Under normal operating conditions a motor exerts a torque of...

Under normal operating conditions a motor exerts a torque of magnitude $T_{F}=1200$ lb\cdotin. at $F .$ Knowing that $r_{D}=8$ in., $r_{G}=3$ in. and the allowable shearing stress is $10.5 \mathrm{ksi}$ in each shaft, determine the required diameter of $(a)$ shaft $C D E$ $(b)$ shaft $F G H$.

Key Concepts

Shaft Diameter Sizing

Shaft diameter sizing involves determining the minimum required diameter of a shaft that can safely transmit a given torque without exceeding an allowable shear stress. This process utilizes the torsion formula and the polar moment of inertia, ensuring that the design maintains structural integrity and meets safety criteria.

Torque

Torque is the rotational equivalent of force, representing a moment that tends to produce rotation about an axis. It is defined as the product of the applied force and the distance from the point of rotation, and it is a fundamental concept in mechanical design and power transmission systems.

Shear Stress in Torsion

When a shaft is subjected to a twisting moment or torque, it experiences shear stress. This stress is distributed over the shaft’s cross-section, and the maximum shear stress typically occurs at the outer surface. Proper consideration of this stress is essential to ensure that the material does not exceed its allowable shear strength, thereby avoiding failure.

Torsion Formula

The torsion formula relates torque, shear stress, the outer radius of the shaft, and the polar moment of inertia. It provides a direct method to calculate the maximum shear stress in a circular shaft, making it crucial for designing shafts that must safely transmit a given torque without yielding.

Polar Moment of Inertia

The polar moment of inertia is a geometrical property of a cross-section that quantifies its resistance to torsion. For circular shafts, it is directly related to the shaft's diameter and is used in the torsion formula to determine how the applied torque will result in shear stress. A larger polar moment of inertia indicates a greater ability of the shaft to resist twisting forces.

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