Determine the relationship between the torque N and the torsion angle `varphi` for
(a) the tube whose wall thickness `Deltar` is considerably less than the tube radius,
(b) for the solid rod of circular cross-section. Their lengh l, radius r, and shear modulus G are supposed to be known.

Explanation

(a) Consider a hollow cylinder of length l, outer radius `r+Deltar` inner radius r, fixed at one end and twisted at the other by means of a couple of moment N. The angular displacement `varphi`, at a distance l from the fixed end, is proportional to both l and N. Consider an element of length `dx` at the twisted end. It is moved by an angle `varphi` as shown. A vertical section is also shown and the twisting of the parallelopipe of length l and area `Deltardx` under the action of the twisting couple can be discussed by elementary means. If f is the tangential force generated then shearing stress is `f//Deltardx` and this must equal
`Gtheta=G(rvarphi)/(l)`, since `theta=(rvarphi)/(l)`.
Hence, `f=GDeltardx(rvarphi)/(l)`.
The force f has moment `f r` about the axis and so the total moment is
`N=GDeltar(varphi)/(l)r^2intdx=(2pir^3Deltarvarphi)/(l)G`
(b) For a solid cylinder we must integrate over r. Thus
`N=overset(0)overset(r)int(2pir^3drvarphiG)/(l)=(pir^4Gvarphi)/(2l)`