Question

Calculate the work done when a wire of length l and area of cross-section A is made of material of young's Modulus Y is stretched by an amount x?

Answer 1

Lets consider,

$Y=$ young's modulus

$A=$ cross-sectional area

$l=$ length of the wire

$x=$ change in length

Young's modulus is given by,

$Y=\frac{\frac{F}{A}}{\frac{x}{l}}$

$F=YA\frac{x}{l}$. . . . .(1)

The work done,

$dW=F.dx$

$dW=\frac{YA}{l}xdx$

$W=\frac{YA}{l}{\int }_0^{l}xdx$

$W=\frac{YA}{l}\frac{l^2}{2}$

$W=\frac{YAl}{2}$