Irodov Problem 1.223

Problem Statement

A uniform thin spherical shell of mass M and radius R. Find the gravitational field (a) inside the shell; (b) outside the shell.

Given

Spherical shell, M, R.

Concepts & Formulas

Shell theorem: Inside a uniform shell, the gravitational field is zero. Outside, the shell acts as a point mass.

Step-by-Step Solution

Step 1: For r < R: by Gauss's law for gravity, enclosed mass = 0 → g = 0.
Step 2: For r > R: enclosed mass = M → g = GM/r² (directed inward).
Step 3: At r = R: g = GM/R².

Worked Calculation

Inside (r < R): g = 0. Outside (r > R): g = GM/r².

Boxed Answer

g = 0 (r < R); g = GM/r² (r > R)

Physical Interpretation

The interior zero-field result means a person inside a hollow spherical shell feels no net gravitational pull from the shell — the contributions from all directions cancel exactly.