Irodov Problem 1.225

May 19, 2026

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Problem Statement

Find the gravitational potential φ at the surface of the Earth and at a distance h above the surface. M_E = Earth mass, R_E = Earth radius.

Given

M_E, R_E, h above surface.

Concepts & Formulas

φ = −GM/r. At surface: φ_s = −GM_E/R_E. At height h: φ_h = −GM_E/(R_E+h).

Step-by-Step Solution

Step 1: φ = −GM/r by definition (zero at infinity).
Step 2: Surface: φ_s = −GM_E/R_E = −gR_E (since g = GM_E/R_E²).
Step 3: Height h: φ_h = −GM_E/(R_E+h).
Step 4: Δφ = φ_h − φ_s = GM_E·h/(R_E(R_E+h)) ≈ gh for h ≪ R_E.

Worked Calculation

φ_s = −GM_E/R_E; φ_h = −GM_E/(R_E+h).

Boxed Answer

φ_surface = −GM_E/R_E; φ_h = −GM_E/(R_E+h)

Physical Interpretation

The negative potential signifies a bound state. The potential energy increases (becomes less negative) as you move away, consistent with gravity doing negative work when lifting objects.