Question

Two rods $A$ and $B$ of same length and made of same material have their radii $R_1$ and $R_2$ respectively. If the rods are rigidly fixed at one end and twisted by the same couple applied at the free end, then what is the ratio $\frac{\text{angle of twist at the end of A}}{\text{angle of twist at the end of B}}$.

Answer 1

Shear modulus$\left(\eta \right)=$$\frac{\frac{F}{a}}{\varphi }$

$\therefore \varphi =\frac{F}{a\times \eta }$

$\therefore \frac{{\varphi }_A}{{\varphi }_B}=\frac{F}{a_A\times \eta }\times \frac{a_B\times \eta }{F}$

$=\frac{a_B}{a_A}=\frac{R_2^2}{R_1^2}$

$\therefore {\varphi }_A:{\varphi }_B=R_2^2:R_1^2$