Question 15.SP.6: The double gear shown rolls on the stationary lower rack; th...

MODELING and ANALYSIS:
a. Angular Velocity of the Gear. Since the gear rolls on the lower rack, its center A moves through a distance equal to the outer circumference  $2 \pi r_1$  for each full revolution of the gear. Noting that  $1  \mathrm{rev}=2 \pi$  rad, and that when A moves to the right  $\left(x_A>0\right)$,  the gear rotates clockwise (θ < 0), you have

$\frac{x_A}{2 \pi r_1}=-\frac{\theta}{2 \pi}               \quad x_A=-r_1 \theta$

Differentiating with respect to the time t and substituting the known values  $v_A=1.2$  m/s and  $r_1=150  \mathrm{~mm}=0.150  \mathrm{~m} \text {, }$  you obtain

$v_A=-r_1 \omega                \quad 1.2  \mathrm{~m} / \mathrm{s}=-(0.150  \mathrm{~m}) \omega                   \quad \omega=-8  \mathrm{rad} / \mathrm{s}$

$\omega=\omega \mathbf{k}=-(8  \mathrm{rad} / \mathrm{s}) \mathrm{k}$

where k is a unit vector pointing out of the page.

b. Velocity of Upper Rack. The velocity of the upper rack is equal to the velocity of point B; you have

$\begin{aligned}\mathbf{v}_R &=\mathbf{v}_B=\mathbf{v}_A  +  \mathbf{v}_{B / A}=\mathbf{v}_A  +  \omega \mathbf{k} \times \mathbf{r}_{B / A} \\&=(1.2  \mathrm{~m} / \mathrm{s}) \mathbf{i}  –  (8  \mathrm{rad} / \mathrm{s}) \mathbf{k} \times(0.100  \mathrm{~m}) \mathbf{j} \\&=(1.2  \mathrm{~m} / \mathrm{s}) \mathbf{i}  +  (0.8  \mathrm{~m} / \mathrm{s}) \mathbf{i}=(2  \mathrm{~m} / \mathrm{s}) \mathbf{i}\end{aligned}$

$\mathrm{v}_R=2  \mathrm{~m} / \mathrm{s} \rightarrow$

Velocity of Point D. The velocity of point D has two components (Fig. 2):

$\begin{aligned}\mathbf{v}_D &=\mathbf{v}_A  +  \mathbf{v}_{D / A}=\mathbf{v}_A  +  \omega \mathbf{k} \times \mathbf{r}_{D / A} \\&=(1.2  \mathrm{~m} / \mathrm{s}) \mathbf{i}  –  (8  \mathrm{rad} / \mathrm{s}) \mathbf{k} \times(-0.150  \mathrm{~m}) \mathbf{i} \\&=(1.2  \mathrm{~m} / \mathrm{s}) \mathbf{i}  +  (1.2  \mathrm{~m} / \mathrm{s}) \mathbf{j}\end{aligned}$

$\mathbf{v}_D=1.697  \mathrm{~m} / \mathrm{s} \text{⦨} 45^{\circ}$

REFLECT and THINK: The principles involved in this problem are similar to those that you used in Sample Prob. 15.3, but in this problem, point A was free to translate. Point C, since it is in contact with the fixed lower rack, has a velocity of zero. Every point along diameter CAB has a velocity vector directed to the right (Fig. 1) and the magnitude of the velocity increases linearly as the distance from point C increases.