Let us consider a hollow cylinder of inner radius \( r_1 \), outer radius \( r_2 \) and length \( l \). \( \eta \) is the rigidity modulus of the material of the hollow cylinder. Then we know that the moment of force or torque about the axis of the cylinder is given by,

\( \displaystyle{\frac{2\pi\eta\theta}{l}x^3{dx}} \), [ Read In Detail ]
where, \( \theta \) is the angle of twist.

So the total twisting couple or torque on the hollow cylinder is given by,

\( \displaystyle{\Gamma=\frac{2\pi\eta\theta}{l}\int_{r_1}^{r_2}x^3{dx}\\=\frac{2\pi\eta\theta}{l}[\frac{x^4}{4}]\vert_{r_1}^{r_2}}\)

or, \( \displaystyle{\Gamma=\frac{\pi\eta\theta}{2l}({r_2}^4-{r_1}^4)} \)