CHAPTER 7.10. 7, PROBLEM 10
315
7.10 Chapter 7, Problem 10
Problem: A disk of radius R is mounted on L-shaped Rod CD and rotates with constant angular velocity as shown. Rod CD rotates with constant angular velocity about the z axis. Determine the absolute velocity and acceleration of Point B, which is the lowest point on the disk relative to the xy plane.
Solution: An observer in a coordinate frame rotating with angular velocity = k about Point D sees the motion of Point B as pure rotation with angular velocity = i about the center of the disk (Point C). Thus, the relative velocity and acceleration vectors are vB aB = rB/C = i (R k) = R j = vB = i R j = 2 R k vB = + vB rB/D
The absolute velocity of Point B is given by where rB/D is given by
1 rB/D = 2R i + R j R k 2 Thus, substituting for vB and taking the indicated cross product, we find vB = R j + R 2 j Point Bs absolute acceleration is where vB is the collars velocity as seen by the rotating observer and rB/D is its position vector relative to the collar. Now, we know that = 0, while vB and aB are as given above. Thus, the absolute acceleration of Point B is 1 aB = 2 R k + 2 k R j + k k 2R i + R j R k 2 1 = 2 R k 2 R i + k R 2 j i 2 1 = 2 R k 2 R i + 2 R 2 i j 2 12 = 2( )R i R j + 2 R k 2 aB = aB + rB/D + 2 vB + ( rB/D ) 1 i 2 1 = R i + ( 2)R j 2