10. Derive an expression for g (acceleration due to gravity) at a dept

1.	10. Derive an expression for g (acceleration due to gravity) at a depth d from the surface of earth. Consider theearth as a sphere of uniform mass density. What happens to 'g' at the centre of earth.
Answer» Consider earth to be a sphere of radius R and mass M. The acceleration due to gravity on the surface of the earth is g = GM/R2 , where G is the Universal gravitational constant. If ρ is the density of the earth, then Mass of earth M = 4πR3ρ/3 ∴ g = G.(4πR3ρ/3)/R2 or, g = 4πRGρ/3. Consider a body taken to a depth d inside the earth's surface. The body will be attracted by the mass of the earth which is enclosed in a sphere of radius (R - d). If the mass of this portion is denoted by M', the acceleration due to gravity at the point by g', Then, g' = GM'/(R − d)2 , where M' = 4π(R−d)3ρ/3. g' = G94π(R−d)3ρ/3)/(R − d)2 g' = 4π(R−d)Gρ/3. g'/g = (R-d)/R or g' = g(1-d/R) This gives the variation of g with depth d.

10. Derive an expression for g (acceleration due to gravity) at a depth d from the surface of earth. Consider theearth as a sphere of uniform mass density. What happens to 'g' at the centre of earth.

Answer»

Consider earth to be a sphere of radius R and mass M.

The acceleration due to gravity on the surface of the earth is g = GM/R2 , where G is the Universal gravitational constant.

If ρ is the density of the earth, then Mass of earth M = 4πR3ρ/3 ∴ g = G.(4πR3ρ/3)/R2 or, g = 4πRGρ/3.

Consider a body taken to a depth d inside the earth's surface. The body will be attracted by the mass of the earth which is enclosed in a sphere of radius (R - d). If the mass of this portion is denoted by M', the acceleration due to gravity at the point by g', Then, g' = GM'/(R − d)2 , where M' = 4π(R−d)3ρ/3. g' = G94π(R−d)3ρ/3)/(R − d)2 g' = 4π(R−d)Gρ/3. g'/g = (R-d)/R or g' = g(1-d/R)

This gives the variation of g with depth d.

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