Three arrangements of spring-mass system are shown in figures (A), (B) and (C), If $T_{1}$ , $T_{2}$ and $T_{3}$ represent the respective periods of oscillation, then correct relation is :

T_{1} > T_{2} > T_{3}

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T_{3} > T_{2} > T_{1}

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T_{2} > T_{1} > T_{3}

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T_{2} > T_{3} > T_{1}

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Solution

The correct option is B $T_{2} > T_{1} > T_{3}$
The restoring force $F = - k x$ , where $x$ is the elongation or compressed length.
Now for the figure A, when the mass $m$ moves a distance $x$ downwards, the spring is also elongated with the same distance $x$ . $P . E . = \frac{1}{2} k x^{2}$ So the restoring force for fig A is $F = - k x$
For the figure B, when the mass $m$ moves a distance $x$ downwards, the spring is elongated by a distance $\frac{x}{2}$ . $P . E . = \frac{1}{2} k (x / 2)^{2} = \frac{1}{2} (\frac{k}{4}) x^{2}$ . So the restoring force for fig B is $F = - (\frac{k}{4}) x$ .
In the similar way, for fig C, the restoring force is $F = - (4 k) x$ .
$∴$ for fig A, B, C the new spring constant are $k, \frac{k}{4}, 4 k$ respectively.
$T_{1} = 2 π \sqrt{\frac{m}{k}}$
$T_{2} = 2 π \sqrt{\frac{4 m}{k}}$
and $T_{3} = 2 π \sqrt{\frac{m}{4 k}}$

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Three arrangements of spring-mass system are shown in figures (A), (B) and (C), If T1, T2 and T3 represent the respective periods of oscillation, then correct relation is :

Three arrangements of spring-mass system are shown in figures (A), (B) and (C), If $T_{1}$ , $T_{2}$ and $T_{3}$ represent the respective periods of oscillation, then correct relation is :