Derive an expression for the variation of acceleration due to gravity (a) above and (b) below the surface of the earth.
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i)variation of g with height : when an object is placed on the earth's surface. It will be at a distance r = R (radius of the earth)
Then, we have g = GM/R² ........(1)
where G is gravitational constant, M is mass of the earth and R is the radius of the earth .
when an object is at a height h above the surface of the earth , then r = (R + h)
so, g' = GM/(R + h)² .......(2)
From equations (1) and (2),
g' = g/(1 + h/R)²
If h << R then, g' = g(1 - 2h/R) [ by using binomial expansion ]
Here it is clear g value decreases with altitude.
ii) variation of g with depth :
Let us assume that earth to be a homogeneous uniform sphere of radius R , Mass M and uniform density p .
We know that, g = GM/R² = 4/3 πGRp ....(1)
consider a body of mass n be placed at a depth d.
g' = 4/3πp(R - d)G .......(2)
from equations (1) and (2),
g' = g(1 - d/R)
Hence, value of g decreases with depth.
Then, we have g = GM/R² ........(1)
where G is gravitational constant, M is mass of the earth and R is the radius of the earth .
when an object is at a height h above the surface of the earth , then r = (R + h)
so, g' = GM/(R + h)² .......(2)
From equations (1) and (2),
g' = g/(1 + h/R)²
If h << R then, g' = g(1 - 2h/R) [ by using binomial expansion ]
Here it is clear g value decreases with altitude.
ii) variation of g with depth :
Let us assume that earth to be a homogeneous uniform sphere of radius R , Mass M and uniform density p .
We know that, g = GM/R² = 4/3 πGRp ....(1)
consider a body of mass n be placed at a depth d.
g' = 4/3πp(R - d)G .......(2)
from equations (1) and (2),
g' = g(1 - d/R)
Hence, value of g decreases with depth.
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