Two blocks are arranged as shown in the figure. What will be the relation between the accelerations $a_{1}$ of block $m_{1}$ and $a_{2}$ of block $m_{2}$ , if the surface and the pulleys are frictionless and the strings are inextensible?

a_{1}

a_{2}

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a_{1} = 6

a_{2}

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a_{1} = 3

a_{2}

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a_{1} = 4

a_{2}

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Solution

The correct option is B $a_{1} = 6$ $a_{2}$
Since the strings are inextensible, therefore their length remains same. Let $a_{A}$ be the acceleration of the pulley A.
$\Rightarrow l_{1} + l_{2} = constant$
Double differentiating w.r.t time $t$ , we get the equation in the form of acceleration i.e.
$\Rightarrow + a_{A} + (a_{A} - a_{1}) = 0$
$\begin{matrix} \Rightarrow 2 a_{A} = a_{1} \\ (1) \end{matrix}$
Also for the other string,
$l_{3} + l_{4} + l_{5} + l_{6} = constant$
Double differentiating w.r.t time $t$ , we get the equation in the form of acceleration i.e.
$\Rightarrow - a_{A} + a_{2} + a_{2} + a_{2} = 0$
$\begin{matrix} \Rightarrow a_{A} = 3 a_{2} \\ (2) \end{matrix}$
From $(1)$ and $(2)$ , $a_{1} = 6 a_{2}$

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