Irodov Problem 1.193

Problem Statement

A block of mass m₁ lies on a horizontal surface (coefficient of kinetic friction μk) and is connected by a string over a pulley to a hanging mass m₂. Find the acceleration and tension.

Given

m₁ on surface, μk friction. m₂ hanging. Massless string/pulley.

Concepts & Formulas

m₂g − T = m₂a (hanging). T − μk·m₁g = m₁a (sliding).

Step-by-Step Solution

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

Worked Calculation

a = g(m₂−μk·m₁)/(m₁+m₂). T = m₁m₂g(1+μk)/(m₁+m₂).

Boxed Answer

a = g(m₂ − μk·m₁)/(m₁+m₂); T = m₁m₂g(1+μk)/(m₁+m₂)

Physical Interpretation

If μk·m₁ ≥ m₂ the system stays at rest — the hanging mass cannot overcome table friction. The system accelerates only when the hanging weight exceeds friction.