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Solve the following :

Question:

Two masses $m_{1}$ and $m_{2}$ are connected by a spring of spring constant $k$ and are placed on a frictionless horizontal surface. Initially the spring is stretched through a distance ${ }^{x_{0}}$ when the distance is released from rest. Find the distance moved by the two masses before they again come to rest.

Solution:

Let ${ }^{x_{1}}$ and ${ }^{x_{2}}$ be travelled by ${ }^{m_{1}}$ and ${ }^{m_{2}}$

$\therefore$ C.O.L. $M \Rightarrow m_{1} x_{1}=m_{1} x_{2}$

$\Rightarrow m_{1} x_{1}=m_{1} x_{2}$ \{intearate\}

Use C.O.E.L,

$\frac{1}{2} k x_{0}^{2}=\frac{1}{2} k\left(x_{1}+x_{2}-x_{0}\right)^{2}$

$\Rightarrow 2 x_{0}=x_{1}+x_{2}$

$\Rightarrow x_{1}=2 x_{0}-x_{2}=2 x_{0}-\frac{m_{1} x_{1}}{m_{2}}$

$\Rightarrow x_{1}=\frac{2 m_{2}}{m_{1}+m_{2}} x_{0}$

Similarly, $\quad x_{2}=\frac{2 m_{1}}{m_{1}+m_{2}} x_{0}$

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