Question

Find the realation between the velocity of block $A$ and velocity of block $B$ in the shown figure.

• A. $8v_A+2v_B=4v$
• B. $4v_A+v_B=2v$
• C. $8v_A+v_B=6v$
• D. $8v_A+3v_B=2v$

Answer 1

The correct option is C $8v_A+v_B=6v$

In the given condition, let the velocities of block $A$ and $B$ are denoted as $v_A$ and $v_B$ respectively.
Also, let the velocities of block $A$, block $B$, pulley $P_1$ and pulley $P_2$ be upward respectively and taking upward positive we have

For pulley $P_2$ we have
$v_{P2}=\frac{v_D+v_E}{2}=\frac{-v-3v}{2}=-2v$
Thus, the velocity of pulley $P_2$ will be $2v$, downward.

Now, as we know that, the string's end attached to the fixed pulley will have zero velocity. So, considering the points $1$ to $6$ in the figure, we have
$v_{P2}-v_A-v_A-v_A-v_A-v_{P1}=0$
$\Rightarrow 2v-v_{P1}=4v_A\Rightarrow v_{P1}=2v-4v_A$......(i)

Again, for pulley $P_1$ we have
$v_{P1}=\frac{v_B+v_C}{2}=\frac{v_B-2v}{2}$.....(ii)
From equation (i) and (ii) we have
$2v-4v_A=\frac{v_B-2v}{2}$
$\Rightarrow 8v_A+v_B=6v$
Hence, the relation between the velocities of block $A$ and block $B$ is $8v_A+v_B=6v$