Irodov Problem 1.192

Problem Statement

Two bodies of masses m₁ and m₂ are connected by a light string passing over a frictionless pulley (Atwood machine). Find: (a) acceleration; (b) tension.

Given

m₁ > m₂. Atwood machine. Frictionless pulley, massless string.

Concepts & Formulas

m₁ descends, m₂ ascends. m₁g − T = m₁a; T − m₂g = m₂a.

Step-by-Step Solution

Step 1: Add: (m₁−m₂)g = (m₁+m₂)a → a = (m₁−m₂)g/(m₁+m₂).
Step 2: T = m₂(g+a) = m₂g(1 + (m₁−m₂)/(m₁+m₂)) = 2m₁m₂g/(m₁+m₂).
Step 3: Check: T = m₁(g−a) = m₁g·2m₂/(m₁+m₂) = 2m₁m₂g/(m₁+m₂). ✓

Worked Calculation

a = (m₁−m₂)g/(m₁+m₂). T = 2m₁m₂g/(m₁+m₂).

Boxed Answer

a = (m₁−m₂)g/(m₁+m₂); T = 2m₁m₂g/(m₁+m₂)

Physical Interpretation

Tension equals the harmonic mean of the two weights, always between m₂g and m₁g — the string carries less than the heavier weight and more than the lighter.