Question

The two circular shafts made of same material, one solid(S) and one hollow(H), have the same length and polar moments of inertia. Both are subjected to same torque. Here ${\theta }_S$ is the twist and ${\tau }_S$ is the maximum shear stress in the solid shaft, whereas ${\theta }_H$ is the twist and ${\tau }_H$ is the maximum shear stress in the hollow shaft. Which one of the following is TRUE?

• A. ${\theta }_S={\theta }_{H}and{\tau }_S={\tau }_H$
• B. ${\theta }_S>{\theta }_{H}and{\tau }_S>{\tau }_H$
• C. ${\theta }_S={\theta }_{H}and{\tau }_S<{\tau }_H$
• D. ${\theta }_S={\theta }_{H}and{\tau }_S={\tau }_H$

Answer 1

The correct option is C ${\theta }_S={\theta }_{H}and{\tau }_S<{\tau }_H$

$L_S=L_H,T_S=T_H=T,I_{pS}=I_{pH}\therefore \frac{\pi }{32}{d}^4=\frac{\pi }{32}[d_o^4-d_i^4]\therefore d_0>d\therefore r_0>r$

Angle of twist,

$\theta =\frac{TL}{G{I}_p}\therefore {\theta }_S={\theta }_H{\tau }_{ma{x}_S}=\frac{T}{I_p}.r{\tau }_{ma{x}_H}=\frac{T}{I_p}.r_0\therefore {\tau }_{ma{x}_H}>{\tau }_{ma{x}_S}$