Figure (a) shows a spring attached to ceiling in its natural length. A mass 2m suspended from a given spring causing it to stretch relative to its natural length as shown in figure (b). The mass and the spring are then each cut into two identical pieces and connected as shown in figure (c). Measured from diagram (a), the displacements covered by bottom most point of spring in (b) is $x_{1}$ and the displacement of bottom most point of lower part of spring in (c) is x2. Find the value of $24 \times \frac{x_{1}}{x_{2}} .$

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answer is 32.

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Detailed Solution

Assuming k to be spring constant of the original spring, for diagram (b)
$k x_{1} = 2 m g \Rightarrow x_{1} = \frac{2 m g}{k}$
For the diagram (c), considering $y_{1} a n d y_{2}$ to be the elongations in upper and lower spring parts,
$2 k y_{2} = m g$ and $2 k y_{1} = 2 m g$
$x_{2} = y_{1} + y_{2} = \frac{3 m g}{2 k}$
Thus, $\frac{x_{1}}{x_{2}} = \frac{4}{3}$