Final answer:
The hollow steel shaft's diameters are calculated using torque, allowable shearing stress, and the angle of twist limits, alongside torsion formulas that incorporate the given shear modulus.
Step-by-step explanation:
The student is tasked with determining the inside and outside diameters of a hollow steel shaft designed to transmit a specific torque, so as not to exceed a certain allowable shearing stress and angle of twist, given the material's shear modulus (G). Employing the Torsion formula, which relates the torque (T), shear stress (τ), shear modulus (G), angle of twist (θ), and the dimensions of the shaft, this problem is solved using mechanics of materials principles.
Firstly, we use the torque and shear stress to find the outer diameter (D) with the torsion formula T = (τ * π * D³) / 16 and the relationship of shear stress τ = T * c / J, where c = D/2 and J is the polar moment of inertia. Using the given maximum shearing stress, we can solve for D. Once we have D, we calculate the polar moment of inertia J for the shaft, factoring in the inner diameter (d), with J = (π/32) * (D⁴ - d⁴).
Next, to prevent the angle of twist (θ) from exceeding the allowed 2.5° for the 3-m-long shaft, we use the angle of twist formula θ = (T * L) / (J * G), to find a relationship between the inner and outer diameters, ultimately determining the size of the hollow section needed (d) such that the shaft can safely transmit the torque within the allowable stress and angle of twist.