Question

Find out the velocity of block $E$ for the arrangement as shown in figure. Consider all pulleys to be light and frictionless. All surfaces are smooth and the string is massless and inextensible.

• A. $\frac{21}{2}\text{m/s}$ (upwards)
• B. $\frac{31}{2}\text{m/s}$ (downwards)
• C. $\frac{31}{2}\text{m/s}$ (upwards)
• D. $\frac{21}{2}\text{m/s}$ (downwards)

Answer 1

The correct option is C $\frac{31}{2}\text{m/s}$ (upwards)

Step-1: We choose the longest string.
Applying string constraint:
Length of string $\left(\text{abcdefghij}\right)=$ constant


i.e $x_{ab}+x_{cd}+x_{ef}+x_{gh}+x_{ij}=$ constant
Differentiating w.r.t time,
$(-v_a)+0+\left(v_f\right)+(v_g+v_h)+(v_i+v_j)=0\dots \left(i\right)$

[$\because v_b=v_c=v_d=v_e=0$ (attached to fixed object)]

Now find the velocity of pulley $k$:
$v_k=\frac{v_A+v_B}{2}$
$\therefore v_k=\frac{8-2}{2}=3\text{m/s}$ (upward)

Simillarly for other pulleys:
$v_a=\frac{v_k+v_c}{2}=\frac{3+2}{2}=\frac{5}{2}\text{m/s}.$ (upward)

$v_x=v_F=4\text{m/s}$
$v_x=\frac{v_y+v_z}{2}$ and $v_z=0$ (fixed)
$\Rightarrow v_y=2v_x=8\text{m/s}$ (upward)

Also
$v_y=\frac{v_h+v_i}{2}=8\text{m/s}$
$\Rightarrow v_h+v_i=2\times v_y$

Let us assume block $E$ move upward then $v_j=-v_E$ ($\because$length decreases )

Putting the above values in eq. $\left(i\right)$:
$(-v_a)+0+v_f+v_g+(v_h+v_i)+v_j=0$
$\Rightarrow -\frac{5}{2}+1+1+2\times 8-v_E=0$
$\therefore v_E=\frac{31}{2}\text{m/s}$ (upward)