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Question

Find the acceleration of mass m1 and m2 in the arrangement shown in figure, if mass m2 is twice that of mass m1, and the angle that the inclined plane forms with the horizontal is equal to 30. Mass of the pulley and string are negligible and friction is absent everywhere.


A
(a)m2=2g18
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B
(a)m2=7g9
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C
(a)m1=7g18
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D
(a)m1=7g9
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Solution

The correct options are
B (a)m2=7g9
C (a)m1=7g18
Here pulley 1 is movable and if it moves by x downwards, then total length of string slackened is 2x.
However, one end of the string is fixed so in order to maintain string length constant, block m2 will move downwards by 2x and block m1 will move x upwards along the incline.

Differentiating the displacement twice w.r.t time:
a1=a for m1, and a2=2a for m2 respectively.


Applying Newton's 2nd law for m1 and m2 along the direction of acceleration:
2Tm1gsinθ=m1a ...(i)
m2gT=m2(2a) ...(ii)

Here m2=2m1, θ=30 ...(iii)

Now, adding (ii)×2+ (i),
2m2gm1gsin30=a(m1+4m2)
4m1gm1g2=a(m1+8m1)
[m2=2m1]
7m1g2=a(9m1)
a=7g18
Therefore,
Acceleration of m1=a=7g18
Acceleration of m2=2a=2×7g18=7g9

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