$*8-151$ The ring in Fig. P$8-151$ slides along the rotating rod $OB.$ In the position shown $r=0\mathrm{.}3m,tan=\frac{4}{3},r$ is increasing at the rate of $0\mathrm{.}5\frac{m}{s},$ and $r^{}$ is decreasing at $0\mathrm{.}6\frac{m}{s^2}.$ Member $AB$ has an angular velocity of $2ra\frac{d}{s}$ clockwise and an angular acceleration of $5ra\frac{d}{s^2}$ clockwise.

$($a$)$ Set up an expression for the vertical component of the position of A and determine the vertical component of the acceleration of $A$ by differentiation.