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

Question

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:

TestHub · Accepted Answer

The formula to calculate the Young's modulus of the material of the wire, in the first case, can be written as

$Y = \frac{\frac{F}{π r^{2}}}{\frac{l}{L}} . . . (1)$

From equation (1), it follows that

$F = Y π r^{2} \times \frac{l}{L} . . . (2)$

For the second case, the Young's modulus is given by

$Y = \frac{\frac{F / 2}{π r^{2} / 4}}{\frac{l'}{L}} . . . (3)$

From equation (3), it follows that

$F = Y \frac{l'}{L} \times 2 \times \frac{π r^{2}}{4} . . . (4)$

From equations (2) and (4), it implies that

$Y π r^{2} \times \frac{l}{L} = Y \frac{l'}{L} \times 2 \times \frac{π r^{2}}{4} \Rightarrow l' = 2 l$

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