Question 15.6: The center of the double gear of Sample Prob. 15.2 has a vel......
The center of the double gear of Sample Prob. 15.2 has a velocity of 1.2 m/s to the right and an acceleration of 3 m/s² to the right. Recalling that the lower rack is stationary, determine (a) the angular acceleration of the gear, (b) the acceleration of points B, C, and D of the gear.
a. Angular Acceleration of the Gear. In Sample Prob. 15.2, we found that x_{A}=-r_{1} \text {u}\text{ and }v_{A}=-r_{1} \text {v}. Differentiating the latter with respect to time, we obtain a_{\mathrm{A}}=-r_{1} \mathrm{a}.
v_A=-r_1 \text v \quad \quad 1.2 ~m / s =-(0.150 ~m ) \text v \quad \quad \text v =-8~ rad / s \\ \\ a_{ A }=-r_1 a \quad\quad 3 ~m / s ^2=-(0.150~ m ) a \quad\quad a =-20~ rad / s ^2A = ak = -(20 rad/s²)k
b. Accelerations. The rolling motion of the gear is resolved into a translation with A and a rotation about A.
Acceleration of Point B. Adding vectorially the accelerations corresponding to the translation and to the rotation, we obtain
Acceleration of Point C
Acceleration of Point D
