Question

A shaft is fixed at one end and free at another end. It carries uniformly distributed torque (t kN - m/m) as shown in the figure. The shaft is prismatic and have constant GJ. The angle of twist at the free end is:

• A. $\frac{t{L}^2}{8GJ}$
• B. $\frac{t{L}^2}{2GJ}$
• C. $\frac{3t{L}^2}{8GJ}$
• D. None of these

Answer 1

The correct option is C $\frac{3t{L}^2}{8GJ}$

We know, $\frac{T}{J}=\frac{G\theta }{L}\int d\theta =\int \frac{T_{x}dx}{GJ}$
Now, $T_x=\{\begin{array}{c}tx0<x<\frac{L}{2}\\ \frac{tL}{2}\frac{L}{2}<x<L\end{array}$
$\theta ={\int }_0^{L/2}\frac{txdx}{GJ}+{\int }_{L/2}^L\left(\frac{tL}{2}\right)\frac{dx}{GJ}$
$=\frac{t}{GJ}{\left(\frac{x^2}{2}\right)}_0^{L/2}+\left(\frac{tL}{2}\right)\frac{1}{GJ}(L-\frac{L}{2})$
$=\frac{t}{2GJ}\left(\frac{L^2}{4}\right)+\frac{tL}{2GJ}\times \frac{L}{2}$
$=\frac{t{L}^2}{8GJ}+\frac{t{L}^2}{4GJ}=\frac{3t{L}^2}{8GJ}$