WEBVTT
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Let's now consider
our rolling wheel,
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and we want to look at
some special conditions.
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So at time t equals 0--
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and we'll have our wheel that's
rolling, here's the ground--
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let's say that our point P
is right up here at the top.
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That's cm.
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And we'll be in the
ground frame now.
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And then at a
later time, time t,
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the wheel has
moved to the right.
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So let's draw the
wheel over here.
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Not the greatest
picture of the wheel,
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but we'll have the
wheel over here.
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And now the point P has
moved some angle delta theta.
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And we'll call this
time interval delta t.
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Now, the center of mass of the
wheel has moved a distance.
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Xcm is the velocity of the
center of mass times delta t.
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And the point P on the
rim, this arc change,
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this length here
on the rim that P
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has moved around in the
center of mass frame,
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is R delta theta.
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Now, we want to ask ourselves--
we'll call this delta x.
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We now have three
possible conditions.
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We call rolling
without slipping.
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That will be our first case 1.
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And that's the case
when the arc length
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delta s is exactly equal to
the distance along the ground.
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So we have delta Xcm is delta s.
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And so we get Vcm delta
t equals R delta theta,
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or Vcm equals R delta
theta over delta t.
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Now, in the limit as
delta t goes to 0,
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we have that delta
theta over delta t
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in this limit as delta t
goes to 0 is d theta dt
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And that's what we
called the angular speed.
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So in our limit as this wheel
is rolling without slipping,
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we have the condition that the
velocity Vcm equals R omega.
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So that's our first condition,
and we call this the rolling
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without slipping.
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Now what is Vcm?
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Vcm?
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That's the velocity of the
center of mass of the wheel,
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and every single point on this
wheel has that same speed.
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And R omega, you can think of
that as the tangential velocity
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in reference frame cm.
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This is just the speed
in the reference frame
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moving with the center of mass.
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So this is our condition for
rolling without slipping.
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Now, our second case is
imagine that the wheel is not
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moving forward at all,
but it's just spinning.
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That's what we call the wheel
is slipping on the ground,
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for instance, if there were ice.
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And so what we call slipping
is a little bit more general.
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It's whenever the
wheel is spinning,
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and the arc length
is much greater
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than the horizontal distance
that the wheel has moved.
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So we have delta s
representing the arc length
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that the point has
moved in the center
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of mass frame is greater than
how far the center of mass
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is moving.
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And so, again, we have R delta
theta is greater than Vcm delta
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t, or in the limit R
omega is greater than Vcm.
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You can say it's spinning
faster than it's translating.
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And finally, the
skidding condition.
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Skidding-- imagine that the
wheel-- you're braking a wheel.
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The wheel is not
spinning at all,
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but it's just sliding
along horizontally.
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So the horizontal
delta x center of mass
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is bigger than delta scm.
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And so this is the case where
delta Xcm, how far it moved
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horizontally, is greater
than the amount of arc length
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that the point moved.
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And so in the same
type of argument,
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when we put our conditions
in we get that Vcm
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is greater than R omega.
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And again, what that corresponds
to in the skidding case,
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imagine the limit where
it's not rotating at all,
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this would be 0, and it's just
skidding along the ground, Vcm.
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So we have our three conditions.
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We have the slipping
condition, where it's spinning
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faster than it's translating.
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We have the skidding
condition, where
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it's translating faster
than it's spinning.
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And we have the rolling
without slipping condition,
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in which the arc
length is exactly
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equal to the distance,
horizontal distance,
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along the ground.