Question

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

• A.
• B.
• C.
• D. None of these

Answer 1

The correct option is A

Let us assume the respective distance of each block as shown in figure. Since the length of the string is constant, $x_1+x_2+2x_3$ = constant

On differentiating twice with respect to time, we get

$\frac{d^2{x}_1}{dt}+\frac{d^2{x}_2}{d{t}^2}+2\frac{d^2{x}_3}{d{t}^2}=0$
As $x_1,x_2$ decrease while $x_3$ increases,
$\frac{d^2{x}_1}{dt}=-a_1$, $\frac{d^2{x}_2}{dt}=-a_2$ and $\frac{d^2{x}_3}{dt}=a_3$
Hence, $a_1+a_2=2a_3$

Alternate solution II:

Displacement of pulley is average displacement from both sides of pulley. We will have acceleration $a_1$ downwards as block $m_1$ is going right with $a_1$, similarly right side of the pulley will have acceleration $a_2$

Now, $a_3=\frac{x_2-x_c}{2}$

$a_1+a_2=2a_3$