Question

A wire of length L and radius r is clamped rigidly at one end. When the other end of the wire is pulled by a force f, its length increases by l, Another wire of the same material of length 2L and radius 2r, is pulled by a force 2f. find the increase in length of this wire.

Answer 1

Young’s Modulus = \(\frac{F}{A}\times \frac{L}{l},\) where l is the increase in length of wire I

Or I wire, Y = \(\frac{f}{πr^2}\) × \(\frac{L}{l}\)               (i)

For II wire, the increase in length be l′.

Then Y = \(\frac{2f}{4πr^2}\) × \(\frac{L}{l'}\)

Y = \(\frac{f}{πr^2}\)  × \(\frac{L}{l'}\)                          (ii)

From eqⁿ (s) (i) and (ii)

\(\frac{F}{πr^2}\)  × \(\frac{L}{l}\) =  \(\frac{f}{πr^2}\)  × \(\frac{L}{l'}\) 

∴ l = l′