Derive an expression for the acceleration due to gravity at a depth $d$ below the Earth's surface, assuming the Earth to be a uniform sphere of density $\rho$. Show that $g$ becomes zero at the Earth's centre.

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Accepted Answer

Model the Earth as a uniform sphere of density $ho$ and consider a point at depth $d$ below the surface, a distance $R-d$ from the Earth's centre. A standard result for a uniform spherical shell is that it exerts NO net gravitational force on any point strictly inside it (the pulls from every part of the shell cancel out by symmetry) -- so only the solid sphere of radius $R-d$ enclosed WITHIN this point contributes any net pull; the spherical shell of material lying between radius $R-d$ and the full radius $R$ contributes nothing.

Since the density $ho$ is uniform, mass scales directly with volume, so the enclosed mass at radius $R-d$ is

$$M' = M\left(\frac{R-d}{R}ight)^3$$

(the ratio of the two spheres' volumes). The acceleration due to gravity at this depth is then

$$g_d = \frac{GM'}{(R-d)^2} = \frac{GM(R-d)^3/R^3}{(R-d)^2} = \frac{GM}{R^3}(R-d) = \frac{GM}{R^2}\left(1-\frac{d}{R}ight) = g\left(1-\frac{d}{R}ight)$$ …

Q.Derive an expression for the acceleration due to gravity at a depth $d$ below the Earth's surface, assuming the Earth to be a uniform sphere of density $\rho$ . Show that $g$ becomes zero at the Earth's centre.

Concept understanding — Variation Of Gravity

Variation of Gravity: Why Your Weight Changes Even When You Don't

The Core Idea

Why Does Gravity Vary? Three Main Reasons

1. Altitude (Height Above Sea Level)

2. Depth (Going Underground)

3. Rotation of the Earth (Latitude Effect)

4. Shape of the Earth (Oblateness)

Q.Derive an expression for the acceleration due to gravity at a depth ddd below the Earth's surface, assuming the Earth to be a uniform sphere of density ρ\rhoρ. Show that ggg becomes zero at the Earth's centre.

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Q.Derive an expression for the acceleration due to gravity at a depth $d$ below the Earth's surface, assuming the Earth to be a uniform sphere of density $\rho$ . Show that $g$ becomes zero at the Earth's centre.

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