Rigid body motion types; double gear velocities

Types of rigid body motion

A rigid body in plane motion can have:

In space, a body can also have rotation about a fixed point (a spinning top) and general three-dimensional motion.

Double gear: angular velocity

The lower rack is stationary, so the contact point $C$ ($r_1=0.150$ m below $A$) is the instantaneous centre:

$$\omega = \frac{v_A}{r_1} = \frac{1.2}{0.150} = 8.00\ \text{rad/s (clockwise)}$$

Velocities of R and D

The upper rack is in contact with the gear at $B$, $r_2 = 0.100$ m above $A$, and moves with point $B$. Distance from $C$ to $B$ is $0.150 + 0.100 = 0.250$ m:

$$v_R = v_B = \omega\,(CB) = 8.00\times0.250 = 2.00\ \text{m/s}\ \to$$

Point $D$ is on the horizontal line through $A$, $0.150$ m to the left of $A$ (on the outer circle). Its distance from $C$ is $CD = \sqrt{0.150^2 + 0.150^2} = 0.2121$ m:

$$v_D = \omega\,(CD) = 8.00\times0.2121 = 1.70\ \text{m/s}$$

Components: $\vec v_D = \vec v_A + \omega\hat k\times\vec r_{D/A}$ with $\omega = -8\hat k$: $\vec v_D = (1.2\,\hat i + 1.2\,\hat j)$ m/s, i.e. 45° above the horizontal towards the right.

Answer: $\omega = 8.00$ rad/s clockwise; $v_R = 2.00$ m/s to the right; $v_D = 1.70$ m/s at $45^\circ$ above horizontal.