Irodov Problem 1.191

Problem Statement

A body of mass m₁ is connected by a string over a pulley to another body of mass m₂ on an inclined plane (angle α). Find the acceleration of the system and the tension in the string. Ignore friction.

Given

m₁ hangs, m₂ on smooth incline α. Massless pulley and string.

Concepts & Formulas

Newton’s 2nd law for each mass. m₁ descends: m₁g − T = m₁a. m₂ up the incline: T − m₂g sinα = m₂a.

Step-by-Step Solution

Step 1: Add equations: m₁g − m₂g sinα = (m₁+m₂)a.
Step 2: a = g(m₁ − m₂ sinα)/(m₁+m₂).
Step 3: T = m₁(g−a) = m₁g·(1 − (m₁−m₂sinα)/(m₁+m₂)) = m₁m₂g(1+sinα)/(m₁+m₂).

Worked Calculation

a = g(m₁−m₂sinα)/(m₁+m₂). T = m₁m₂g(1+sinα)/(m₁+m₂).

Boxed Answer

a = g(m₁ − m₂sinα)/(m₁+m₂); T = m₁m₂g(1+sinα)/(m₁+m₂)

Physical Interpretation

When m₁ = m₂sinα the system is in equilibrium. The tension is always less than m₁g, confirming the hanging mass accelerates downward.