EXERCISE $3\mathrm{.}52$ Bent shaft $\backslash ($ A B C $\backslash )$ is welded to the vertical shaft, which rotates at constant speed $\backslash (\backslash$Omega $\backslash ).$ The angle of rotation of the square plate about axis $\backslash ($ B C $\backslash )$ is $\backslash (\backslash$phi $\backslash ),$ which is an arbitrary function of time. The configuration depicted in the sketch, in which the plate is situated in the vertical plane, corresponds to $\backslash (\backslash$phi$=0\backslash ).$ The $\backslash ($ x y z $\backslash )$ coordinate system is attached to the plate. $($a$)$ Derive expressions for the angular velocity and angular acceleration of the plate, valid for arbitrary $\backslash (\backslash$phi $\backslash ),$ in terms of components with respect to $\backslash ($ x y z $\backslash ).($b$)$ Derive expressions for the velocity and acceleration of the center of the plate in terms of $\backslash ($ x y z $\backslash )$ components.