Irodov Problem 1.228

Problem Statement

Find the binding energy of a satellite of mass m orbiting Earth at height h.

Given

m, M_E, R_E, h.

Concepts & Formulas

Total energy of orbit E = −GM_Em/(2(R_E+h)). Binding energy = −E = GM_Em/(2(R_E+h)).

Step-by-Step Solution

Step 1: KE = GM_Em/(2(R_E+h)) (from virial: KE = −E_total).
Step 2: PE = −GM_Em/(R_E+h).
Step 3: E_total = KE + PE = −GM_Em/(2(R_E+h)).
Step 4: Binding energy (to free it) = |E_total| = GM_Em/(2(R_E+h)).

Worked Calculation

E_binding = GM_Em/(2(R_E+h)).

Boxed Answer

E_binding = GM_Em / [2(R_E + h)]

Physical Interpretation

Binding energy is exactly half the magnitude of potential energy (virial theorem). To escape from orbit costs additional energy equal to the current binding energy.