\mathit{W}}_{\mathbf{h}}\mathbf{=}\frac{\mathbf{2}}{\mathbf{9}}\mathbf{\times }\mathbf{900} \\ {\mathit{W}}_{\mathbf{h}}\mathbf{=}\mathbf{200}\mathbf{}\mathit{N} \end{gathered}$$

Hence, the weight of the book on this heavy body is 200 N.