The angle of rotation of end A of the gear-and-shaft system...

The angle of rotation of end $A$ of the gear-and-shaft system shown must not exceed $4^{\circ} .$ Knowing that the shafts are made of a steel for which $\tau_{\text {all }}=65 \mathrm{MPa}$ and $G=77.2 \mathrm{GPa}$, determine the largest torque T that can be safely applied at end $A$.

Key Concepts

Material Properties (Shear Modulus and Yield Limit)

The material's shear modulus (G) describes its stiffness in shear, while the yield limit in shear (allowable shear stress) indicates the maximum stress the material can withstand before plastic deformation. Both properties are critical when determining the safe operating conditions under torsional load.

Torsion in Circular Shafts

This concept covers the behavior of circular shafts when subjected to a twisting moment (torque). It involves understanding how the applied torque produces a distribution of shear stress over the cross?section and results in angular deformation along the shaft's length.

Shear Stress in Torsion

Under torsion, the shear stress in a circular shaft is not uniformly distributed – it is highest at the outer surface and zero at the center. The relationship is often expressed as ? = T*r/J, where T is the applied torque, r is the radius, and J is the polar moment of inertia.

Polar Moment of Inertia

The polar moment of inertia is a geometric property of a cross?section that measures its resistance to torsion. It plays a crucial role in determining both the maximum shear stress and the angle of twist in a shaft when a torque is applied.

Angle of Twist (Torsional Deformation)

This concept relates the applied torque to the angular deformation (twist) experienced by the shaft. The angle of twist can be calculated using the formula ? = T*L/(J*G), where L is the length of the shaft and G is the shear modulus of the material, linking the applied load to the resulting deformation.

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