So I would like to now consider the wheel that
is rolling without slipping.
And what I'd like to do is consider--
let's draw the wheel rolling without slipping.
And I'd like to consider the contact point between the wheel
and the ground.
And I'd like to understand what the velocity of that contact
point is.
And the result is surprising.
Now, we know by our law of the addition of velocities--
let's call that the point P--
that the velocity of P is the velocity of the center of mass,
and that pointed this way, plus the velocity of the point
P as seen in the center of mass frame.
And in that frame, every single point on the rim
was just doing circular motion.
So that velocity is in the opposite direction.
And the vector sum is the velocity of this point P
as seen in the ground frame.
So we want to consider the magnitudes of these two terms.
The velocity in the center of mass frame
of a point on the rim has a magnitude equal to r omega.
And we said that the velocity of the center of mass,
we call that V cm.
Now, the rolling without slipping condition
was that these two magnitudes were equal.
V cm equals r omega, or the velocity
of the point on the rim in the center of mass frame
is equal to the center of mass velocity of the wheel.
And therefore, the velocity P, which
is the sum of these two vectors for the contact point--
so that's when P is in contact with the ground--
is equal to 0.
So what we say is that the contact point
is instantaneously at rest with respect to the ground.
And that's what really is the mystery of a wheel that's
rolling without slipping-- that this contact point is
the sum of these two vectors, and it's
instantaneously at rest with the ground
when the wheel is rolling without slipping.
The other points are certainly not.
Remember, up here, these two vectors pointed
in the same direction, so that vector has twice the magnitude
of either one.