Question

Let $m_1=1kg,m_2=2kg$ and $m_3=3kg$ in figure. Find the accelerations of $m_1,m_2$ and $m_3$. The string from the upper pulley to $m_1$ is $20cm$ when the system is released from rest. How long will it take before $m_1$ strikes the pulley?

Answer 1

Given,

$\begin{array}{c}{m}_1=1kg\\ m_2=2kg\\ m_3=3kg\end{array}$

Now,

$\begin{array}{c}{a}_1=\frac{a_3-a_2}{2}........\left(i\right)\\ \Rightarrow a_2-a_3+2a_1=\mathrm{0.......}\left(i\right)\\ 2T-M_{1}g=M_1{a}_1\\ \Rightarrow 2T-g=a_1.........\left(ii\right)\\ \Rightarrow T-2g=2a_2.........\left(iii\right)\\ \Rightarrow 3g-T=3a_3........\left(iv\right)\\ \Rightarrow a_1=\frac{19g}{2g}\\ a_2=\frac{-\pi g}{2g}\\ a_3=\frac{21g}{2g}\end{array}$

For $m_1:t=\sqrt{\frac{2\times 20\times 2g}{100\times 19g}}$

$=\sqrt{\frac{58}{95g}}$