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[tex]\huge{\green{\tt{\underline{\underline{QUESTION:-}}}}}[/tex]
Why value of 'g' is zero at center of Earth?
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[tex]\huge{\red{\tt{\underline{\underline{EXPLANATION:-}}}}}[/tex]
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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[tex]\huge{\green{\tt{\underline{\underline{PROOF:-}}}}}[/tex]
The general equation of acceleration due to gravity is;-
g = [tex]\frac{GM}{r^2} [/tex]
Assuming earth to be a perfect sphere and considering its uniform density.
We make the following adjustments in the following equation.
g = [tex]\frac{GM}{r^2} [/tex]
Multiplying and dividing RHS by volume 'V'.
g = [tex]\frac{GM}{V} × V × \frac{1}{r^2} [/tex]
We know that;-
[tex]\frac{M}{V}[/tex] = Density, ρ
Therefore,
g = [tex]\frac{ρGV}{r^2}[/tex]
For sphere;-
V = [tex]\frac{4}{3}[/tex]π[tex]r^{3}[/tex]
This makes
g = [tex]\frac{4}{3}[/tex]π[tex]r^{3}[/tex] × [tex]\frac{ρG}{r^2}[/tex]
g = [tex]\frac{4πρG}{3}[/tex]
So, at center of Earth
since, r = 0
so, g = 0.
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[tex]\boxed{\underline{\green{\star{HOPE \: YOU\:GOT \:THE \:ANSWER.}}}}[/tex]
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