$6\mathrm{.}44$ A circular shaft $\backslash ($ A E $\backslash )$ of length $\backslash (5$ a $\backslash ),$ diameter $\backslash ($ d $\backslash ),$ and shear modulus $\backslash ($ G $\backslash )$ is welded to a support at end $\backslash ($ A $\backslash )$ and has a rigid arm of length $\backslash ($ b $\backslash )$ welded to its other end, $\backslash ($ E $\backslash ).$ A pointer is fastened to the shaft at $\backslash ($ C $\backslash ).$ When the shaft is unloaded, the

$376$ An Introduction to the Mechanics of Solids

pointer and the rigid arm are both horizontal, as shown in $($a$).$ A weight $\backslash ($ W $\backslash )$ is now hung on the end of the rigid arm and, at the same time, moments $\backslash ($ M$\_\{$B$\}\backslash )$ and $\backslash ($ M$\_\{$D$\}\backslash )$ are applied to the shaft at points $\backslash ($ B $\backslash )$ and $\backslash ($ D $\backslash )$ such that the pointer and the rigid arm remain horizontal, as shown in $($b$).$ Find the values of $\backslash ($ M$\_\{$B$\}\backslash ),$ and $\backslash ($ M$\_\{$D$\}\backslash )$ required to keep the pointer and rigid arm horizontal when the weight $\backslash ($ W $\backslash )$ is hung on the arm.