Question

A torque T is applied at the free end of circular cross-sections as shown in the figure. The shear modulus of the material of the rod is G. The expression for d to produce an angular twist $\theta$ at the free end is

• A. ${\left(\frac{32TL}{\pi \theta G}\right)}^{\frac{1}{4}}$
• B. ${\left(\frac{18TL}{\pi \theta G}\right)}^{\frac{1}{4}}$
• C. ${\left(\frac{16TL}{\pi \theta G}\right)}^{\frac{1}{4}}$
• D. ${\left(\frac{2TL}{\pi \theta G}\right)}^{\frac{1}{4}}$
  

Answer 1

The correct option is B ${\left(\frac{18TL}{\pi \theta G}\right)}^{\frac{1}{4}}$

By using torsion equation, we have,

$\frac{T}{J}=\frac{\tau }{r}=\frac{G\theta }{L}\therefore \Sigma \theta =\Sigma \frac{TL}{GJ}\therefore \theta =\frac{T\left(\frac{L}{2}\right)}{G\left(\frac{\pi }{32}\right)d^4}[\frac{1}{2}+\frac{1}{16}]\theta =\frac{TL}{G\left(\frac{\pi }{32}{d}^4\right)}\times \left(\frac{9}{16}\right)=\frac{18TL}{G\pi d^4}$

$ord={\left(\frac{18TL}{\pi \theta G}\right)}^{\frac{1}{4}}$