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Question

An arrangement of pulley, string and blocks is shown in figure. Given m1=4 kg,m2=2 kg and m3=4 kg. All surfaces are frictionless, string and pulleys are massless. Find the acceleration of block m1. Assume g=10 m/s2.


A
3611 m/s2
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B
4011 m/s2
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C
367 m/s2
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D
407 m/s2
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Solution

The correct option is D 407 m/s2
Assuming the acceleration of pulley and blocks, as shown in the figure.
.
Applying string contraint :
l1+l2=constant
On double differentiating with time gives:
a0+ap=0ap=a0 ...(i)
(ve and +ve sign are used for decrease and increase of length of string respectively.)

Similarly,
l3+l4= constant
Double differentiate w.r.t time
+(a1ap)+(a2ap)=0
Or, a1+a22ap=0
2ap=2a0=a1+a2....(ii)

From the FBD of blocks and massless pulley:


Equation of dynamics in direction of acceleration;
2T=m1a0
2T=4a0 ...(iii)


m2gT=m2a1
20T=2a1 ...(iv)
and m3gT=m3a2
40T=4a2 ...(v)

(2×Eq.iv) + Eq.v gives:
803T=4(a1+a2)
Putting a0 from Eq. (ii) 803T=4(2a0)=8a0 ...(vi)

Adding 2×Eq. (vi) and 3× Eq. (iii) gives:
160=28a0
a0=407 m/s2

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