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Two rods A and B of same material and length have their radii r₁ and r₂ respectively. When they are rigidly fixed at one end and twisted by the same couple applied at the free end, the ratio
$\frac{\emptyset_{A}}{\emptyset_{B}} = \frac{angle of twist at the end of A}{angle of twist at the end of B}$

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$\frac{r_{1}^{2}}{r_{2}^{2}}$

$\frac{r_{1}^{3}}{r_{2}^{3}}$

$\frac{r_{2}^{4}}{r_{1}^{4}}$

$\frac{r_{1}^{4}}{r_{2}^{4}}$

answer is C.

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Detailed Solution

The angle of twist for a rod of length L and radius r that has been fastened at one end and applied a torque τ to at the other end is given by

$θ = \frac{2 τL}{{πr}^{4} N}$

As L,τ,η are same for both the wires.

$θ \propto \frac{1}{r^{4}} \Rightarrow \frac{θ_{1}}{θ_{2}} = {(\frac{r_{2}}{r_{1}})}^{4} \Rightarrow \frac{θ_{1}}{θ_{2}} = \frac{{r_{2}}^{4}}{{r_{1}}^{4}}$

Hence the correct answer is $\frac{{r_{2}}^{4}}{{r_{1}}^{4}} .$

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