The acceleration due to gravity on the surface of the moon is 1.62 ms⁻². The radius of the moon is 1740 km. Find the mass of the moon. Ans. (7.35x10²² kg)


Given:  

Value of  Gravitational  acceleration on the surface of the Moon = gₘ =  1.62 m s⁻² 

Radius of the Moon = Rₘ = 1740 Km = 1.740 ｘ 10³ｘ 10³ m = 1.740 ｘ 10⁶ m
Gravitational Constant = G = 6.67 ｘ 10⁻¹¹N m² Kg⁻²

To Find:
Mass of the Moon = Mₘ = ? kg

Solution:   

Let m be the mass of the body on the surface of the Moon and Mₘ is the mass of the Moon. The distance between their centres is equal to the radius Rₘ of the Moon and  then According to the Law of Gravitation we have 

F = `\frac {G m Mₘ}{Rₘ^2 }`   -----eqn (1)

But according to the 2nd law of Newton

F = W = mgₘ ----eqn (2)

By comparing these two Eqns 1 & 2 we have 

mgₘ = `\frac {G m Mₘ}{R^2 }`

m will cancel from both sides hence 

gₘ = `\frac {G Mₘ}{Rₘ^2 }`

or

Mₘ = `\frac {gₘ Rₘ^2}{ G }`

By putting values

Mₘ = `\frac {1.62 m s⁻²ｘ (1.740 ｘ 10⁶ m)^2}{6.67 ｘ 10⁻¹¹N m² Kg⁻² }`

Mₘ = `\frac {1.62 m s⁻²ｘ 3.0276 ｘ 10¹² m² }{6.67 ｘ 10⁻¹¹N m² Kg⁻² }`

Mₘ = `\frac {1.62 ｘ 3.0276}{6.67}` ｘ 10¹²⁺¹¹ Kg

Mₘ = 0.735ｘ 10²³ Kg

or

Mₘ = 7.35ｘ 10²² Kg

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