A torque T of 150 N-m is transmitted by a thaft A (see Fig....

A torque $T$ of $150 \mathrm{~N}-\mathrm{m}$ is transmitted by a thaft $A$ (see Fig. P. 14.11) at an angular speed $\omega_A$ of 500 rpm . The diameter of shaft $A$ is .15 m . If the same maximum stress is to be maintained in shaft $B$ as in shaft $A$, what should its diameter be? Take $D_1=.6 \mathrm{~m}$ and $D_2=3 \mathrm{~m}$. Figure P.14.11.

Key Concepts

Shear Stress in Torsion

This concept involves understanding that when a torque is applied to a circular shaft, it produces a shear stress that varies across the cross?section. The maximum shear stress occurs at the outer surface and is given by the formula ? = Tc/J, where T is the applied torque, c is the outer radius, and J is the polar moment of inertia. This relationship is critical in ensuring that shafts are designed to withstand operational loads without exceeding material limits.

Polar Moment of Inertia

The polar moment of inertia, J, for a circular section is a measure of the shaft's resistance to torsion and is calculated as J = (?d^4)/32, where d is the diameter of the shaft. This parameter links the geometry of the shaft to its ability to resist twisting and is a key factor in calculating the shear stresses induced by an applied torque.

Scaling of Shaft Dimensions with Torque

In mechanical design, when two shafts are required to maintain the same maximum stress under different loading conditions, the dimensions—especially the diameter—must be scaled appropriately. Since the torsional shear stress is inversely proportional to the cube of the diameter, the shaft diameter must be chosen such that the torque-to-diameter-cubed ratio remains constant, ensuring similar stress levels in different designs.

Power Transmission via Pulleys

When transmitting power from one shaft to another using belts and pulleys, the diameters of the pulleys determine the relationship between the angular speeds and the torques on the connected shafts. A larger pulley will result in increased torque transfer with a reduction in angular speed, which is essential for matching the operational requirements between different components in a mechanical system.

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