Two Masses & Pulley on Inclined Plane

Two Masses on Inclined Plane with Pulley

Mass m₁ (brown) slides along a plane inclined at angle θ, connected by a rope over a pulley to mass m₂ (green) hanging vertically. The rope is inextensible and the pulley is frictionless.

Key Equations

• Acceleration (m₁ uphill): a = g(m₂ − μm₁cosθ − m₁sinθ) / (m₁ + m₂)
• Tension: T = m₁m₂g(1 + μcosθ + sinθ) / (m₁ + m₂)
• Equilibrium when: m₁(sinθ − μcosθ) ≤ m₂ ≤ m₁(sinθ + μcosθ)

Common Mistakes

• T ≠ m₂g: When m₂ accelerates downward, T = m₂(g−a) < m₂g. When m₁ slides downhill and m₂ accelerates upward, T = m₂(g+|a|) > m₂g. Only at equilibrium (a=0) does T = m₂g.
• Friction flips: When the system reverses direction, friction on m₁ flips sign. Watch the red arrow change direction!
• Dead zone: There's a range of m₂ where nothing moves. Static friction adjusts within this range.

Try These

1. Balanced preset: System at exact equilibrium. Increase m₂ by 0.1 kg and watch it start.
2. Frictionless: Pure Atwood on incline. a = g(m₂ − m₁sinθ)/(m₁+m₂).
3. Equal masses: On a 30° ramp, who wins? m₂ always wins since sin(30°) = 0.5 < 1.
4. Slides Down preset: m₁ is heavy, gravity pulls it down, m₂ goes up. Watch friction flip!
5. Energy tab: KE + PE + heat = constant. The green line stays flat.