Determine the smallest allowable diameter of a 10 -ft-long...

Determine the smallest allowable diameter of a 10 -ft-long steel rod $\left(G=11.2 \times 10^{6} \mathrm{psi}\right)$ if the rod is to be twisted through $90^{\circ}$ without exceeding a shearing stress of 15 ksi.

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Key Concepts

Shear Modulus

The shear modulus, also known as the modulus of rigidity, is a material property that relates shear stress to shear strain. It is used to predict the elastic deformation (the twist) in a member under an applied torsional load.

Angle of Twist

The angle of twist is the measure of rotation experienced by a shaft due to applied torque. It depends on factors such as the applied moment, the material’s shear modulus, the length of the shaft, and its polar moment of inertia, and is a key parameter in the design of torsional systems.

Torsion

Torsion refers to the twisting of an object due to an applied moment or torque. In mechanical engineering, analyzing torsion is essential to ensure that the induced shear stresses in a structural member remain within safe limits during operation.

Polar Moment of Inertia

The polar moment of inertia is a geometric property that reflects how resistant a cross-section is to torsional deformation. For circular shafts, it is calculated based on the diameter and plays a crucial role in determining the distribution of shear stress when the shaft is twisted.

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