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
212m/s (upwards)
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B
312m/s (downwards)
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C
312m/s (upwards)
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D
212m/s (downwards)
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Solution
The correct option is C312m/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=8−22=3m/s (upward)
Simillarly for other pulleys: va=vk+vc2=3+22=52m/s. (upward)
vx=vF=4m/s vx=vy+vz2 and vz=0 (fixed) ⇒vy=2vx=8m/s (upward)
Also vy=vh+vi2=8m/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+2×8−vE=0 ∴vE=312m/s (upward)