The pulleys $A$ and $C$ are fixed while the pulley $B$ is moveable. $A$ mass $M_2$ attached to pulley $B$ , while the strings has masses $M_1$ and $M_3$ at the two ends. Find the acceleration of the each mass.

 

Let at any instant the position of the masses are as shown.

The total length of the string

                  $y_1+2y_2+y_3+{\mathcal{l}}_0 = \mathcal{l}$     …(i)

Differentiating w.r.t. time, we get

                   $\frac{d{y}_1}{dt}+\frac{d{y}_2}{dt}+\frac{d{y}_3}{dt} = 0$

 or                 $v_1+2v_2+v_3 = 0$        ….(ii)

Also              $a_1+2a_2+a_3 = 0$         …(iii)

Suppose $a_1$ is upward and $a_3$ is upward, then

            $a_2 = \left(\frac{a_1+a_3}{2}\right)$ downward.

By Newton’s second law, we have

for block $M_1$ ;

                         $T-M_{1}g = M_1{a}_1$   …(i)

for block $M_3$ ;

                         $T-M_{3}g = M_3{a}_3$    …(ii)

for block $M_2$ ;

                         $M_{2}g-2T = m_2{a}_2$   …(iii)

Solving above equations we get

$a_1 = \left[\frac{-M_1{M}_2+3M_2{M}_3-4M_1{M}_3}{M_1{M}_2+M_2{M}_3+4M_1{M}_3}\right]g$ ,

$a_2 = \left[\frac{M_1{M}_2+M_2{M}_3-4M_1{M}_3}{M_1{M}_2+M_2{M}_3+4M_1{M}_3}\right]g$ ,

$a_3 = \left[\frac{3M_1{M}_2-M_2{M}_3-4M_1{M}_3}{M_1{M}_2+M_2{M}_3+4M_1{M}_3}\right]g$ .