A system consists of three masses m₁ , m₂ and m₃ connected by a string passing over a pu

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4-2.Friction

hard

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 $(Assume\, m = m_2 = m_3 = m)$

A$\frac{{g\left( {1 - g\mu } \right)}}{9}$

B$\frac{{2g\mu }}{3}$

C$\;\frac{{g\left( {1 - 2\mu } \right)}}{3}$

D$\;\frac{{g\left( {1 - 2\mu } \right)}}{2}$

(AIPMT-2014)

Solution

$\begin{array}{l}
\,\,\,\,\,\,\,\,\,\,\,\,\,Force\,of\,friction\\
on\,mass\,{m_2} = \mu {m_2}g\\
Force\,of\,friction\,on\,mass\\
{m_3} = \mu {m_3}g\\
Let\,a\,be\,common\,acceleration\\
of\,the\,system.\\
\therefore \,a = \frac{{{m_1}g – \mu {m_2}g – \mu {m_3}g}}{{{m_1} + {m_2} + {m_3}}}
\end{array}$
$\begin{array}{l}
Here,\,{m_1} = {m_2} = {m_3} = m\\
\therefore a = \frac{{mg – \mu mg – \mu mg}}{{m + m + m}} = \frac{{mg – 2\mu mg}}{{3m}}\\
\,\,\,\,\,\,\,\, = \frac{{g\left( {1 – 2\mu } \right)}}{3}\\
Hence,\,the\,downward\,acceleration\,of\,mass\,\\
{m_1}\,is\,\frac{{g\left( {1 – 2\mu } \right)}}{3}.
\end{array}$

Standard 11

Physics