A system consists of three masses m₁, m₂ and m₃ connected by a string passing over a pulley P. The mass m1 hangs freely and m₂ and m₃ are on a rough horizontal table (the coefficient of friction = µ). The pulley is frictionless and of negligible mass. The downward acceleration of mass m₁ is (Assume m₁ = m₂ = m₃ = m)

\(\frac{g(1-2\mu)}{3}\)

\( m_1g - T_1 = m_1a \) ---------(1)

\( T_1 - f_1 - T_2 = m_2a \) ------------(2)

\( T_3 - f_1 = m_3a \) -----------(3)

On solving above equation we get

m₁ = m₂ = m₃ = m

mg - 2μmg = 3ma

⇒ a = \(\frac{g(1-2\mu)}{3}\)