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Thermal Stress

The value of ...

Question

The value of tension in a long, thin metal wire has been changed from $T_{1}$ to $T_{2}$ . The lengths of the metal wire at two different values of tension $T_{1}$ and $T_{2}$ are $l_{1}$ and $l_{2}$ respectively. The actual length of the metal wire is :

\frac{T_{1} l_{2} - T_{2} l_{1}}{T_{1} - T_{2}}

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\frac{T_{1} l_{1} - T_{2} l_{2}}{T_{1} - T_{2}}

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\frac{l_{1} + l_{2}}{2}

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\sqrt{T_{1} T_{2} l_{1} l_{2}}

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Solution

The correct option is A $\frac{T_{1} l_{2} - T_{2} l_{1}}{T_{1} - T_{2}}$
We know that,

$Y = \frac{F L}{A Δ L}$

$\Rightarrow Y = \frac{T_{1} l_{0}}{A (l_{1} - l_{0})} = \frac{T_{2} l_{0}}{A (l_{2} - l_{0})}$

$\frac{T_{1} l_{0}}{(l_{1} - l_{0})} = \frac{T_{2} l_{0}}{(l_{2} - l_{0})}$

$1 = \frac{T_{1} (l_{2} - l_{0})}{T_{2} (l_{1} - l_{0})}$

$T_{2} l_{1} - T_{2} l_{0} = T_{1} l_{2} - T_{1} l_{0}$

$(T_{1} - T_{2}) l_{0} = T_{1} l_{2} - T_{2} l_{1}$

$∴ l_{0} = \frac{T_{1} l_{2} - T_{2} l_{1}}{T_{1} - T_{2}}$

Hence, option $(A)$ is the correct answer.

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