The problem involves a stepped rod of circular cross-sections, which is subjected to a torque T at the free end. The objective is to determine the value of d that produces an angular twist θ at the free end. The material of the rod has a shear modulus of G.


Torsion is a type of deformation in which a material is twisted by an applied torque. The magnitude of the torque is proportional to the angular twist and is given by the equation:

T = GθJ/r

where T is the torque, θ is the angular twist, G is the shear modulus of the material, J is the polar moment of inertia, and r is the distance from the center of the cross-section to the outer surface.


The polar moment of inertia of a circular cross-section is given by the equation:

J = πd^4/32

where d is the diameter of the cross-section.

Since the rod has a stepped cross-section, the value of d will vary at different sections. Let di and di+1 be the diameters of two adjacent sections. The polar moment of inertia for each section can be calculated using the above equation. The total polar moment of inertia for the entire rod can be calculated by summing up the polar moments of inertia for each section.

J = Σ(πdi^4/32)


The distance from the center of the cross-section to the outer surface can be calculated using the equation:

r = d/2

Since the diameter varies for different sections, the value of r will also vary. Let ri and ri+1 be the values of r for two adjacent sections. The average value of r for these sections can be calculated as:

(ri + ri+1)/2 = di+1/4 + di/4


Using the equation for torque, we can rearrange the terms to obtain the equation for d:

d = (Tr/(Gθ))(Σ(πdi^4/32))/Σ(di+1/4 + di/4)

Substituting the given values of torque, shear modulus, and angular twist, we can calculate the value of d that produces the desired angular twist at the free end.


To summarize, the value of d that produces an angular twist θ at the free end of a stepped rod of circular cross-sections can be calculated by determining the polar moment of inertia and the distance from the center to the outer surface for each section, and then using the equation for torque to solve for d.