Question

Equal torsional torques act on two rods X and Y having equal length but the diameter of Y is twice that wire X. If ${\theta }_x$ and ${\theta }_y$ are angles of twist,

• A. ${\theta }_x=\frac{1}{2}{\theta }_y$
• B. ${\theta }_x=\frac{1}{4}{\theta }_y$
• C. ${\theta }_y=\frac{{\theta }_x}{8}$
• D. ${\theta }_y=\frac{1}{16}{\theta }_x$

Answer 1

The correct option is D ${\theta }_y=\frac{1}{16}{\theta }_x$

Torsional torque for rod X is ${\tau }_x=\frac{\pi \eta r_x^4}{2l}{\theta }_x$ where $\eta$ shear modulus.
Torsional torque for rod Y is ${\tau }_y=\frac{\pi \eta r_y^4}{2l}{\theta }_y$
Here diameter : $d_y=2d_x\Rightarrow 2r_y=2\left(2r_x\right),\frac{r_y}{r_x}=2$
also ${\tau }_x={\tau }_y\Rightarrow r_x^4{\theta }_x=r_y^4{\theta }_y$
$\therefore {\theta }_y=(r_x/r_y{)}^4{\theta }_x=(1/2{)}^4{\theta }_x=\frac{1}{16}{\theta }_x$