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\(\huge{\blue{\mathbb{HELLO\:BUDDY}}}\)
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\(\huge{\green{\tt{\underline{\underline{QUESTION:-}}}}}\)
Why value of 'g' is zero at center of Earth?
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\(\huge{\red{\tt{\underline{\underline{EXPLANATION:-}}}}}\)
When we move towards centre of earth, the mass is equally distributed in all directions.
The mass beneath you = Mass in front of you = Mass behind you
Thus, all the gravitational forces applied cancel each other and acceleration due to gravity (g) at centre of earth on centre of earth becomes zero (0).
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\(\huge{\green{\tt{\underline{\underline{PROOF:-}}}}}\)
The general equation of acceleration due to gravity is;-
g = \(\frac{GM}{r^2}\)
Assuming earth to be a perfect sphere and considering its uniform density.
We make the following adjustments in the following equation.
g = \(\frac{GM}{r^2}\)
Multiplying and dividing RHS by volume 'V'.
g = \(\frac{GM}{V} × V × \frac{1}{r^2}\)
We know that;-
\(\frac{M}{V}\) = Density, ρ
Therefore,
g = \(\frac{ρGV}{r^2}\)
For sphere;-
V = \(\frac{4}{3}\)π\(r^{3}\)
This makes
g = \(\frac{4}{3}\)π\(r^{3}\) × \(\frac{ρG}{r^2}\)
g = \(\frac{4πρG}{3}\)
So, at center of Earth
since, r = 0
so, g = 0.
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