Question

In the pulley system shown in the figure the movable pulleys $A,B$ and $C$ are of mass $1kg$ each. $D$ and $E$ are fixed pulleys. The strings are light and inextensible. Find tension in the string.

Answer 1

Since, all strings are connected to each other, it will have some tension at all point, let it be $T$.

Considering the acceleration of pulley, $A,B$ and $C$ is $a_1,a_2$ and $a_3$ downward.

$\to$ according to constrain equation,

$\sum T.a=0$

$-2T{a}_3-T.a_2-T{a}_1=0$

$2a_3+a_2+a_1=0\to \left(1\right)$

Applying Newton''s laws on each block,

$10-2T=a_3\to \left(2\right)$

$10-T=a_2\to \left(3\right)$

$10-T=a_1\to \left(4\right)$

From equation $\left(2\right)$ and $\left(3\right)$

$-T=a_3-a_2$

$T=a_2-a_3\to \left(5\right)$

putting it in

$10-a_2+a_3=a_2$

$2a_2=10+a_3\to \left(6\right)$

Solving $\left(2\right)$ and $\left(4\right)$, we get,

$a_1=a_2=\frac{10}{3},a_3=-\frac{10}{3}$

$T=\frac{20}{3}$Newton