What is the condition for pure rolling ?

Explanation

(i) In pure rolling, the point of the rolling object which comes in contact with the surface is at momentary rest.
(ii) This is the case with every point that is on the edge of the rolling object. As the rolling proceeds, all the points on the edge, one by one come in contact with the surface, remain at momentary rest at the time of contact and then take the path of the cycloid.
Consider the pure rolling in two different ways.
(a) The combination of translational motion and rotational motion about the center of mass.
(or)
(b) The momentary rotational motion about the point of contact.
(iii) As the point of contact is at momentary rst in pure rolling, its resultant velocity v is zero (v = 0). For example, at the point of contact, `v_("TRANS")` is forward (to right) and `v_("ROT")` is backwards (to the left).

(iv) That implies that, `v_("TRANS") and v_("ROT")` are equal in magnitude and opposite in directon `(v = v_("TRANS")-v_("ROT") = 0)`. Hence, we conclude that in pure rolling, for all the points on the edge, the magnitudes of `v_("TRANS") and v_("ROT")` are equal `(v_("TRANS")=v_("ROT"))`. As `v_("TRANS") = v_(CM) and v_("ROT") = R omega`, in pure rolling we have, `v_(CM) = R omega`
(v) For the topmost point, the two velocities `v_("TRANS") and v_("ROT")` are equal in magnitude and in the same direction (to the right). Thus, the resultant velocity v is the sum of these two velocities, `v = v_("TRANS") + v_("ROT")` In other form, `v = 2 v_(CM)`
Sliding
(i) Sliding is the case when `v_(CM) gt R omega (or v_("TRANS") gt v_("ROT))`. The translation is more than the rotation. This kind of motion happens when sudden break is applied in a moving vehicles, or when the vehicle enters into a slippery road. In this case, the point of contact has more of `v_("TRANS")` than `v_("ROT")`
(ii) Hence, it has a resultant velocity v in the forward direction. The kinetic frictional force `(f_(k))` opposes the relative motion. Hence, it acts in the opposite direction of the relative velocity.
(iii) This frictional force reduces the translational velocity and increases the rotational velocity till they become equal and the object sets on pure rolling. Sliding is also referred as forward slipping.
Slipping
(i) Slipping is the case when `v_(CM) lt R omega (or v_("TRANS") lt v_("ROT"))`. The rotation is more than the translation. This kind of motion happens when we suddenly start the vehicle from rest or the vehicle is stuck in mud.
(ii) In this case, the point of contact has more of `v_("ROT") "then" v_("TRANS")`. It has a resultant velocity v in the backward direction.
(iii) The kinetic frictional force `(f_(k))` opposes the relative motion. Hence it acts in the opposite direction of the relative velocity.
(iv) This frictional force reduces the rotational velocity and increases the translational velocity till they become equal and the object sets pure rolling. Slipping is sometimes empahasised as backward slipping.