Rolling motion can be regarded as the combination of pure rotation and pure translation.
The wheels of all vehicles running on a road have rolling motion.
Let a disc of radius R rolling on a road without slipping. This means that at any instant of time the bottom of the disc which is in contact with the surface is at rest.
The rolling motion of the disk has two simultaneous motions.
(I) Translational Motion
The translation velocity of the disc is the velocity Vcm of its centre of mass. Since, centre of mass of the disc is C, So Vcm is the velocity of C. It is parallel to the level surface of road.
(II) Rotational Motion
The disc rotates with angular velocity ω about its symmetry axis through C. The linear velocity of a particle P at distance r from the axis due to the rotational motion is Vrot= rω
The velocity Vrot is directed perpendicular to the radius vector CP.
The effective linear velocity Vp of particle P is the resultant of velocity Vrot and Vcm, then Vp is perpendicular to BP.
Therefore, the line passing through the bottom point B and parallel to the axis through C is called the instantaneous axis of rotation.
At the bottom B, the linear velocity Vrot, due to rotation is directed exactly opposite to the translational velocity Vcm.
Also Vrot = Rω at the point B [Since, v=rω].
The point be will be instantaneously at the rest if
Vcm = Vrot ⇒Vcm = Rω
Hence, for the disc the condition for rolling without sleeping is Vcm = rω
At the top point A of the disc, the linear velocity Rω due to the rotational motion and translation velocity Vcm are in the same direction, parallel to the level surface.
Therefore, Vtop = Rω + Vcm Vtop = Vcm + Vcm [Since, Vcm = Rω] Vtop = 2Vcm
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