Example $3$

The solid shaft of radius $\text{'}c\text{'}$ is subjected to a torque T$.$ Determine the fraction of T that is resisted by the material contained within the outer region of the shaft, i$.$e$.\frac{c}{2}c$

First draw the problem figure

Shear stress due to torque T is given by$,$

$=0$ when $r=0$

$=\frac{T}{J}=\frac{Tc}{J}={}_{\text{m}\text{a}\text{x}}$ when $r=c$ or $=\frac{}{c}{}_{\text{m}\text{a}\text{x}}$

Shear stress increases linearly from origin to outer radius

We want to find the torque carried by outer region. We can use stress to find force.