Question

In the figure shown, the acceleration of $A$ is $2\, m / s ^{2}$ to left and acceleration of $B$ is $1\, m / s ^{2}$ to left.

The magnitude of acceleration of $C$ in $m / s ^{2}$ is__.

Answer 1

Let displacement of $A$ is $x_{1}, B$ is $x_{2}$ and that of $C$ is $x_{3}$ in a given time.
Now length of string should remain same.

So initial length $=$ final length
$\Rightarrow l_{1}+l_{2}+l_{3}+l_{4}+l_{5}$
$=\left(l_{1}+x_{1}-x_{2}\right)+\left(l_{2}-x_{2}\right)+\left(l_{3}+x_{1}\right)+\left(l_{4}+x_{3}\right)+\left(l_{5}+x_{3}\right)$
$\Rightarrow x_{3}=x_{2}-x_{1}$ Differentiating twice:
$a_{3}=a_{2}-a_{1} $
$\Rightarrow a_{3}=1-2=-1\, m / s ^{2}$
So acceleration of $C$ is $1\, m / s ^{2}$ upwards.