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Existence of Limit

The length of...

Question

The length of a metal wire is $l_{1}$ when the tension in it is $T_{1}$ and is $l_{2}$ when the tension is $T_{2}$ . The natural length of the wire is :

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

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

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

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

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Solution

The correct option is C $\frac{l_{1} T_{2} - l_{2} T_{1}}{T_{2} - T_{1}}$
If $Δ l$ be the elongation due to tension T, then $T \propto Δ l$
Thus, $T_{1} \propto Δ l_{1}$ and $T_{2} \propto Δ l_{2}$ where $l =$ natural length of wire.
So, $\frac{T_{1}}{T_{2}} = \frac{Δ l_{1}}{Δ l_{2}} . . . . (1)$
Now, $l_{1} = l + Δ l_{1}$ and $l_{2} = l + Δ l_{2}$
So, $l_{1} T_{2} = T_{2} l + T_{2} Δ l_{1} . . . (2)$ and
$l_{2} T_{1} = T_{1} l + T_{1} Δ l_{2} . . . (3)$
using (1), (3) becomes $l_{2} T_{1} = T_{1} l + T_{2} Δ l_{1} . . . (4)$
$(2) - (4) \to l_{1} T_{2} - l_{2} T_{1} = T_{2} l - T_{1} l$
or $l = \frac{l_{1} T_{2} - l_{2} T_{1}}{T_{2} - T_{1}}$

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