Question

In the pulley system shown in 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. Choose the correct alternative (s). (Assume $g=9.8m{s}^{-2}$)

• A. Tension in the string is nearly equal to $6.5N$
• B. Acceleration of pulley $A$ is $\frac{g}{3}$ downward
• C. Acceleration of pulley $B$ is $\frac{g}{6}$ upward
• D. Acceleration of pulley $C$ is $\frac{g}{3}$ upward

Answer 1

Answer: (A), (B), (D)

The correct options are
A Tension in the string is nearly equal to $6.5N$
B Acceleration of pulley $A$ is $\frac{g}{3}$ downward
D Acceleration of pulley $C$ is $\frac{g}{3}$ upward



As the string is inextensible, total change in length of the string should be zero.
From the figure
$(y_B-y_A)+y_B+2y_A+(y_C-y_B)+y_C=\text{constant}$

Take the second derivative of this equation w.r.t. time to get
$a_A+a_B+2a_C=0...\left(1\right)$

Here $(a_A,a_B,a_C)$ are accelerations( acting downwards) of $A,B,C$ respectively.

Equations of motion for each movable pulley can be written as
$T+mg-2T=m{a}_A$
$T+mg-2T=m{a}_B$
$mg-2T=m{a}_C$
Substituting $m=1kg$ , we get
$g-T=a_A....\left(2\right)$
$g-T=a_B....\left(3\right)$
$g-2T=a_C....\left(4\right)$

Substituting $\left(2\right),\left(3\right),\left(4\right)$ in $\left(1\right)$

$g-T+g-T+2(g-2T)=0$
$4g-6T=0$
$T=\frac{2g}{3}=6.53N$
Substituting $T$ in $\left(2\right),\left(3\right),\left(4\right)$, we get
$a_A=a_B=-a_C=\frac{g}{3}$
$\Rightarrow$ $A,B$ has downward accelration of $\frac{g}{3}$ and $C$ has upward acceleration of $\frac{g}{3}$