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 [NEET 2014]

negligible mass. The downward acceleration of mass \[{{m}_{1}}\] 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 :

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)

\[\Rightarrow \]   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),

\[\Rightarrow \]   \[mg-{{T}_{1}}=m\times a\]

\[{{T}_{1}}-2\mu mg=2m\times a\]

\[mg(1-2\mu )=3m\times a\]

\[a=\frac{2}{3}(1-2\mu )\]