Question

Here three blocks A, B and C are connected with strings and pulleys as shown in figure. The constraint relation between the acceleration of masses A, B, and C can be formulated as?

• A. + + = 0
• B. + 2 + = 0
• C. + 2 - = 0
• D. + = 2

Answer 1

The correct option is B

+ 2 + = 0

Take the ceiling as reference Now assume the distance of block A is $y_A$, B is $y_B$ and C is $y_C$

Now we can say that $y_A$ + 2$y_B$ + $y_C$ = $l_0$ where $l_0$ is the length of the string. Assume acceleration of blocks A, B and C as shown and differentiate once with respect to time

$\frac{d{y}_A}{dt}=-v_A$ & $\frac{d{y}_B}{dt}=-v_B$ as length is decreasing by the assumed acceleration

-$v_A$ - 2$v_B$ + $v_C$ = 0 $(\frac{d{l}_0}{dt}=0as{l}_{0}isconstant)$ differentiating again. -$a_A$ - 2$a_B$ + $a_C$ = 0 $\Rightarrow$ $a_C$ = ($a_A$ + 2$a_B$) But C is going down while A & B are going up so $a_C$ = - ($a_A$ + 2$a_B$)

Alternate solution I Let us assume that masses A and B would go up by distance $x_A$ and $x_B$, respectively. This length of the string will slack as length (ab+cd) above the pulley P. Thus the block B will go up by a distance $x_B$ as shown in figure. Thus we have

(ab + cd) + $x_A$ = -$x_C$ or 2$x_B$ + $x_A$ = - $x_C$ (-ve sign due to sign convention)

Differentiating with respect to time, we get 2$v_B$ + $v_A$ = -$v_C$ ...(i)

Differentiating again with respect to time, 2$a_B$ + $a_A$ = -$a_C$ ...(ii)

Equations (i) and (ii) are the constraint relations for motion of masses A, B, and C.

Alternate solution II: Displacement of pulley is average displacement of both sides

Pulley 0 = $\frac{x_A-x_1}{2}$

= $X_A$ = $X_1$

For Pulley z

$X_B=\frac{x_2-x_1}{2}$

2$X_B+X_1=X_2$

$x_2$ = $x_C$

$x_C$ = 2$x_B$ + $x_A$

$\because$ $x_2$ is up and $x_C$ is down so with sign

$x_C$ = -(2$x_B$ + $x_A$)

$\Rightarrow$ $x_A$ + 2$x_B$ + $x_C$ = 0

$\Rightarrow$ $v_A$ + 2$v_B$ + $v_C$ = 0

$\Rightarrow$ $a_A$ + 2$a_B$ + $a_C$ = 0