A cylindrical shaft is supported by spherical ball bearings that freely rotate. There is no slipping between the center cylinder and the bearings, or between the bearings and the outside surface. If the cylinder rotates at 4 rad/s, what is the velocity magnitude of a ball bearing center (point C)?

Shaft and Bearing Orientation

At any given instant, both points A and D are not moving. Thus, use those two points as pinned or fixed points (of course, an instant later, point D will be at a different location).

Since there is no slipping, the velocity of point B will be the same on shaft or ball, thus,
     v_B = ω_A r_AB = ω_C r_BD
     4(6) = ω_C (2)
     ω_C (2) = 12 rad/s

Now point C can be found by using the velocity of point D and B.
     v_C = v_D + v_C/D = 0 + ω_C × r_DC
     v_C = ω_C r_DC
         = 12 (1) = 12 cm/s

v_C = 12 cm/s