Prob. $3\mathrm{.}7-5.$ A two$-$segment linearly elastic rod has a cir$-$

cular cylindrical cross section of area $A_2=1\mathrm{.}5A.$ Half of

the rod, from $A$ to $B,$ is turned down so that $A_1=A.($Note:

$E_1=E_2=E,$ and $L_1=L_2=L.)$ At end A the rod is connected

to a rigid wall, but at end $C$ there is a small gap $\frac{?}{\text{b}\text{a}\text{r}}()$ between

the original position of end $C$ and the rigid wall, as shown

in Fig. P$3\mathrm{.}7-5.($a$)$ Derive an expression for the width of the

gap if$,$ when an axial force $P$ is applied at $B$ as shown in the

figure, the magnitude of the reaction at $C$ is equal to the

reaction at $A.($b$)$ Derive an expression for the amount, $u_B,$

that joint $B$ moves under the conditions stated in part $($a$).$

$($Note: The gap $\frac{?}{\text{b}\text{a}\text{r}}()$ shown in the figure below is the gap

before load $P$ is applied at $B.)$