The length of a uniform metal wire is observed to be $l_{1}$ and $l_{2}$ under the stretching forces $F_{1}$ and $F_{2}$ . The natural length of the wire is

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

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

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

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

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Solution

The correct option is C $\frac{l_{1} F_{2} - l_{2} F_{1}}{F_{2} - F_{1}}$
Let, natural length be L.
And, $Y = \frac{F L}{A l}$
when, force $F_{1}$ and $l = (l_{1} - L)$ then, $Y = \frac{F_{1} L}{A (l_{1} - L)} ⟶ (i)$
and when force $F_{2}$ and $l = (l_{2} - L)$ then, $Y = \frac{F_{2} L}{A (l_{2} - L)} ⟶ (i i)$
from equation (i) and (ii),
$\frac{F_{1} L}{A (l_{1} - L)} = \frac{F_{2} L}{A (l_{2} - L)}$
or, $F_{1} (l_{2} - L) = F_{2} (l_{1} - L)$
or, $F_{1} l_{2} - F_{1} L = F_{2} l_{1} - F_{2} L$
or, $L (F_{2} - F_{1}) = F_{2} l_{1} - F_{2} L$
or, $L = \frac{l_{1} F_{2} - l_{2} F_{1}}{F_{2} - F_{1}}$

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The length of a uniform metal wire is observed to be l1 and l2 under the stretching forces F1 and F2. The natural length of the wire is

The length of a uniform metal wire is observed to be $l_{1}$ and $l_{2}$ under the stretching forces $F_{1}$ and $F_{2}$ . The natural length of the wire is