To determine the angle of twist of wheel B with respect to wheel A, we first need to calculate the torque being transmitted through the shaft. We can do this using the formula:
T = J * τ / R
where T is the torque, J is the polar moment of inertia of the shaft, τ is the shear stress, and R is the radius of the shaft. The polar moment of inertia of a solid shaft is given by:
J = π * d^4 / 32
where d is the diameter of the shaft.
Substituting the given values, we get:
J = π * (40 mm)^4 / 32 = 1.02 x 10^7 mm^4
The torque being transmitted through the shaft is not given in the problem statement, so we cannot calculate the angle of twist without this information.
To determine if yielding occurs using the maximum distortion energy theory, we need to calculate the von Mises stress using the formula:
σ_vm = sqrt(3/2 * τ^2)
where σ_vm is the von Mises stress and τ is the shear stress.
The maximum shear stress occurs at the surface of the shaft, and is given by:
τ = T * R / J
Substituting the given values, we get:
τ = T * (20 mm) / (1.02 x 10^7 mm^4) = 0.0196 T MPa
where T is the torque in Nm.
Substituting this value of τ into the formula for von Mises stress, we get:
σ_vm = sqrt(3/2 * (0.0196 T MPa)^2) = 0.034 T MPa
For yielding to occur, the von Mises stress must be equal to or greater than the yield strength of the material. The given yield strength of the steel is 250 MPa. Therefore, yielding will not occur using the maximum distortion energy theory as long as the torque T is less than 7352 Nm (i.e., 250 MPa / 0.034 MPa/Nm^2).