The figure shows a system consisting of (i) a ring of outer radius 3R rolling clockwise without slipping on a horizontal surface with angular speed ω and (ii) an inner disc of radius 2R rotating anti-clockwise with angular speed ω2. The ring and disc are separated by frictionless ball bearings. The system is in the x - z place. The point P on the inner disc is at distance R from the origin O, where OP makes an angled of 30∘ with the horizontal. Then with respect to the horizontal surface,
The point P has a linear velocity 114Rω^i−√34Rω^k
If the ring of radius '3R' does pure rolling, its centre moves with velocity,
→V0=3ωR^i i.e., towards right.
Now as the disc and bearings are contained in the ring, the whole system moves as one ring with the above velocity.
→VPG=→VPO+→VOG
where, G → ground
O → centre of the disc
Clearly, |→VPO|=ωR2 as shown,
→VPG=ωR2cos30∘^k+3ωR^i−ωR2sin30∘^i
→VPG=114Rω^i+√34Rω^k
Now the velocity of 'P' with respect to ground can be formulated as.