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00:00:03,750 --> 00:00:06,450
So I would like to now
consider the wheel that

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00:00:06,450 --> 00:00:09,330
is rolling without slipping.

3
00:00:09,330 --> 00:00:13,250
And what I'd like
to do is consider--

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00:00:13,250 --> 00:00:15,900
let's draw the wheel
rolling without slipping.

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00:00:15,900 --> 00:00:24,170
And I'd like to consider the
contact point between the wheel

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and the ground.

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00:00:26,910 --> 00:00:30,330
And I'd like to understand what
the velocity of that contact

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00:00:30,330 --> 00:00:30,830
point is.

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00:00:30,830 --> 00:00:32,820
And the result is surprising.

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00:00:32,820 --> 00:00:36,350
Now, we know by our law of
the addition of velocities--

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00:00:36,350 --> 00:00:38,840
let's call that the point P--

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00:00:38,840 --> 00:00:44,180
that the velocity of P is the
velocity of the center of mass,

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00:00:44,180 --> 00:00:49,160
and that pointed this way,
plus the velocity of the point

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00:00:49,160 --> 00:00:53,150
P as seen in the
center of mass frame.

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00:00:53,150 --> 00:00:56,690
And in that frame, every
single point on the rim

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00:00:56,690 --> 00:00:58,520
was just doing circular motion.

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00:00:58,520 --> 00:01:03,200
So that velocity is in
the opposite direction.

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00:01:03,200 --> 00:01:07,520
And the vector sum is the
velocity of this point P

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00:01:07,520 --> 00:01:12,210
as seen in the ground frame.

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00:01:12,210 --> 00:01:16,700
So we want to consider the
magnitudes of these two terms.

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00:01:16,700 --> 00:01:20,570
The velocity in the
center of mass frame

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00:01:20,570 --> 00:01:27,060
of a point on the rim has a
magnitude equal to r omega.

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00:01:27,060 --> 00:01:31,789
And we said that the velocity
of the center of mass,

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00:01:31,789 --> 00:01:34,580
we call that V cm.

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00:01:34,580 --> 00:01:39,590
Now, the rolling without
slipping condition

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00:01:39,590 --> 00:01:42,270
was that these two
magnitudes were equal.

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00:01:42,270 --> 00:01:47,690
V cm equals r omega,
or the velocity

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00:01:47,690 --> 00:01:50,150
of the point on the rim in
the center of mass frame

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00:01:50,150 --> 00:01:54,710
is equal to the center of
mass velocity of the wheel.

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00:01:54,710 --> 00:01:58,880
And therefore, the
velocity P, which

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00:01:58,880 --> 00:02:05,630
is the sum of these two
vectors for the contact point--

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00:02:05,630 --> 00:02:10,639
so that's when P is in
contact with the ground--

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00:02:10,639 --> 00:02:12,950
is equal to 0.

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00:02:12,950 --> 00:02:19,910
So what we say is
that the contact point

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00:02:19,910 --> 00:02:34,130
is instantaneously at rest
with respect to the ground.

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00:02:37,210 --> 00:02:40,260
And that's what really is
the mystery of a wheel that's

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00:02:40,260 --> 00:02:43,680
rolling without slipping--
that this contact point is

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00:02:43,680 --> 00:02:45,750
the sum of these two
vectors, and it's

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00:02:45,750 --> 00:02:47,910
instantaneously at
rest with the ground

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00:02:47,910 --> 00:02:50,280
when the wheel is
rolling without slipping.

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00:02:50,280 --> 00:02:52,890
The other points
are certainly not.

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00:02:52,890 --> 00:02:56,760
Remember, up here, these
two vectors pointed

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00:02:56,760 --> 00:03:02,280
in the same direction, so that
vector has twice the magnitude

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00:03:02,280 --> 00:03:04,310
of either one.