Question

In the arrangement of three blocks as shown in $Fig.6.135$, the string is inextensible. If the directions of accelerations are as shown in the figure, then determine the constraint relation.

Answer 1

Let us assume the respective distance of each block as shown in $Fig6.136$ Since the length of the string is constant, $x_1+x_2+2x_3$= constant.
On differentiating twice w.r.t. time, we get,
$\frac{d^2{x}_1}{dt}+\frac{d^2{x}_1}{{dt}^2}+2\frac{d^2{x}_3}{{dt}^2}=0$
Since $x_1$ and $x_2$ are assumed to be decreasing with time,
$\frac{d^2{x}_1}{{dt}^2}=-a_1$ and $\frac{d^2{x}_2}{{dt}^2}=-a_1$
and $x_3$ is assumed to be increasing with time. Therefore,
$\frac{d^2{x}_3}{{dt}^2}=+a_1$
Thus, $-a_1-a_2+2a_3=0$ or $a_1+a_2=2a_3$