Question

Within the elastic limit, find the work done by a stretching force on a wire.

Answer 1

a.

Let,
L = length of wire
A = area of cross section of wire
r = radius of cross section of wire
l = elongation of the wire by applying load.

b.

If the wire is perfectly elastic then,
Young’s modulus,

`Y=(F/A)/(l/L)`

`=F/AxxL/l`

`F=(YAl)/L`..............(1)

c.   Let ‘f’ be the restoring force and ‘x’ be its corresponding extension at certain instant during the process of extension.

`f=(YAx)/L`................(2)

d.  Let ‘dW’ be the work done for the further small extension ‘dx’.

 dW=fdx

`dW=(YAx)/Ldx` ..........(3)

e.  The total amount of work done in stretching the wire from 0 to l can be found out by integrating equation (3).

`W=int_0^ldW=int_0^l(YAx)/Ldx=(YA)/L int_0^lx dx`

`W=(YA)/L[x^2/2]_0^l`

`W=(YAl^2)/2L`

`W=(YAl)/L.l/2`

But,`(YAl)/L=F`

`therefore W=1/2xxfxxl` ..................(4)

Equation (4) represents the work done by stretching a wire.