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

A system consists of three masses $m_{1}, m_{2} a n d m_{3}$ 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
$(Assume m_{1} = m_{2} = m_{3} = m)$

A
$\frac{g (1 - g μ)}{9}$
B
$\frac{2 g μ}{3}$
C
$\frac{g (1 - 2 μ)}{3}$
D
$\frac{g (1 - 2 μ)}{2}$

Solution:

Force of friction on mass $m_{2} = μ m_{2} g$

Force of friction on mass $m_{3} = μ m_{3} g$

Let $α$ be common acceleration of the system.

$∴ a = \frac{m_{1} g - μ m_{2} g - μ m_{3} g}{m_{1} + m_{2} + m_{3}}$

$Here, m_{1} = m_{2} = m_{3} = m$

$\begin{aligned} Hence, the downward acceleration of mass m_{1} is \\ \frac{g (1 - 2 μ)}{3} \end{aligned}$

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