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Question:

A system consists of three identical masses m1, m2 and m3 connected by a string passing over a pulley P. The mass m1 hangs freely and m2 and m3 are on a rough horizontal table (the coefficient of friction = μ). The pulley is frictionless and of negligible mass. The downward acceleration of mass m1 is

2gμ3

g(1−μ)2

g(1−μ)9

g(1−μ)3

Solution:

For the motion of horizontal block,(m2+m3)a=T−f2−f3 or (m+m)a=T−μmg−μmg or 2ma=T−2μmg (1)
For the motion of vertical block, m1a=m1g−T or ma=mg−T.. (2)
From (1) and (2), 2ma=mg−ma−2μmg or 3a=g−2μg or a=g(1−2μ)3