Question

Equal torsional torques act on two rods $x$ and $y$ having equal length. The diameter of rod $y$ is twice the diameter of rod $x$. If ${\theta }_x$ and ${\theta }_y$ are the angle of twist, then $\frac{{\theta }_x}{{\theta }_y}=$

Answer 1

Torsional torque for rod $x$ is
${\tau }_x=\frac{\pi \eta r_x^4{\theta }_x}{2l}$
for rod $y$ is ${\tau }_y=\frac{\pi \eta r_y^4}{2l}{\theta }_y$
Given $d_y=2d_x\Rightarrow \frac{r_y}{r_x}=2$
Also ${\tau }_x={\tau }_y\Rightarrow r_x^4{\theta }_x=r_y^4{\theta }_y$
$\therefore {\theta }_y={\left(\frac{r_x}{r_y}\right)}^4{\theta }_x={\left(\frac{1}{2}\right)}^4{\theta }_x=\frac{1}{16}{\theta }_x$
${\theta }_y=\frac{1}{16}{\theta }_x$
$\frac{{\theta }_x}{{\theta }_y}=16$