Question

show that the weight of an object on the moon is 1/6 of its weight on the earth

Answer 1

The force acting on a particle by the planet is it's weight.

So, The force of attraction between Earth and the object is the weight of it on earth.

Let W be the weight of the object on Earth.

So,

$W = \frac{ GM_{e}m} { \: R_{e} ^2}$

We know that,

Mass of earth ≈ 5.98 * 10^24Kg
Radius of earth ≈ 6.37 * 10^6m.

The equation becomes,
W = G(5.98 * 10^24) *m / (6.37 * 10^6)²

Consider w is the weight of the object on moon.

So,

$w = \frac{ GM_{moon}m} { \: R_{moon} ^2}$
We know that,
Mass of the moon ≈ 7.36 * 10^22 Kg
Radius of moon ≈ 1.74 * 10^6 m.

So, w = G(7.36 * 10^22)m / (1.74 * 10^4)²

Taking ratio of W & w.

$\large{\frac{W}{w} = \frac{ \frac{ (5.98 \times 10^{24})} {(6.37 * 10^{6})^{2}} } { \frac{ 7.36 \times 10^{22}} {( 1.74 * 10^{6})^{2} } }}$

= ( 1.474 * 10^11) / 2.431 * 10^10

≈ 6 .

So, W = 6w

w = 1/6 W.

Therefore , Weight of an object on the moon is 1/6 of its weight on the earth.

Answer 2

$\huge \red{Hello \: Friends}$

$\green{Weight : }$

The weight of an object is defined as the force with which the earth or any other planet,stars,moons etc. attracts the object . Weight of an object keeps changing according in change of the value of gravity. The weight of an object on moon is 1/6 that on the earth .

$\green{Proof}$

Suppose the weight of an object :

Wₑ = On earth

Wₘ = on moon

$\bf{For \: earth}$

F = Wₑ = G mₑm/R²e

Equation becomes,

$We \: = \frac{GMem}{R {}^{2}e } ...........(1)$

$\bf{For \: moon : }$

Wₘ = Gmₘm/R²ₘ

Equation becomes,

$Wm = \frac{GMmm}{R {}^{2} m} .............(2)$

Now, divide equation (1) and (2)

$\frac{We = \frac{GMem}{R {}^{2} e} }{Wm = \frac{GMmm}{R {}^{2}m } }$

Constant G and mass(m) got cancelled
we get,

$\frac{We \: = \ mer {}^{2}e }{Wm \: = mmr {}^{2}m }$

So, we got a ratio wich show the weight on earth or weight on moon.

Now, put the values of mass of earth and radius of earth similarly, mass of moon and radius of moon. we get,

$\frac{We}{Wm} = \frac{6}{1}$

Assume that the weight of an object on earth is 60 kg. So the weight of that object on moon will be ,

$\frac{We}{Wm} = \frac{6}{1}$

$\frac{60}{Wm} = \frac{6}{1}$

After cross multiplication we get,

$6Wm \: = 60$

$Wm \: = 6$

Therefore, we can say that the weight of an object on moon is 1/6 that on the earth .

$\boxed{ \boxed{hence \: proved}}$