Work done in twisting a wire or cylinder:

Let us consider a cylindrical wire of radius r and length l. The wire is clamped at the upper end and a twisting couple is applied at the lower end in a plane perpendicular to its length with its axis coinciding with that of the wire. This produces a twist of an angle \( \theta \) at this end.

The couple to produce a twist \( \theta \) in this wire is \( \displaystyle{\tau\theta} \).
Where \( \tau \) is the twisting couple per unit twist, i.e., torsional rigidity of the wire.

Now, the work done in twisting the couple through a small angle \( d\theta \) is \( dw=\tau\theta{d\theta} \).

Therefore the total work done in twisting the wire through an angle \( \theta \) is given by,

\( \displaystyle{W=\int_0^\theta\tau\theta{d\theta}=\frac{1}{2}\tau{\theta}^2} \)

Now we know that \( \displaystyle{\tau=\frac{\pi\eta{r^4}}{2l}} \) [Read In Detail],
where \( \eta \) is the modulus of rigidity of the material of the wire.

Therefor, \( \displaystyle{W=\frac{1}{2}\frac{\pi\eta{r^4}}{2l}{\theta}^2} \)
or, \( \displaystyle{W=\frac{\pi\eta{r^4}}{4l}{\theta}^2} \)