Prob. $4\mathrm{.}5-11.$ A plumber is cutting threads on the end of a $2-$ ft $-$long section of $1-$in$.-$diameter$*{?}^{**}$ pipe. This

section, $AB,$ has already been attached to a $2-$in$.-$diameter$*$ section, $BC,$ by a reducing coupler and, in

turn, to a $3-$in$.-$diameter$*$ section $CD.$ The shear modulus of the steel pipe is $G=11\mathrm{.}5{10}^3$ksi.

Neglecting the dime ns and flexibility of the couplers, $($a$)$ determine the maximum shear stress in the

assembly, $($b$)$ determinc angle of rotation at end $A,{}_A,$ and $($c$)$ determine the torsional stiffness, $k_t,$ of

the assembly. That is$,$ determ the torque that must be applied at $A$ to produce a unit rotation $($i$.$e$.,1$ rad

rotation$)$ at $A.(*$The stated dian ters are nominal pipe diameters. See Table D$.7$ for section properties of

the pipe.$)$