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Stress-Strain Graph

Two blocks of...

Question

Two blocks of masses $m_{1}$ and $m_{2}$ joined to a non deformed spring of length $l_{0}$ and stiffness $k$ as shown in figure. If a force $F$ is applied on block of mass $m_{2}$ . Find the maximum separation between the blocks.

l_{0} + \frac{m_{1} F}{k (m_{1} + m_{2})}

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l_{0} + \frac{m_{2} F}{k (m_{1} + m_{2})}

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l_{0} + \frac{2 m_{2} F}{k (m_{1} + m_{2})}

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l_{0} + \frac{2 m_{1} F}{k (m_{1} + m_{2})}

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Solution

The correct option is B $l_{0} + \frac{2 m_{1} F}{k (m_{1} + m_{2})}$
$x_{1}$ and $x_{2}$ be the maximum displacement in $m_{1}$ and $m_{2}$ respectively.

$a_{C O M} = \frac{F}{m_{1} + m_{2}}$

$\frac{1}{2} k (x_{1} + x_{2})^{2} = \frac{m_{1} F}{m_{1} + m_{2}} x_{1} + (F - \frac{m_{2} F}{m_{1} + m_{2}}) x_{2}$

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

or $x_{1} + x_{2} = \frac{2 m_{1} F}{k (m_{1} + m_{2})}$

Thus maximum separation $= l_{0} + x_{1} + x_{2}$

$= l_{0} + \frac{2 m_{1} F}{k (m_{1} + m_{2})}$

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