The value of g is 4.0 m s⁻² at a distance of 10,000 km from the centre of the Earth. Find the mass of the Earth. Ans. (5.99x10²⁴ kg)


Given:  

Value of  Gravitational  acceleration = g =  4.0 m s⁻² 

Distance from the centre of the Earth = r = 10,000 km = 1 ｘ 10⁴⁷ｘ 10³ m = 1 ｘ 10⁷ m

Gravitational Constant = G = 6.67 ｘ 10⁻¹¹ N m² kg⁻²



To Find:
Mass of the Earth = Mₑ = ? kg



Solution:   

According to the Law of Gravitation's, we have 

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

But according to the 2nd law of Newton

F = W = mg ----eqn (2)

By comparing these two Eqns 1 & 2 we have 

mg = `\frac {G m Mₑ}{r^2 }`

m will cancel from both sides hence 

g = `\frac {G Mₑ}{r^2 }`

or

Mₑ = `\frac {g r^2}{ G }`

By putting values

Mₑ = `\frac {4.0 m s⁻²ｘ (1 ｘ 10⁷ m)^2}{6.67 ｘ 10⁻¹¹ N m² kg⁻² }`

Mₑ = `\frac {4.0 ms⁻²ｘ 1 ｘ 10¹⁴ m² }{6.67 ｘ 10⁻¹¹ N m² kg⁻² }`

Mₑ = `\frac {4.0}{6.67}` ｘ 10¹⁴⁺¹¹ kg

Mₑ = 0.5997ｘ 10²⁵ kg

or

Mₑ = 5.997ｘ 10²⁴ kg

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