The electric motor exerts a 500-N ·m torque on the aluminum shaft A B C D when it is rotating at

step 1 So let's solve this question. So in this question, we have J equals to pi by 30, 2, T04 minus D

The electric motor exerts a 500-N ·m torque on the aluminum...

The electric motor exerts a $500-\mathrm{N} \cdot \mathrm{m}$ torque on the aluminum shaft $A B C D$ when it is rotating at a constant speed. Knowing that $G=27 \mathrm{GPa}$ and that the torques exerted on pulleys $B$ and $C$ are as shown, determine the angle of twist between $(a) B$ and $C$ (b) $B$ and $D$.

Key Concepts

Torque

Torque is a measure of the force that causes an object to rotate about an axis. It is defined as the product of the applied force and the distance from the point of rotation, and it plays a central role in analyzing the rotational effects on structures such as shafts in mechanical systems.

Torsion in Circular Shafts

Torsion in circular shafts refers to the twisting of a shaft when subjected to an applied torque. This phenomenon is governed by the shaft's material properties and geometric characteristics. When a torque is applied, shear stresses are induced over the cross-section, leading to an angular deformation along the length of the shaft.

Angle of Twist

The angle of twist is the measure of the rotation experienced by a cross-section of a shaft under torsional loading. It is determined by the applied torque, the shaft’s length, its geometry, and the material's shear modulus. The twist angle is critical in assessing the performance and integrity of rotating mechanical components.

Shear Modulus

The shear modulus, also known as the modulus of rigidity, is a material property that quantifies its ability to resist shear deformation. In the context of torsion, it relates the applied shear stress to the resulting shear strain, playing a key role in the calculation of the angle of twist in shafts.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that indicates its resistance to torsion. It depends on the shape and size of the cross-section and is used alongside the applied torque and shear modulus in determining the angular deformation or twist of the shaft.

Superposition Principle in Torsion

The superposition principle in torsion allows for the determination of the overall twist in a shaft by summing the individual twist contributions from each segment. This is useful when different sections of a shaft experience different torques, such as when multiple loads or components influence the twist along the shaft's length.

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