question_answer

A system consists of three masses \[{{m}_{1}},{{m}_{2}}\]and \[{{m}_{3}}\] connected by a string passing over a pulley P. The mass w; 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 mass m, is (Assume\[{{m}_{1}}={{m}_{2}}={{m}_{3}}=m\])

A) \[\frac{g(1-g\mu )}{9}\]

B) \[\frac{2g\mu }{3}\]

C) \[\frac{g(1-2\mu )}{3}\]  

D) \[\frac{g(1-2\mu )}{2}\]

Correct Answer: C

Solution :

 \[a=\frac{{{m}_{1}}g-\mu ({{m}_{2}}+{{m}_{3}})g}{{{m}_{1}}+{{m}_{2}}+{{m}_{3}}}=\frac{m[g-2\mu g]}{3m}\] \[=\frac{g}{3}[1-2\mu ]\]