Angle of Twist: Problem Solving

To calculate the angle of twist between pulleys, make a cut between them and analyze the cut cross-section using a free-body diagram. Apply equilibrium principles to find the internal torque, then calculate the polar moment of inertia, which is proportional to the fourth power of the shaft radius. Substitute all known parameters—torque, shaft length, diameter, and modulus of rigidity—into the angle of twist formula to determine the result in radians, then convert to degrees if needed.

The polar moment of inertia is a geometric property that quantifies a shaft's resistance to twisting. It is proportional to the fourth power of the shaft radius, meaning small increases in diameter significantly increase resistance to torsion. This parameter is essential in the angle of twist equation; larger polar moments result in smaller twist angles for the same applied torque and material properties.

The modulus of rigidity, also called the shear modulus, measures a material's resistance to shear deformation under torsional loading. It directly appears in the angle of twist formula; materials with higher modulus values resist twisting better and produce smaller twist angles. For the aluminum shaft in this problem, the modulus of rigidity of 25 GPa determines how much the shaft twists under the applied torques.

When a cut is made between two pulleys, the equilibrium principle states that internal torques must balance external torques. If pulley B exerts an anticlockwise torque, the internal torque at the cut cross-section must be equal in magnitude but opposite in direction to maintain equilibrium. This equilibrium torque is then used in the angle of twist calculation for that shaft section.

Shaft diameter dramatically affects resistance to twisting because the polar moment of inertia depends on the fourth power of the radius. Doubling the diameter increases the polar moment by a factor of 16, substantially reducing the angle of twist for the same applied torque. This relationship makes diameter a critical design parameter in transmission shaft applications.

The angle of twist formula naturally produces results in radians because it uses standard engineering equations based on radian measure. Converting to degrees makes the result more intuitive for practical interpretation and communication. Both units represent the same physical rotation; degrees simply provide a more familiar scale for engineers and technicians reviewing the calculations.

Multiple pulleys create different torque values in different shaft sections. In this problem, the motor applies 700 N-m, but pulleys B and C extract 300 N-m and 400 N-m respectively. Each section between pulleys experiences different internal torques, requiring separate free-body diagram analysis and angle of twist calculations for each segment to fully characterize the shaft's deformation.