Rolling is a very important type of motion for in nature but perhaps even more so for human technology. What we shall talk about here is rolling without slipping. It is a coupled rotation and translational motion.
The first thing to determine is the relationship between the angular
displacement and translational displacement. If we unroll a string
wound around a wheel by rolling the wheel forward we note that
the amount of string rolled off equals the translational displacement.
Thus we can write:
This simple geometrical relationship is central to understanding
not only rolling but also mechanical problems involving strings rolling off pulleys!
Taking a time derivative we get a relationship between
translational and angular velocity:
From this formula we can derive the velocity of a point on the rim of the wheel with respect to the ground. To keep it simple we consider just the velocity at the top and bottom of the rim where the velocity
of the rim with respect to the center of the wheel is parallel to the velocity of the center of the wheel with respect to the ground. We have
Thus the bottom of the rim does not move with repsect to the ground.
It is challenging to come to grips with this but it is the
crux of rolling without sliding. In fact we could have turned things around
and chosen to derive Eq. 17 from the requirement that
. Pursuing this line of reasoning it can be shown that
at any instant during the rolling motion every point in the wheel
moves as though it were in pure rotation about the point of contact
with the surface on which it rolls. Even though the wheel moves on
and the point of contact thus is constantly changing this can be a
very useful result to bear in mind.