Question

Which of the following is the variation of acceleration due to gravity at a depth “d” below the earth’s surface? Let “R” be the radius of the earth and “M1” the mass of earth. (Assume the density of the earth to be constant)

(a) g = (G*M1)/(R – d)

(b) g = [(G*M1) x density]/d

(c) g = (G x M1/R^3) / (R – d)

(d) g = (G x M1/R^3) x (R – d)

The question was posed to me in an interview.

Query is from Acceleration due to Gravity below and above the Surface of Earth topic in section Gravitation of Physics – Class 11

Answer 1

The correct option is (d) g = (G x M1/R^3) x (R – d)

Easy explanation: g = (G*M)/r^2;

M = Mass contained in an enclosed volume

r = distance from centre of the earth to the depth “d”

Therefore; r = R – d; where, R = Radius of the earth

M = (density) x (volume)

 = (density) x [(4/3) x (pi) x (r^3)]

 = (density x 4 x pi x r^3) / 3

 = (M1 x r^3) / R^3

Therefore;

g = (G x density x 4 x pi x r^3) / (3 x r^2)

 = (G x density x 4 x pi x r^1) / 3

Therefore;

g = [(G x density x 4 x pi) / 3] x (R – d)

 = (G x M1/R^3) x (R – d).