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A system consists of three 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 = $\mu$). The pulley is frictionless and of negligible mass. The downward acceleration of mass $m_1$ is: (Assume $m_1 = m_2 = m_3 = m$)
A
$\frac{g(1 - g\mu)}{9}$
B
$\frac{2g\mu}{3}$
C
$\frac{g(1 - 2\mu)}{3}$
D
$\frac{g(1 - 2\mu)}{2}$
Detailed Solution
Acceleration $= \frac{\text{Net force in the direction of motion}}{\text{Total mass of system}}$
$a = \frac{m_1g - \mu(m_2 + m_3)g}{m_1 + m_2 + m_3} = \frac{g}{3}(1 - 2\mu)$
$a = \frac{m_1g - \mu(m_2 + m_3)g}{m_1 + m_2 + m_3} = \frac{g}{3}(1 - 2\mu)$