A system consists of three masses m _ 1 , m _ 2 and m _ 3 connected...

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}_{3}

hangs freely and

{m}_{2}

and

{m}_{1}

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}_{3}

is: (Assume

{m}_{1} = {m}_{2} = {m}_{3} = m )

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 _

Options

A\dfrac{g $\left(1 - \mu\right)}{9}$
B $\dfrac{2g\mu}{3}$
C\dfrac{g $\left(1 - 2\mu\right)}{3}$
D\dfrac{g $\left(1 - 2\mu\right)}{2}$

Correct answer

C. \dfrac{g $\left(1 - 2\mu\right)}{3}$

Step-by-step solution

Let the tension force between ${m}_{1}$ and ${m}_{2}$ is ${T}_{1} and$ {m}_{2} $and$ {m}_{3} $is$ {T}_{2} $$ {T}_{1} -$ $\mumg = ma . . .$ \left(1\right) $$ {T}_{2} - {T}_{1} - \mumg = ma . . . \left(2\right) $$ mg - {T}_{2} = ma . . . \left(3\right) $From equation$ 1, 2 \text{ and } 3 $,$ 3amg = mg - 2\mumg $$ ⇒ a = \dfrac{mg - 2\mumg}{3mg} $ $ ⇒ a = \dfrac{g - 2\mumg}{3} = g\left(\dfrac{1 - 2\mu}{3}\right)$

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