(Solved): A shaft of solid circular cross-section consisting of two segments is shown in figure QB2a (the top ...

A shaft of solid circular cross-section consisting of two segments is shown in figure QB2a (the top part of Figure QB2). The left-hand segment has diameter $80\mathrm{mm}$ and length $1.2\mathrm{m}$; the right-hand segment has diameter $60\mathrm{mm}$ and length $0.9\mathrm{m}$. Shown in Figure QB2b (the bottom part of the Figure QB2) is a tubular shaft made of the same material and having the same length. The thickness of the tube is $t=d/5$, where $d$ is the external diameter. Both shafts are subjected to the same torque $T$. a. If the hollow shaft is to have the same torsional stiffness as the solid shaft, what should be its external diameter $d$ ? [10 marks] b. If the shear stress is not to exceed $30\mathrm{MPa}$ what is the allowable torque? [10 marks] c. Assuming that tubes are made of steel with shear modulus $G=$ $76\mathrm{GPa}$ determine the angle of twist ${\theta }_s$ in the solid shaft, subjected to the maximum allowable torque; demonstrate 2 ways of determining this angle of twist. [10 marks] Figure QB2: Cross-sections of tubes