Question

A solid shaft with a 30 mm diameter is to be used in lieu of a hollow one that has an outside diameter of 50 mm and an interior diameter of 25 mm. What is the ratio of the maximum resistible torque in the hollow to that of solid shaft if the maximum shear stresses produced in the two shafts are to be equal?

Answer 1

To find the ratio of the maximum resistible torque in the hollow shaft to that of the solid shaft, we need to equate their maximum shear stresses and solve for the ratio of the torques.

Step-by-step explanation:

To determine the ratio of the maximum resistible torque in the hollow shaft to that of the solid shaft, we need to consider their respective maximum shear stresses. The maximum shear stress in a solid shaft is given by the equation:

τ = T / (2π r3 G)

where τ is the shear stress, T is the torque, r is the radius of the shaft, and G is the shear modulus. For the hollow shaft, the maximum shear stress is given by:

τ = T / (2π (R4 - r4) G)

where R is the outer radius of the shaft and r is the inner radius of the shaft.

By equating the maximum shear stresses in the two shafts, we can solve for the ratio of the torques:

Thollow / Tsolid = (R4 - r4) / r3

Given the dimensions of the hollow shaft (R = 25 mm, r = 12.5 mm) and the solid shaft (r = 15 mm), we can calculate the ratio of the maximum resistible torques.