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Spring Block's Time Period

Let m1=1 kg...

Question

Let $m_{1} = 1 k g, m_{2} = 2 k g$ and $m_{3} = 3 k g$ in figure. Find the accelerations of $m_{1}, m_{2}$ and $m_{3}$ . The string from the upper pulley to $m_{1}$ is $20 c m$ when the system is released from rest. How long will it take before $m_{1}$ strikes the pulley?

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Solution

Given,
$\begin{matrix} m_{1} = 1 k g m_{2} = 2 k g m_{3} = 3 k g \end{matrix}$
Now,
$\begin{matrix} a_{1} = \frac{a_{3} - a_{2}}{2} . . . . . . . . (i) \Rightarrow a_{2} - a_{3} + 2 a_{1} = 0....... (i) 2 T - M_{1} g = M_{1} a_{1} \Rightarrow 2 T - g = a_{1} . . . . . . . . . (i i) \Rightarrow T - 2 g = 2 a_{2} . . . . . . . . . (i i i) \Rightarrow 3 g - T = 3 a_{3} . . . . . . . . (i v) \Rightarrow a_{1} = \frac{19 g}{2 g} a_{2} = \frac{- π g}{2 g} a_{3} = \frac{21 g}{2 g} \end{matrix}$
For $m_{1} : t = \sqrt{\frac{2 \times 20 \times 2 g}{100 \times 19 g}}$
$= \sqrt{\frac{58}{95 g}}$

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Figure

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