Question

The value of acceleration due to gravity at a point p inside the earth and at another point q outside the earth is g/2 (g being acceleration due to gravity at the surface of earth). Maximum possible distance in terms of radius of earth r between p and q is?

Answer 1

Maximum possible distance in terms of radius of earth r between p and q is 2√2R

Point inside the earth = p (Given)

Point outside the earth = q (Given)

Gravity = g/2 (Given)

For above the surface, the equation will be -

g/2 = GM/(R+h)²

For below the surface,

g/2 = GM/(R-d)²

Equating both we get,

R² = h² + 2Rh  and  R² = d² – 2Rd

Solving quadratic equation for both d and h as -  

h = (√2 – 1)R

and

d = (√2 + 1)R

= h + d = (√2 + 1)R + (√2 - 1)R

= 2√2R

Answer 2

Answer:

R(2√2 + 1)/2

Explanation:

imagine it in 3d in your head

you cannot use the formula of

g' = GM/(R-d)² because at depth d below the surface the mass is not M(The whole mass of earth) it becomes

g' = GM'/(R-d)²

where M' = M(R-d)³/R³ (assume the mass of earth to be distributed uniformly)

and this way

you'll get g' = g(1 - d/R) which is always valid