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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. |
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Answer» Solution :(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)` 
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