What is the limiting speed of a shaft transmitting 7.5 ×10^5 W if the allowable stress is 1.4 ×

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What is the limiting speed of a shaft transmitting 7.5 ×10^5...

What is the limiting speed of a shaft transmitting $7.5 \times 10^5 \mathrm{~W}$ if the allowable stress is $1.4 \times 10^8 \mathrm{~Pa}$ and the diameter is .15 m ? If $G=1.4 \times 10^{11} \mathrm{~Pa}$, what is the minimam kagth to allow for a rotation of $6^{\circ}$ between the ends?

Key Concepts

Torsional Shear Stress

Torsional shear stress refers to the internal stress generated within a shaft or any cylindrical member when a torque is applied. It is calculated using the applied torque, the geometry of the shaft (specifically the polar moment of inertia), and the radial distance from the center. This concept is crucial for ensuring that the induced stress remains below the material’s allowable limit, thus preventing failure.

Relationship Between Power, Torque, and Angular Velocity

In rotational dynamics, power is transmitted through rotating shafts by the interplay of torque and angular velocity. This relationship is expressed as power equaling torque multiplied by angular velocity. Understanding this concept enables engineers to design systems with suitable dimensions and operational speeds so that neither the torque nor the resulting stresses exceed the safe limits.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross-section that quantifies its resistance to torsion. For circular shafts, it can be derived from the shaft’s diameter. It plays a central role in linking the applied torque to the resulting shear stresses and deformations, thereby influencing both the strength and stiffness of the component.

Angle of Twist and Torsional Rigidity

When a torque is applied to a shaft, it exhibits rotational deformation called the angle of twist, which is a measure of how much one end of the shaft rotates relative to the other. The angle of twist depends on the applied torque, the length of the shaft, the polar moment of inertia, and the material’s shear modulus. Torsional rigidity, defined as the product of the shear modulus and the polar moment of inertia, determines how resistant a shaft is to twisting under a given load. This concept is vital for ensuring that the rotational deformation remains within acceptable limits for proper function.

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