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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
212 m/s (upwards)
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B
312 m/s (downwards)
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C
312 m/s (upwards)
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D
212 m/s (downwards)
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Solution

The correct option is C 312 m/s (upwards)
Step-1: We choose the longest string.
Applying string constraint:
Length of string (abcdefghij)= constant


i.e xab+xcd+xef+xgh+xij= constant
Differentiating w.r.t time,
(va)+0+(vf)+(vg+vh)+(vi+vj)=0(i)

[vb=vc=vd=ve=0 (attached to fixed object)]

Now find the velocity of pulley k:
vk=vA+vB2
vk=822=3 m/s (upward)

Simillarly for other pulleys:
va=vk+vc2=3+22=52 m/s. (upward)

vx=vF=4 m/s
vx=vy+vz2 and vz=0 (fixed)
vy=2vx=8 m/s (upward)

Also
vy=vh+vi2=8 m/s
vh+vi=2×vy

Let us assume block E move upward then vj=vE (length decreases )

Putting the above values in eq. (i):
(va)+0+vf+vg+(vh+vi)+vj=0
52+1+1+2×8vE=0
vE=312 m/s (upward)

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