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Two springs of equal lengths and equal cross-sectional areas are made of materials whose Young's modulii are in the ratio of 3 : 2. They are suspended and loaded with the same mass. When stretched and released, they will oscillate with time periods in the ratio of

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3:2

$\sqrt{3} : \sqrt{2}$

$3 \sqrt{3} : 2 \sqrt{2}$

9:4

answer is A.

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Detailed Solution

Young's modulus $Y = \frac{F}{A} . \frac{L}{l}$

Force constant $k = \frac{F}{l} = \frac{Y A}{L}$

where l is the extension in the spring of original length L and cross-sectional area A when. a force F = Mg is applied. Now, the time period of vertical oscillations is given by

$T = 2 π \sqrt{\frac{M}{K}} = 2 π \sqrt{\frac{ML}{YA}} ∴ \frac{T_{1}}{T_{2}} = \sqrt{\frac{Y_{2}}{Y_{1}}} = \sqrt{\frac{3}{2}}$

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