use Castigliano's first theorem to determine the slopes and deflections indicated. $E=29{10}^6$ psi for the problems having customary units and $200000$MPa for those with SI units. Similarly circled values are cross$-$sectional areas in square inches or in $m{m}^2.$

Problem $21\mathrm{.}2$ Slope and dellection at free end; $I=1300i{n}^4($Ans$.=0\mathrm{.}0129$ rad $=3\mathrm{.}09in1)$

Problem $21\mathrm{.}9$ Slope and deflection at ${60}^t$ load; $I-2700m^4$