19.5:
Angle of Twist: Problem Solving
The electric motor exerts a 700 N-m torque on an aluminium shaft, causing it to rotate at a constant speed. Pulleys B and C experience torques of 300 N-m and 400 N-m, respectively. The modulus of rigidity is given as 25 GPa. If the length and diameter of each section are known, calculate the angle of twist between the pulleys B and C.
First, a cut is made between pulleys B and C, and a free-body diagram of the cut cross-section is considered.
The torque acting on the pulley B is anticlockwise. So, using the principle of equilibrium, the torque at the cut cross-section of the shaft will be equal and opposite to the torque at pulley B.
Next, the polar moment of inertia at the cut cross-section, which is proportional to the fourth power of the radius of the shaft, is calculated.
By substituting all the known parameters, the angle of twist between pulleys B and C is determined. The angle obtained is in radians and can be converted to degrees.
19.5:
Angle of Twist: Problem Solving
An electric motor applies a torque of 700 N·m to an aluminum shaft, triggering a stable rotation. Two pulleys, B and C, are subjected to torques of 300 N·m and 400 N·m, respectively. The modulus of rigidity is provided as 25 GPa. With the knowledge of the length and diameter of each segment, the twist angle between the two pulleys can be computed. First, a section cut is made between pulleys B and C, and the cut cross-section is analyzed using a free-body diagram. Given that the torque exerted on pulley B is in an anticlockwise direction, the equilibrium principle dictates that the torque at the cut cross-section of the shaft should match this but in the opposite direction.
The polar moment of inertia is then calculated at the cut cross-section. This value is proportional to the radius of the shaft raised to the fourth power. Then, by inserting all the known parameters into the equation for calculating the angle of twist, it is possible to determine the twist angle between pulleys B and C.
The result is initially in radians but can be converted into degrees for easier interpretation. This process allows for understanding the system's mechanical behavior under the applied torques.