Category : JEE Main & Advanced
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
\[W=\frac{1}{2}C{{\theta }^{2}}=\frac{\pi \eta {{r}^{4}}{{\theta }^{2}}}{4l}\]
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