In the figure shown, suppose m₂ = 2.0 kg and m₃ = 3.0 kg . What should be the mass m₁ so that it remains at rest?

Step-by-step solution

Let T₁ be the tension in the string connected to m₁ , and T₂ be the tension in the string connecting m₂ and m₃ . For m₁ to remain at rest, the upward tension must balance its weight: T₁ = m₁ g Since m₁ is at rest, the movable pulley is also stationary. Balancing the forces on the massless movable pulley gives: T₁ = 2T₂ With the movable pulley at rest, m₂ and m₃ form a simple Atwood machine. The tension T₂ in this lower string is: T₂ = 2 m₂ m₃ m₂ + m₃ g Substitute this T₂ back into the T₁ equation: T₁ = 2 ( 2 m₂ m₃