A wire of length L and radius r is clamped at one end. If its other end is pulled by a force F, its length increases by l. If the radius of the wire and the applied force both are reduced to half of their original values keeping original length constant,

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Solution

Q.
A wire of length $L$ and radius $r$ is clamped at one end. If its other end is pulled by a force $F$ , its length increases by $l$ . If the radius of the wire and the applied force both are reduced to half of their original values keeping original length constant, the increase in length will become [2024]

1 3 times

2 4 times

3 $\frac{3}{2}$ times

4 2 times

Ans.
(4)

Young's modulus, $Y = \frac{s t r e s s}{s t r a i n} = \frac{F / A}{∆ l / L} = \frac{F L}{A ∆ l}$

In case I : $Y = \frac{F L}{{πr}^{2} l}$ ...(i)

In case II : $Y = \frac{(F / 2) L}{π {(\frac{r}{2})}^{2} ∆ l} = \frac{2 F L}{π r^{2} ∆ l}$ ...(ii)

From (i) and (ii), $\frac{F L}{π r^{2} l} = \frac{2 F L}{π r^{2} ∆ l} \Rightarrow ∆ l = 2 l$

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