A proving ring is a highly reliable device for verifying weight and force in the laboratory as well as in the field. Figure $2($a$)$ is a sketch of a proving ring subjected to compression load $\backslash (\backslash$boldsymbol$\{$P$\}\backslash ).$ If fabricated properly and the working stresses are in the elastic range, the ring's lateral diameter will change with a consistent linear relationship of load versus deflection. Structural analysis of the ring is indeterminate to the $\backslash (3^\{\backslash$text $\{$rd $\}\}\backslash )$ degree, but symmetry simplifies the problem to one redundant, which is the internal moment at $\backslash (\backslash$boldsymbol$\{$A$\}\backslash )$ or $\backslash (\backslash$boldsymbol$\{$C$\}\backslash ).$ Figure $2($b$)$ shows one free$-$body diagram for the ring, cut across horizontal diameter $\backslash (\backslash$boldsymbol$\{$A B$\}\backslash ).$ Figure $2($b$)$ shows the analysis results for the internal shear force $\backslash (\backslash$boldsymbol$\{$H$\}\backslash )$ and bending moment $\backslash (\backslash$boldsymbol$\{$M$\}\_\{\backslash$boldsymbol$\{$A$\}\}\backslash )$ at the cut. The shear force and internal moment act in the directions shown.

Problem.

$($a$)$ Determine the limiting load $\backslash (\backslash$boldsymbol$\{$P$\}\backslash )$ so that the maximum circumferential stresses at $\backslash (\backslash$boldsymbol$\{$A$\}\backslash )$ and $\backslash (\backslash$boldsymbol$\{$C$\}\backslash )$ do not exceed $\backslash (\backslash$mathbf$\{12,000\}\backslash$mathbf$\{$~ p s i$\}\backslash ).$ The table below provides additional information about the proving ring.

$($b$)$ Based on your calculated value for the limiting load $\backslash (\backslash$boldsymbol$\{$P$\}\backslash ),$ discuss the advantage that the proving ring offers versus relying on a set of standard deadweights for calibrating forces.

Analysis Results

Force $\backslash (\backslash$mathrm$\{$H$\}=\backslash$mathrm$\{$O$\}\backslash )$

Moment $\backslash ($ M$\_\{$A$\}=\backslash$left$(\backslash$frac$\{1\}\{2\}-\backslash$frac$\{1\}\{\backslash$pi$\}\backslash$right$)$ P$\_\{$F$\}\backslash )$

where $\backslash (\backslash$bar$\{$r$\}\backslash$equiv $\backslash )$ radius of centroidal axis.

Figure $2($a$)$

Figure $2($b$).$ give me the answer on paper