Figure (a) shows a spring of force constant k fixed at one end and carrying a mass m at the other end placed on a horizontal frictionless surface. The spring is stretched by a force F. Figure (b) shows the same spring with both ends free and a mass m fixed at each free end. Each of the spring is stretched by the same force F. The mass in case (a) and the masses in case (b) are then released. Which of the following statements are true ?

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The time period of oscillation is the same in both cases.

The time period of oscillation in case (a) is $\sqrt{2}$ times that in case (b).

While oscillating, the maximum extension of the spring is more in case (a) than in case (b)

The maximum extension of the spring is same in both cases.

answer is B, D.

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Detailed Solution

Extension ; x = F / k ; as F is same in both cases, x will be the same. time period $= 2 π \sqrt{\frac{m}{k}}$

$\Rightarrow \frac{T^{'}}{T} = \sqrt{\frac{k}{k^{'}}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}}$

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