A system consists of three masses `m_(1)` , `m_(1)` , `m_(1)` , `m_(2)` and `m_(3)` connected by a string passing over a pulley `P` . The mass `m_(1)` hangs freely and `m_(2)` and `m_(3)` are on a rough horizontal table (the coefficient of friction=mu) The pulley is frictionless and of negligible mass. The downward acceleration of m`1` is (Assume `m_(1)=m_(2)=m_(3)=m`).

Options

A. `g(1-3mu)/(9)m`
B. `(2gmu)/(3)`
C. `g (1-2mu)/(3)`
D. `g(1-2mu)/(2)`

Answer

c

Explanation

Let a be the downward acceleration of mass m pulling force on all the three masses
`F = (m_(1) + m_(2) +m_(3))a`
force of friction on masses `m_(2)` and `m_(3)`
`f = mu (m_(2) + m_(3))g`
Equation of motion of mass `m_(1)` is
`(m_(1) + m_(2) + m_(3)) a = m_(1) g - mu (m_(2) + m_(3))g`
or `(m + m + m)a = mg - mu (m + m)g`
or `a = (mg (1 -2mu))/(3m) = g(1-2mu)/(3)` .