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Spring Force

Two smooth bl...

Question

Two smooth blocks of masses $m_{1}$ and $m_{2}$ attached with an ideal spring of stiffness $k$ and kept on a horizontal surface. If $m_{1}$ is projected with a horizontal velocity $ν_{0}$ , find the maximum compression of the spring.

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Solution

Method 1.
Decide system: Block and spring.
Observation: No external force acting on system in horizontal direction.
Conclusion: Linear momentum as well as mechanical energy of system will be conserved. At the time of maximum compression both the blocks will move with common velocity.
Problem solving: using conservation of linear momentum at initial time and at time of maximum compression of spring. Let the common velocity of system be $V$ .
$P_{i} = m_{1} ν_{0} + 0$ and $P_{f} = (m_{1} + m_{2}) V$
A $P_{i} = P_{f} ⟹ m_{1} ν_{0} = (m_{1} + m_{2}) V$
or $V = \frac{m_{1} ν_{0}}{(m_{1} + m_{2})}$ (i)
Now using conservation of mechanical energy.
$Δ K + Δ U = 0$
$(K_{f} - K_{i}) + Δ U_{s p r i n g} = 0$
$[(m_{1} + m_{2}) V^{2} - \frac{1}{2} m_{1} ν_{0}^{2}] + \frac{1}{2} k x^{2} = 0$ (ii)
From (i) and (ii), we get
$x = [\sqrt{\frac{m_{1} m_{2}}{(m_{1} + m_{2}) k}}] ν_{0}$

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