Textbook Problem

-10. A tapered bar AB with a solid circular cross section is twisted by torques T(see figure). The diameter of the bar varies linearly from d_Aat the left-hand end to d_Bat the right-hand end.

(a) Confirm that the angle of twist of the tapered bar is

  

(b) For what ratio d_Bld₄will the angle of twist of the tapered bar be one-half the angle of twist of a prismatic bar of diameter d₄1 (The prismatic bar is made of the same material- has the same length, and is subjected to the same torque as the tapered bar.)