Rolling without slipping: Why point of contact has zero velocity

[AlonsoMcLaren]

In rolling without slipping, why does the point of contact have zero velocity relative to the ground? If so, how can the point of contact move relatively to the ground? (It has to move relative to the ground, otherwise it cannot roll and must stay in its original position)

 

[AlephZero]

The point of contact does move relative to the ground. It moves along a curve which is vertical at the point where it touches the ground and lifts off again

See http://en.wikipedia.org/wiki/Cycloid

 

[AlonsoMcLaren]

The point of contact does move relative to the ground. It moves along a curve which is vertical at the point where it touches the ground and lifts off again

See http://en.wikipedia.org/wiki/Cycloid

Then I found something on the Internet:
http://www.columbia.edu/~crg2133/Files/Physics/8_01/noSlip.pdf

Which claims that "When a body is rolling on a plane without slipping,
the point of contact with the plane does not move".

I am really confused.

 

[AlephZero]

Suppose you throw a ball verrtically upwards. When it reaches its highest point, its velocity is 0, so it isn't moving. But the ball still managed to get to its highest point somehow, and then it falls down again.

The explanation is that at the highest point its velocity is 0, but its acceleration is not 0 (it is 9.8m/s^2 downwards, because of gravity).

The same is troe of the point of contact on the wheel. When the point is touchng the ground it's velocity is 0, but it is accelerating vertically upwards. In fact every point on the circumference of the wheel is accelerating towards the center of the wheel, so there is nothing "special" about the fact that the contact point is accelerating.

 

[andrien]

the point of contact always has zero velocity with respect to surface on which it is moving to define the condition of rolling.that's all.It has centripetal acceleration towards centre of curvature.

 

[rcgldr]

A specific point on an object rolling on the ground has zero velocity relative to the ground during the instant in time that point on the object is touching the ground. For a circular object, a specific point on the surface of the object follows the path of a cycloid, and is only in contact with the ground for one instant for every revolution of the rolling object (so it's only the "point of contact" one instant per revolution).

The term "point of contact" normally means the continuous point of contact between rolling object and ground, where the point of contact moves with respect to the ground as the object rolls (and/or skids).

A common real world term similar to this is "contact patch" which is the area of a tire in contact with pavement. "Contact patch" refers to the area of contact between tire and pavement, even if the tire is rolling and/or sliding along the pavement, so the "contact patch" moves as the tire moves with respect to the pavement.

Getting back to the path followed by specific points on the surface of a rolling object, you can sum up the vectors related to rotation and lateral movement, as shown in this web article in the "rolling motion as combined motion" section:

http://cnx.org/content/m14311/latest