고체역학 (Gere) — Torsion: Problem 3.4-14

3.4-14 A uniformly tapered tube \(AB\) with a hollow circular cross section is shown in the figure. The tube has constant wall thickness \(t\) and length \(L\). The average diameters at the ends are \(d_A\) and \(d_B = 2d_A\). The polar moment of inertia may be represented by the approximate formula \(I_p \approx \pi d^3 t / 4\) [see Eq. (3-21)]. Derive a formula for the angle of twist \(\phi\) of the tube when it is subjected to torques \(T\) acting at the ends.