Question

In the arrangement shown in the figure, accelerations of blocks $A,B$ and $C$ are $a_A,a_B$ and $a_C$ respectively. Consider string and pulley to be mass-less and friction absent everywhere in the arrangement. Then

• A. $2a_A+2a_B+a_C=0$
• B. $a_A+a_B+2a_C=0$
• C. $a_A+a_B+a_C=0$
• D. $2a_A+a_B+a_C=0$

Answer 1

The correct option is A $2a_A+2a_B+a_C=0$

$\text{I}-$ Method

Determining relation of accelerations of blocks using constraint of string length i.e total length of the string is constant.
${\text{X}}_0$ is the nearest distance of the pulley from the ceiling
$x_A+x_A-x_0+x_B-x_0+x_B-x_0+x_C-x_0=L$
$\Rightarrow 2x_A+2x_B+x_C=L+3x_0=\text{constant}$
Differentiating two times, we have:
$\Rightarrow 2a_A+2a_B+a_C=0$
Hence, option (a) is correct

$\text{II}-$ Method

By work done method, total work done by the tention on the block is zero and $A$ will have $2T$ , $B$ will have $2T$, $C$ will have $T$ as single string is connected to block $C$

Total work done by tention $=\left(2T\right).s_A+\left(2T\right).s_B+\left(T\right).s_C=0$
where $s_A$ is displacement of block $A$

Differentiating two times, we have:
$\Rightarrow 2a_A+2a_B+a_C=0$