Question

Two rods $A$ and $B$ of the same material and length have radii $r_1$ and $r_2$ respectively. When they are rigidly fixed at one end twisted by the same torque applied at the other end, the ratio $\left[\frac{theangleoftwistattheendofA}{theangleoftwistattheendofB}\right]$ equals to:

• A. $\frac{r_1^2}{r_2^2}$
• B. $\frac{r_1^3}{r_2^3}$
• C. $\frac{r_2^4}{r_1^4}$
• D. $\frac{r_1^4}{r_2^4}$

Answer 1

The correct option is C $\frac{r_2^4}{r_1^4}$

A rod of length $L$ and radius $r$ fixed at one end and twisted by applying a torque $\tau$ at the other end, then the angle of twist $\theta$ is given by
$\theta =\frac{2\tau L}{\pi r^{4}N}$
where $\eta$ is shear modulus of the material of the wire
As $L,\tau ,\eta$ are same for both the wires.
$\therefore \theta \propto \frac{1}{r^4}or\frac{{\theta }_1}{{\theta }_2}={\left(\frac{r_2}{r_1}\right)}^4$