Irodov Problem 1.224

Problem Statement

A uniform solid sphere of mass M and radius R. Find the gravitational field at distance r < R (inside the sphere).

Given

Uniform solid sphere M, R. Point inside at r < R.

Concepts & Formulas

By Gauss’s theorem for gravity: only mass within radius r contributes. M_enc = M(r/R)³.

Step-by-Step Solution

Step 1: M_enc = M·(4πr³/3)/(4πR³/3) = M(r/R)³.
Step 2: g = G·M_enc/r² = GM·r/R³.
Step 3: Field grows linearly with r inside.

Worked Calculation

g = GMr/R³ (inside, r < R).

Boxed Answer

g = GMr/R³ (inside sphere)

Physical Interpretation

Inside a uniform sphere, gravity increases linearly from zero at the center to GM/R² at the surface — like a harmonic restoring force, which is why a tunnel through Earth gives SHM.