A system consists of three identical masses $m_{1}, m_{2}$ and $m_{3}$ connected by a string passing over a pulley P. The mass $m_{1}$ hangs freely and $m_{2}$ and $m_{3}$ are on a rough horizontal table (the coefficient of friction $= μ$ ). The pulley is frictionless and of negligible mass. The downward acceleration of mass $m_{1}$ is

\frac{g (1 - g μ)}{9}

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\frac{2 g μ}{3}

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\frac{g (1 - 2 μ)}{3}

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\frac{g (1 - 2 μ)}{2}

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Solution

The correct option is D $\frac{g (1 - 2 μ)}{3}$
For the motion of horizontal block, $(m_{2} + m_{3}) a = T - f_{2} - f_{3}$
or $(m + m) a = T - μ m g - μ m g$
or $2 m a = T - 2 μ m g . . . . (1)$
For the motion of vertical block, $m_{1} a = m_{1} g - T$ or $m a = m g - T . . . (2)$
From (1) and (2), $2 m a = m g - m a - 2 μ m g$
or $3 a = g - 2 μ g$
or $a = \frac{g (1 - 2 μ)}{3}$

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