Scenario: A large truck's drive shaft catastrophically fails while carrying a heavy load up a steep hill.

Analysis: The drive shaft transmits power from the transmission to the rear axle, subjecting it to immense torsional moments (torque). As torque increases, torsional shear stress develops within the shaft, reaching its maximum at the outer surface. If this maximum stress exceeds the material's shear strength, the shaft fails. Forensic engineers examine the broken shaft and observe a clean, $45^\circ$ helical fracture pattern. This is a classic signature of torsional failure in brittle materials (or ductile materials that failed under fatigue). Understanding the relationship between transmitted power, rotational speed, and resulting torque is critical for sizing drive shafts correctly.

A solid circular steel shaft has a diameter of $40\text{ mm}$ and is subjected to a torque of $500\text{ N}\cdot\text{m}$. Determine the maximum torsional shear stress in the shaft.

A hollow steel shaft has an outer diameter of $60\text{ mm}$ and an inner diameter of $40\text{ mm}$. If it transmits a torque of $1.5\text{ kN}\cdot\text{m}$, find the maximum and minimum shear stresses.

A solid steel shaft ($G = 83\text{ GPa}$) is $2\text{ m}$ long and has a diameter of $30\text{ mm}$. If it is subjected to a torque of $300\text{ N}\cdot\text{m}$, calculate the angle of twist in degrees.

A stepped steel shaft ($G = 80\text{ GPa}$) consists of two solid segments. Segment AB has a diameter of $50\text{ mm}$ and length $1.2\text{ m}$. Segment BC has a diameter of $30\text{ mm}$ and length $0.8\text{ m}$. A torque of $800\text{ N}\cdot\text{m}$ is applied at end C, while end A is fixed. Calculate the total angle of twist at C.

A flanged bolt coupling is used to connect two solid shafts. The coupling uses 6 bolts, each $12\text{ mm}$ in diameter, arranged on a bolt circle of diameter $150\text{ mm}$. If the allowable shear stress in the bolts is $60\text{ MPa}$, what is the maximum torque the coupling can transmit?

A solid steel shaft is required to transmit $50\text{ kW}$ of power at $600\text{ rpm}$. The allowable shear stress is $50\text{ MPa}$. Determine the required minimum shaft diameter.