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 and twisted by the same torque applied at the other end, the ratio 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 and twisted by the same torque applied at the other end, the ratio $\left(\frac{\text{the angle of twist at the end of A}}{\text{the angle of twist at the end of B}}\right)$

• A. $\frac{r_{1}^{2}}{r_{1}^{2}}$
• 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

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 $\theta$ $=\frac{2\tau L}{\pi r^{4}\eta}$ where $\eta$ is shear modulus of the material of the wire.As $L$, $\tau$ and $\eta$ are same for both the wires $\therefore$ $\quad$$\theta \propto\frac{1}{r^{4}}$ or $\frac{\theta_{1}}{\theta_{2}}$ $=\left(\frac{r_{2}}{r_{1}}\right)^{4}$