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If 2 wire o...

Question

If $2$ wire of length $L_{1}$ and $L_{2}$ and Young's modulus $Y_{1}$ and $Y_{2}$ are in series then effective Young's modulus is

A

\frac{y_{1} L_{1} + y_{2} L_{2}}{L_{1} + L_{2}}

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B

\frac{y_{1} L_{2} + y_{2} L_{1}}{L_{1} + L_{2}}

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C

\frac{y_{1} y_{2} (L_{1} + L_{2})}{L_{1} y_{2} + L_{2} y_{1}}

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D

\frac{y_{1} + y_{2}}{2}

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Solution

The correct option is C $\frac{y_{1} y_{2} (L_{1} + L_{2})}{L_{1} y_{2} + L_{2} y_{1}}$
$\begin{matrix} we know & k = \frac{F}{Δ L} k = & \frac{y_{A}}{L} \end{matrix}$
$\begin{matrix} ∴ y & = \frac{F L}{A Δ L} Δ L & = \frac{F L}{A Y} \end{matrix}$
$\Rightarrow \frac{\frac{1}{k e q}}{\Rightarrow \frac{L e q}{K_{1}} + \frac{1}{K_{2}}} Y_{e q} A = \frac{L_{1}}{y_{1} A} + \frac{L_{2}}{Y_{2} A} \Rightarrow Y_{e q} = \frac{(L_{1} + L_{2}) y_{1} y_{2}}{L_{1} Y_{2} + L_{2} Y_{1}}$

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