Sample Problem $5/16$

At the instant represented, the disk with the radial slot is rotating about $O$ with a counterclockwise angular velocity of $4ra\frac{d}{sec}$ which is decreasing at the rate of $10ra\frac{d}{se{c}^2}.$ The motion of slider $A$ is separately controlled, and at this instant, $r=6in.,r^{}=5in\frac{.}{sec},$ and $r^{}=81in\frac{.}{se{c}^2}.$ Determine the absolute velocity and acceleration of A for this position.

Solution. We have motion relative to a rotating path, so that a rotating coordinate system with origin at $O$ is indicated. We attach $x-y$ axes to the disk and use the unit vectors i and $j.$