Example $5$

A solid shaft and a hollow shaft are made of the same material and are of the same weight and length. The hollow shaft has internal and external radii $c_1$ and $c_2$ respectively. Denoting by n the ratio $\frac{c_1}{c_2},$ show that the ratio $\frac{T_s}{T_h}$ of the torque $T_s$ in the solid shaft to the torque $T_h$ in the hollow shaft is $($a$)\frac{(1-n^2{)}^{0\mathrm{.}5}}{1+n^2}$ if the maximum shearing stress is the same in each shaft, $($b$)\frac{1-n^2}{1+n^2}$ if the angle of twist is the same for each shaft.