
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
