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Universal Law of Gravitation

A body weighs...

Question

A body weighs 160 N on the earth. Find its weight on the other planet whose:
(a) mass is $\frac{1}{4}$ times the mass of earth and radius $\frac{1}{3}$ times that of the earth.

(b) mass is $\frac{5}{2}$ times the mass of earth and radius $\frac{4}{5}$ times that of the earth.

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Solution

Given $M_{p l a n e t} = \frac{1}{4} M_{e a r t h}$ and $R_{p l a n e t} = \frac{4}{3} R_{e a r t h},$
(a) $W = 160 N = m g_{e a r t h}$
$g_{p l a n e t} = \frac{G (\frac{1}{4} M_{e a r t h})}{(\frac{1}{3} R_{e a r t h})^{2}}$

$= \frac{9}{4} \frac{G M_{e a r t h}}{R_{e a r t h}^{2}} = \frac{9}{4} g_{e a r t h}$
$W_{p l a n e t} = m g_{p l a n e t}$
$= m (\frac{9}{4} g_{e a r t h})$
$= \frac{9}{4} \times 160 = 360 N$
(b) Given $M_{p l a n e t} = \frac{5}{2} M_{e a r t h}$ and $R_{p l a n e t} = \frac{4}{5} R_{e a r t h},$
$W = 160 N = m g_{e a r t h}$
$g_{p l a n e t} = \frac{G (\frac{5}{2} M_{e a r t h})}{(\frac{4}{5} R_{e a r t h})^{2}}$

$= \frac{125 G M_{e a r t h}}{32 R_{e a r t h}^{2}} = \frac{125}{32} g_{e a r t h}$
$W_{p l a n e t} = m g_{p l a n e t}$
$= m (\frac{125}{32} g_{e a r t h})$
$= \frac{125}{32} \times 160 = 625 N$

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