If A is the area of cross-section of a spring | vedclass

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(D) According to the definition of Young's modulus $(E)$:$E = \frac{F L}{A \Delta L}$where $F$ is the force,$L$ is the original length,$A$ is the cros

Vedclass · Accepted Answer

$T = 2\pi \sqrt {\frac{{ML}}{{EA}}} ,k = \frac{{EA}}{L}$

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(D) According to the definition of Young's modulus $(E)$:$E = \frac{F L}{A \Delta L}$where $F$ is the force,$L$ is the original length,$A$ is the cross-sectional area,and $\Delta L$ is the extension.Rearranging for force $(F)$:$F = \left( \frac{EA}{L} \right) \Delta L$  --- $(1)$According to Hooke's law,the restoring force is given by:$F = k \Delta L$  --- $(2)$Comparing equations $(1)$ and $(2)$,we get the force constant $(k)$:$k = \frac{EA}{L}$The time period $(T)$ of a spring-mass system is given by:$T = 2\pi \sqrt{\frac{M}{k}}$Substituting the value of $k$:$T = 2\pi \sqrt{\frac{M}{EA/L}} = 2\pi \sqrt{\frac{ML}{EA}}$Thus,the time period is $2\pi \sqrt{\frac{ML}{EA}}$ and the force constant is $\frac{EA}{L}$.

If $A$ is the area of cross-section of a spring,$L$ is its length,$E$ is the Young's modulus of the material of the spring,then the time period and force constant of the spring will be respectively:

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