Problem Statement:

A random$-$orbit sander is a useful power tool for certain applications. In the design pictured, the motor shaft is connected rigidly to gear $1,$ which has $14$ teeth. Eccentric shaft ABCD is connected rigidly to

gear $2,$ which has $42$ teeth. Segment $AB$ of the eccentric shaft is concentric with gear $2,$ so that when gear $2$ spins, $AB$ is on the axis of rotation. Because of $0\mathrm{.}59-$inch offset $BC,$ segment $CD$ travels in a

circle around the axis of rotation. A sanding pad is then loosely attached to segment $CD$ so that the pad also travels in a circluar path, but is free to rotate $($or not rotate$)$ independently of the eccentric shaft.

This problem is interested in the motion of the center of segment $CD,$ which will also be the motion of the pad's center point. If the motor shaft spins at an angular velocity of $23(t^3+12t)$ rad$/$s$,$ determine

as functions of time:

Symbolic answers $($do not include units$)$:

The speed of segment CD:

$($in$/$s$)$

The acceleration vector of segment $CD$ :

hat$(u{)}_t+$

hat$(u{)}_n(i\frac{n}{s^2})$

Numerical answer $($include units$)$:

The time at which the tangential and normal acceleration components have equal value: