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

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Solution

Mass of the two blocks are $m_{1}, m_{2}$

Initially the spring is stretched by $x_{0}$ with spring constant K.

For the block to come to rest again,

let the distance travelled by $m_{1}$ and $m_{2}$ be $x_{1} a n d x_{2}$ towards right and left respectively.

As no external force acts in horizontal direction, $m_{1} x_{1} = m_{2} x_{2}$ ....(i)

Again,the energy would be conserved in the spring.

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

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

$\Rightarrow x_{1} + x_{2} = 2 x_{0} . . . (i i)$

$x_{1} = 2 x_{0} - x_{2}$

Similarly $x_{1} = (\frac{2 m_{2}}{m_{1} + m_{2}}) x_{0}$

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

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

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

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Two masses m1 and m2 are connected by a spring of spring constant k and are placed on a friction less horizontal surface.Initially the spring is stretched through a distance x0 when the system is released from rest.Find the distance moved by the two masses before they again come to rest.