A block $A$ of mass $m_1$ rests on a horizontal table. A light string connected to it passes over a frictionless pully at the edge of table and from its other end another block $B$ of mass $m_2$ is suspended. The coefficient of kinetic friction between the block and the table is $\mu _k.$ When the block $A$ is sliding on the table, the tension in the string is
AIPMT 2015, Difficult
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${m_2}g - T = {m_2}a$
$T - \mu k{m_1}g = {m_1}a$
$\Rightarrow a = \frac{{\left( {{m_2} - \mu \ km} \right)g}}{{{m_1} + {m_2}}}$
For the block of mass $'m{_2}'$
${m_2}g - T = {m_2}\left[ {\frac{{m - 2 - {\mu _k}}}{{{m_1} + {m_2}}}} \right]$
$g{m_2}g - T = {m_2}\left[ {\frac{{{m_2} - {\mu _k}m - 1}}{{{m_1} + {m_2}}}} \right]$
${m_2}g = {m_2}g\left[ {\frac{{{m_2} - {\mu _k}m - 1}}{{{m_1} + {M_2}}}} \right]$
$T - \mu k{m_1}g = {m_1}a$
$\Rightarrow a = \frac{{\left( {{m_2} - \mu \ km} \right)g}}{{{m_1} + {m_2}}}$
For the block of mass $'m{_2}'$
${m_2}g - T = {m_2}\left[ {\frac{{m - 2 - {\mu _k}}}{{{m_1} + {m_2}}}} \right]$
$g{m_2}g - T = {m_2}\left[ {\frac{{{m_2} - {\mu _k}m - 1}}{{{m_1} + {m_2}}}} \right]$
${m_2}g = {m_2}g\left[ {\frac{{{m_2} - {\mu _k}m - 1}}{{{m_1} + {M_2}}}} \right]$
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