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Question

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


A
3:2
loader
B
3 : 2
loader
C
33:22
loader
D
9 : 4
loader

Solution

The correct option is A $$\sqrt{3} : \sqrt{2}$$

Young modulus $$Y=\dfrac{(F/A)}{l/L}=\dfrac{FL}{lA}$$  where $$l$$ is the extension of spring of original length L and A be the cross-sectional area and $$F=mg$$ , force applied.

now force constant $$k=\dfrac{F}{l}=\dfrac{AY}{L}$$
Time period $$T=2\pi\sqrt{\dfrac{m}{k}}=2\pi\sqrt{\dfrac{mL}{AY}}$$
thus, $$\dfrac{T_1}{T_2}=\sqrt {\dfrac{Y_2}{Y_1}}=\sqrt{\dfrac{3}{2}}$$ 


Physics

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