Question 15.2: The double gear shown rolls on the stationary lower rack; th......

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 $2pr_1$ for each full revolution of the gear. Noting that 1 rev = 2p rad, and that when A moves to the right ($x_A$ > 0) the gear rotates clockwise (u < 0), we write

Differentiating with respect to the time t and substituting the known values

$v_A=1.2~ m / s ~\text { and }~ r_1=150~ mm =0.150~ m$, we obtain

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

b. Velocities. The rolling motion is resolved into two component motions: a translation with the center A and a rotation about the center A. In the translation, all points of the gear move with the same velocity $\text v_A$. In the rotation, each point P of the gear moves about A with a relative velocity $\text v _{P / A}=\text v k \times r _{P / A}~ \text {, where }~ r _{P / A}$ is the position vector of P relative to A.

Velocity of Upper Rack. The velocity of the upper rack is equal to the velocity of point B; we write

= (1.2 m/s)i – (8 rad/s)k × (0.100 m)j

= (1.2 m/s)i + (0.8 m/s)i = (2 m/s)i

Velocity of Point D

= (1.2 m/s)i – (8 rad/s)k × (-0.150 m)i

= (1.2 m/s)i + (1.2 m/s)j