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 \[{{m}_{1}}\] 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 mass \[{{m}_{1}}\] is (Assume, \[{{m}_{1}}={{m}_{2}}={{m}_{3}}=m\]) [AIPMT 2014]

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 :

First of all consider the forces on the blocks For the 1st block,           \[[\because \,{{m}_{1}}={{m}_{2}}={{m}_{3}}]\] \[mg-B=m\times a\] ?(ii) Let us consider 2nd and 3rd block as a system So,      \[{{T}_{1}}-2\mu mg\,=2m\times a\] ?(i) Solving Eqs. (i) and (ii), Þ         \[\,mg-{{T}_{1}}=m\times a\] \[{{T}_{1}}-2\mu mg=2m\times a\] \[mg(1-2\mu )=3m\times a\] \[a=\frac{2}{3}(1-2\mu )\]