Question

The acceleration due to gravity $g$ and density of the earth $p$ are related by which of the following relations? (Here $G$ is the gravitational constant and $R$ is the radius of the earth).

• A. $p=\frac{4\pi }{3GRd}$
• B. $p=\frac{3g}{4\pi GR}$
• C. $p=\frac{3G}{4\pi GR}$
• D. $p=\frac{4\pi GR}{3G}$

Answer 1

The correct option is A $p=\frac{4\pi }{3GRd}$

Let us take earth to be a sphere of radius R and density d.

Then the mass of the earth M is given by,

M = 4/3 π R³ d. According to law of gravitation the force on a body of mass m, near the surface of earth is given by,

F = G (mM/R²), in magnitude, and acting along the line joining the centre of the earth and the centre of the body. Let this force produce an acceleration g in the body of mass m. Then

G (mM/R²) = mg, or g = G M/R² = 4π/3 G R d.

Acceleration due to gravity is proportional to mean density of earth, and its radius.