Question

Find the constraint relation between acceleration of block 1, 2 and 3.

• A. $2a_1+a_2+a_3=0$
• B. $a_1+2a_2+a_3=0$
• C. $a_1+a_2+2a_3=0$
• D. $-2a_1+a_2+a_3=0$

Answer 1

The correct option is A $2a_1+a_2+a_3=0$

Let the acceleration of the block 1, 2 and 3 be $a_1,a_2\text{and}{a}_3$ in downward direction.

Let the displacements of the ends of the strings be $x_1,x_2,x_3$ and $x_4$
Length of first string,
$x_1+x_4=l_1$ ...(i)
Length of second string,
$(x_2-x_4)+(x_3-x_4)=l_2$ ...(ii)
On double differentiating (i) and (ii) with respect to time, we get
$a_1+a_4=0...\left(iii\right)$
and $a_2+a_3-2a_4=0...\left(iv\right)$

But since $a_4=-a_1$ (From Eq.$\left(iii\right)$)
we have, $a_2+a_3+2a_1=0$

This is the required constraint relation between $a_1,a_2\text{and}{a}_3$.