Question 15.12: The rod AB, of length 7 in., is attached to the disk by a ba......
The rod AB, of length 7 in., is attached to the disk by a ball-and-socket connection and to the collar B by a clevis. The disk rotates in the yz plane at a constant rate \text v_1 = 12 rad/s, while the collar is free to slide along the horizontal rod CD. For the position u = 0, determine (a) the velocity of the collar, (b) the angular velocity of the rod.
a. Velocity of Collar. Since point A is attached to the disk and since collar B moves in a direction parallel to the x axis, we have
\mathrm{v}_{A}=\mathrm{V}_{1} \times {r}_{A}=12 {i} \times 2{k}=-24 {j} \quad\quad\quad \mathrm{v}_{B}=v_{B} {i}Denoting by V the angular velocity of the rod, we write
Equating the coefficients of the unit vectors, we obtain
v_B=\quad-2 \text v _y-3 \text v _z (1)
24=2 \text v _x \quad+6 \text v _z (2)
0=3\text v _x-6\text v _y (3)
Multiplying Eqs. (1), (2), (3), respectively, by 6, 3, -2 and adding, we write
6 v_{B}+72=0 \quad\quad v_{B}=-12 \quad\quad \mathrm{v}_{B}=-(12 ~\mathrm{in} . / \mathrm{s}) {i}b. Angular Velocity of Rod AB. We note that the angular velocity cannot be determined from Eqs. (1), (2), and (3), since the determinant formed by the coefficients of \text v_x, \text v_y,\text{ and } \text v_z is zero. We must therefore obtain an additional equation by considering the constraint imposed by the clevis at B. The collar-clevis connection at B permits rotation of AB about the rod CD and also about an axis perpendicular to the plane containing AB and CD. It prevents rotation of AB about the axis EB, which is perpendicular to CD and lies in the plane containing AB and CD. Thus the projection of V on r_{E/B} must be zero and we write†
\mathrm{V} \cdot {r}_{E / B}=0 \quad \quad \left(\mathrm{V}_{x} {i}+\mathrm{V}_{y} {j}+\mathrm{V}_{z} {k}\right) \cdot(-3 {j}+2 {k})=0 \\ -3 \mathrm{v}_{y}+2\mathrm{V}_{z}=0 (4)
Solving Eqs. (1) through (4) simultaneously, we obtain
†We could also note that the direction of EB is that of the vector triple product {r}_{B / C} \times\left({r}_{B / C} \times {r}_{B / A}\right)\mathrm {~and ~write~} \mathrm{V} \cdot\left[{r}_{B / C} \times\left({r}_{B / C} \times {r}_{B / A}\right)\right]=0. This formulation would be particularly useful if the rod CD were skew.
