고체역학 (Gere) — Torsion: Problem 3.4-20

3.4-20 A magnesium-alloy wire of diameter \( d = 4 \text{ mm} \) and length \( L \) rotates inside a flexible tube in order to open or close a switch from a remote location (see figure). A torque \( T \) is applied manually (either clockwise or counterclockwise) at end \( B \), 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 \( T_0 = 0.2 \text{ N}\cdot\text{m} \) 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 \( t = 0.04 \text{ N}\cdot\text{m/m} \) (torque per unit distance) acting along the entire length of the wire. (a) If the allowable shear stress in the wire is \( \tau_{allow} = 30 \text{ MPa} \), what is the longest permissible length \( L_{max} \) of the wire? (b) If the wire has length \( L = 4.0 \text{ m} \) and the shear modulus of elasticity for the wire is \( G = 15 \text{ GPa} \), what is the angle of twist \( \phi \) (in degrees) between the ends of the wire?