Chapter 3, Problem 3.4.20P

A magnesium-alloy wire of diameter d = 4mm and length L rotates inside a flexible tube in order to open or close a switch from a remote location (see figure). A torque Tis applied manually (either clockwise or counterclockwise) at end 5, thus twisting the wire inside the tube. At the other end A, the rotation of the wire operates a handle that opens or closes the switch.

A torque T₀ = 0.2 N · m is required to operate the switch. The torsional stiffness of the tube, combined with friction between the tube and the wire, induces a distributed torque of constant intensity t = 0.04N − m/m (torque per unit distance) acting along the entire length of the wire.

(a)

If the allowable shear stress in the wire is T _allow = 30 MPa, what is the longest permissible length L_maxof the wire?