P$6\mathrm{.}13$ A two$-$segment shaft is used to transmit power at constant speed through the pulleys shown in Figure P$6\mathrm{.}13.$ Power is input to the shaft at $B$ through a torque $T_B=260N*m.$ A torque $T_A=90N*m$ is removed from the shaft at $A,$ and a torque $T_C=170N*m$ is removed from the shaft at $C.$ The external torques act in the directions indicated in the figure. Both shaft segments are made of phosphor bronze $[G=42$GPa $],$ and both segments are solid shafts. Segment $(1)$ has a diameter $d_1=25mm$ and a length $L_1=300mm.$ Segment $(2)$ has a diameter $d_2=30mm$ and a length $L_2=900mm.$ Determine the rotation angle of pulley $C$ relative to pulley $A.$

FIGURE P$6\mathrm{.}13$