Question

Calculate the depth below the surface of the earth where acceleration due to gravity becomes half of its value at the surface of the earth . Radius of the earth = 6400 km.

Answer 1

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Answer 2

Explanation:

Acceleration due to gravity = GM/R²

where M is the mass of sphere of radius R.

Let x be the new distance of point from center. Hence the mass of earth till radius x is Mx³/R³.

The earth can be split into two parts now. The spherical part of radius x, and the remaining spherical shell of mass outside the radius x. The force of gravity due to the shell at distance x is zero. Hence the only force is due to the sphere of radius x.

$g = \frac{Gm}{ {x}^{2} } = \frac{GM \frac{ {x}^{3} }{ {R}^{3} } }{ {x}^{2} } = \frac{GMx}{ {R}^{3} }$

For this to be half of GM/R²,

x/R³ = 0.5 1/R²

x= R/2 = 3200km