Shell Theorem — Two Parts

Part 1 (Interior): A uniform spherical shell exerts zero net gravitational force on any mass placed inside it, regardless of where inside.

Part 2 (Exterior): A uniform spherical shell acts exactly like a point mass (all mass concentrated at centre) for any mass outside it.

Mathematical Proof Sketch (Part 1)

Consider a small area element dA on the shell. It creates a small force on mass m inside. The diametrically opposite element dA' creates an equal and opposite force (by symmetry of the inverse-square law). Integrating over the entire shell gives zero.

Applications in ME-06

Application 1: Deriving g at depth d

At depth d, the sphere of radius (R−d) acts like a point mass. The shell of thickness d above contributes zero force.

Mass below: M' = M × (R−d)^{3}/$R^{3}$

g' = GM'/(R−d)^{2} = GM(R−d)/$R^{3}$ = g(1−d/R)

This derives the linear depth formula from the Shell Theorem.

Application 2: g at Earth's surface

The entire Earth (approximately) acts like a point mass at its centre. This justifies g = GM/$R^{2}$.

Application 3: g = 0 at centre

At the centre, the entire mass of Earth forms a shell around the point. By Part 1, net force = 0, so g = 0.

Important Note for Non-Uniform Earth

The shell theorem applies exactly only to uniform spherical shells. Earth has:

• Non-uniform density (denser core)
• Slightly oblate shape

For NEET, we always treat Earth as a uniform sphere, so the shell theorem applies exactly.