What will happen to the contact point atom after it leaves the contact point of a Pure rolling object?

Question

If an object is pure rolling then its contact point has zero velocity because both tangential and translational velocity cancel out and there will be only centripetal acceleration, however what i don't understand is what happens the next instant to that atom that was previously the contact point, according to vector addition law it no longer has transational and tangential velocity, only it has velocity upwards (because previously we had centripetal acceleration) so if it no longer has translational motion but all of the other atoms of the object have translational motion then how come it doesn't disconnect from the object or just crash into other atoms?

Answer 1

First off, this is a very clever question! I think the confusion here stems from to the fact that there are two possible reference frames you can take when you consider a rolling object.

1. The reference frame of an observer; you see the object roll past you.
2. The reference frame of the object; the object rolls in place and the world moves around it, so to speak. You can visualize this as an object spinning in midair.

You began this problem by working in the first reference frame, and you correctly added up the translational and rotational velocities such that contact point has a velocity of zero. However, a constant translational velocity is a result of the reference frame we choose and is not "seen" by the object itself. It isn't something we consider when we study the forces acting on the body, because we should not be able to change the laws of physics just by changing our reference frame. Consider what it feels like to be in a train that is moving at a constant velocity, and consider why we do not take into account the speed of the earth moving around the sun (67,000 mph!) when we do basic experiments like rolling objects down an incline.

If we consider the problem using second reference frame, where the object is still and the world is moving around it, you can see that from the object's perspective every single point on its surface, including the contact point, is moving with a constant tangential velocity. This removes the contradiction you observed.

This is a very nuanced problem so please let me know if you need any additional clarification!