JEE Main & Advanced Physics Elasticity Torsion of Cylinder

If the upper end of a cylinder is clamped and a torque is applied at the lower end the cylinder gets twisted by angle \[\theta \]. Simultaneously shearing strain \[\phi \] is produced in the cylinder.

(i) The angle of twist \[\theta \]  is directly proportional to the distance from the fixed end of the cylinder.

At fixed end \[\theta ={{0}^{o}}\] and at free end \[\theta =\]  maximum.

(ii) The value of angle of shear \[\phi \] is directly proportional to the radius of the cylindrical shell.

At the axis of cylinder \[\phi =0\] and at the outermost shell \[\phi =\] maximum.

(iii) Relation between angle of twist \[(\theta )\] and angle of shear \[(\phi )\]           

\[AB=r\theta =\phi l\] \[\therefore \]\[\varphi =\frac{r\theta }{l}\]

(iv) Twisting couple per unit twist or torsional rigidity or torque required to produce unit twist.

\[C=\frac{\pi \eta {{r}^{4}}}{2l}\] \[\therefore \] \[C\propto {{r}^{4}}\propto {{A}^{2}}\]

(v) Work done in twisting the cylinder through an angle \[\theta \]  is

\[W=\frac{1}{2}C{{\theta }^{2}}=\frac{\pi \eta {{r}^{4}}{{\theta }^{2}}}{4l}\]