First slide

Elastic behaviour of solids

Question

Two rods A and B of the same material and length have radii r₁ and r₂ respectively. When they are rigidly fixed at one end and twisted by the same torque applied at the other end, the ratio $[\frac{the angle of twist at the end of A}{the angle of twist at the end of B}]$ equals to

Difficult

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

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

Solution

A rod of length L and radius r are fixed at one end and twisted by applying a torque $τ$ at the other end, then the angle of twist $θ$ is given by
$θ = \frac{2 τL}{{πr}^{4} η}$
where $η$ is shear modulus of the material of the wire.
As L, $τ$ and $η$ are same for both the wires
$∴ θ \propto \frac{1}{r^{4}} or \frac{θ_{1}}{θ_{2}} = {(\frac{r_{2}}{r_{1}})}^{4}$

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