A system consists of three masses m 1 , m 2 and m 3 connected by a string passing over a pulley P . The mass…

Let the tension force between $m_{1 }$and $m_2$ is $T_1$ and $m_2$ and $m_3$ is $T_2$

$T_1-$$\mu mg=ma ...\left(1\right)$ 

$T_2-T_1-\mu mg=ma ...\left(2\right)$

$mg-T_2=ma ...\left(3\right)$

From equation $1,2 \& 3$,

$3amg=mg-2\mu mg$

$\Rightarrow a=\frac{mg-2\mu mg}{3mg}$
$\Rightarrow a=\frac{g-2\mu mg}{3}=g\left(\frac{1-2\mu }{3}\right)$