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00:00:03,122 --> 00:00:06,260
DEEPTO CHAKRABARTY: A pivoted
rod held horizontal, parallel
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to the ground, and
released from rest
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00:00:08,000 --> 00:00:12,590
will simply fall, rotating
about the pivot point.
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00:00:12,590 --> 00:00:14,720
In particular, suppose
we have a mass attached
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to one end of a pivoted rod.
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00:00:17,340 --> 00:00:20,650
So here is my pivot.
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00:00:20,650 --> 00:00:24,020
Here's my pivoted rod, which
we'll assume is massless.
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00:00:24,020 --> 00:00:26,720
And I have-- and
it has a length d,
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00:00:26,720 --> 00:00:30,860
and I attach a
mass m to one end.
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If I let this go from
rest, it will simply
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00:00:34,670 --> 00:00:39,532
fall, rotating about the pivot
point, which I'll call s.
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00:00:39,532 --> 00:00:41,240
And it's easy to
understand that in terms
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00:00:41,240 --> 00:00:45,110
of the action of gravity
which is acting downward
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00:00:45,110 --> 00:00:47,660
and the resulting
torque about point s.
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00:00:50,220 --> 00:01:02,490
If I replace this point mass
with a wheel of the same mass,
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00:01:02,490 --> 00:01:04,590
so this disk is a
wheel with, let's say,
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a radius r and the same mass
m attached to a pivoted rod.
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The rod is massless
and has length d.
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If I hold this horizontal,
parallel to the ground,
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and release it from
rest, it will still just
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fall to the ground.
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Not a surprising result.
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What is surprising is if
I then spin up the wheel,
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so if I have the
wheel rotating about
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00:01:30,900 --> 00:01:36,090
its axle with some large
angular velocity, little omega,
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and then I hold it horizontal,
parallel to the ground,
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and release it from rest, then
remarkably this wheel plus axle
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will not fall.
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It will remain horizontal,
parallel to the ground.
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But the center of
mass of the wheel
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will execute a small circular
orbit about the vertical axis
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through the pivot point.
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This remarkable and
very non-intuitive
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motion is called
precession and the system
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that's undergoing precession
is called a gyroscope.
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Let's see if we can
understand this behavior
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00:02:24,470 --> 00:02:27,090
in terms of the angular
momentum of the system.
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00:02:27,090 --> 00:02:33,270
So what I've drawn here is
a side view of the system.
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00:02:33,270 --> 00:02:35,330
So let's define
some coordinates.
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00:02:35,330 --> 00:02:41,520
So we have r-hat in the
radial outward direction,
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00:02:41,520 --> 00:02:46,770
k-hat in the direction of the
z-axis, the vertical axis.
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And then we'll define theta-hat
pointing into the screen.
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OK, so that's a side view.
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00:02:55,250 --> 00:02:57,480
I'd like to now draw a top view.
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So let's say we're looking
down along the z-axis
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from the top on the system.
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00:03:02,580 --> 00:03:09,230
So now here is my
pivot point, and here
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is a top view of my wheel.
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00:03:14,840 --> 00:03:17,240
Again, this is a distance d.
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00:03:17,240 --> 00:03:23,220
Now this is still
the r-hat direction.
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This is the theta-hat direction.
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And the k-hat direction. is
pointing out of the screen.
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00:03:32,480 --> 00:03:34,910
Let's draw the forces
acting on our diagram here.
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00:03:34,910 --> 00:03:38,560
So the weight is acting at the
center of mass of the wheel.
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That's mg downward.
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00:03:41,400 --> 00:03:44,210
And there's a normal
force acting upwards
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at the pivot point.
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The torque is just
given by r cross f,
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and so relative to
point s at the distance
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d times the weight mg.
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So the torque is mgd.
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00:04:02,000 --> 00:04:04,400
And by doing r cross f
with the right hand rule,
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we see that it's directed in
the plus theta-hat direction.
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00:04:11,640 --> 00:04:13,500
Now again, let's
suppose that I'm
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00:04:13,500 --> 00:04:17,190
holding the wheel horizontal
and release it from rest.
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00:04:17,190 --> 00:04:19,910
If the wheel is not spinning--
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so this is not rotating,
this is a stationary wheel
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00:04:22,200 --> 00:04:26,220
that I'm holding horizontal,
and I release it from rest,
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then its initial
angular momentum,
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00:04:28,980 --> 00:04:32,160
with respect to point s, is 0.
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00:04:32,160 --> 00:04:35,280
Over a short time
interval, delta t,
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00:04:35,280 --> 00:04:37,740
the torque which is acting
in the theta-hat direction,
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00:04:37,740 --> 00:04:41,760
will cause an angular impulse or
change in the angular momentum.
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00:04:41,760 --> 00:04:46,470
That change in angular
momentum, delta l vector,
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00:04:46,470 --> 00:04:51,240
is equal to the torque
times that short time
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00:04:51,240 --> 00:04:52,710
interval, delta t.
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00:04:52,710 --> 00:04:54,792
And it will act in the
theta-hat direction.
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00:04:54,792 --> 00:04:56,250
The change in
angular momentum will
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00:04:56,250 --> 00:04:59,070
be in the theta-hat direction
because that's the direction
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00:04:59,070 --> 00:05:00,060
that the torque is in.
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00:05:00,060 --> 00:05:02,220
So for this case--
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00:05:02,220 --> 00:05:05,220
and by the way, this view here,
I should have labeled this.
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00:05:05,220 --> 00:05:08,660
This is a top view.
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00:05:08,660 --> 00:05:09,710
So this is the side view.
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00:05:09,710 --> 00:05:11,570
This is the top view
of the same system.
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00:05:11,570 --> 00:05:14,880
So the torque is mgd in
the theta-hat direction.
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00:05:14,880 --> 00:05:18,290
So in the side view, that's
going into the screen.
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00:05:18,290 --> 00:05:24,260
And in the top view, that's
going pointing upward.
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00:05:24,260 --> 00:05:26,130
That's the theta-hat direction.
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00:05:26,130 --> 00:05:30,890
So in the case where
the wheel is not
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00:05:30,890 --> 00:05:35,770
spinning, if we think
about the top view,
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the initial angular
momentum is 0.
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00:05:39,201 --> 00:05:39,700
Right?
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00:05:39,700 --> 00:05:41,200
So I'll just draw that as a dot.
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00:05:41,200 --> 00:05:47,590
So that's l initial equals 0.
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00:05:47,590 --> 00:05:54,080
And then I add a small torque,
or small angular impulse
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00:05:54,080 --> 00:05:55,540
due to the torque,
delta l, which
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00:05:55,540 --> 00:05:57,290
is in the theta-hat direction.
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00:05:57,290 --> 00:06:01,490
So that's pointing in
the theta-hat direction.
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00:06:01,490 --> 00:06:05,260
So that's my delta l.
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00:06:05,260 --> 00:06:08,920
And when I sum those
together, I start out with 0.
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00:06:08,920 --> 00:06:10,750
I add a small delta l.
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00:06:10,750 --> 00:06:15,220
So my final angular momentum,
a time delta t later,
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is just equal to my delta l.
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So this is l final.
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00:06:21,880 --> 00:06:26,090
And that's pointing in
the theta-hat direction.
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So a torque in the
theta-hat direction,
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into the screen
for the side view,
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is consistent with the
wheel falling down.
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00:06:33,550 --> 00:06:36,970
So with theta-hat
pointing into the screen,
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the wheel will
basically fall this way.
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It's rotating about point s.
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And that's consistent
with the torque pointing
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in the theta-hat direction.
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Now as the wheel
falls, the torque
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continues to point in
the theta-hat direction.
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And so the angular
acceleration will increase.
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00:06:57,860 --> 00:07:01,840
Now what happens if this wheel
is not stationary, but instead
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is spinning rapidly?
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00:07:04,360 --> 00:07:07,010
In that case, the
torque remains the same.
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00:07:07,010 --> 00:07:10,510
It's still mgd in the
theta-hat direction.
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00:07:10,510 --> 00:07:14,350
But now the initial
angular momentum is not 0.
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00:07:14,350 --> 00:07:19,000
Rather it is a very large vector
pointing along the spin axis.
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Let's choose the
sense of rotation
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such that l points in
the plus r-hat direction.
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That's actually the
way I've drawn it here.
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So in that case, the
angular momentum vector
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00:07:28,360 --> 00:07:33,100
initially points in the
plus r-hat direction.
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00:07:33,100 --> 00:07:35,680
So then what happens over
a short time, delta t?
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00:07:35,680 --> 00:07:39,705
So now let's consider the case
where the wheel is spinning.
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So now my initial
angular momentum
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is a large vector pointing
in the r-hat direction.
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That's l initial.
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I'm adding a small
perpendicular vector, delta l,
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in the theta-hat direction.
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And so the sum of
those two things,
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if this is the original r-hat
direction, what's happened
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is that my new vector is at
a small angle with respect
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to the original r-hat direction.
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I'll call that
angle delta theta.
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00:08:23,260 --> 00:08:26,890
So what I've done is I've
rotated my initial angular
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momentum vector by a small angle
without changing its length.
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00:08:32,909 --> 00:08:35,289
Notice the two very
different situations.
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00:08:35,289 --> 00:08:39,490
In one case, I start out
with 0 angular momentum.
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And all the angular
momentum I end up with
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00:08:41,500 --> 00:08:44,800
comes from the angular
impulse due to the torque.
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00:08:44,800 --> 00:08:47,020
In the second case, where
the wheel is spinning,
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I start out with a very large
initial angular momentum.
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00:08:51,390 --> 00:08:53,890
I then add a small
angular impulse,
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00:08:53,890 --> 00:08:56,320
small compared to my
initial angular momentum,
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in the perpendicular direction.
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And that causes not a change
in the length of the vector,
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but a change in its direction.
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00:09:03,850 --> 00:09:07,390
Which means that the angular
momentum vector rotates.
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00:09:07,390 --> 00:09:10,930
That's why the
system precesses when
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the wheel is rotating rapidly.
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Before we can understand
precession more carefully,
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it'll be useful to review
the mathematics of rotating
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vectors.
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So first, suppose we have
a vector that I'll call r1.
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And I'm going to add a
vector delta r to this.
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00:09:33,562 --> 00:09:35,020
And I'm going to
have the condition
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that the length of
delta r is much,
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much smaller than the length
of my original vector r1,
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and that delta r is
perpendicular to R1.
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So here's my delta r.
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00:10:03,830 --> 00:10:08,120
And so if I add
these two vectors,
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let's say r2 is the
sum of r1 plus delta r,
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so that's my vector r2.
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And I'm going to call
this angle delta theta.
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00:10:24,210 --> 00:10:26,340
And now few things to note.
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First of all, since this
is a right triangle,
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notice that the length
r2 is equal to r1 divided
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by the cosine of delta theta.
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00:10:41,640 --> 00:10:45,920
But if that angle is small, if
delta theta is a small angle,
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then cosine theta is
well approximated by 1,
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00:10:49,110 --> 00:10:54,320
and so this is just
my original length r1.
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00:10:54,320 --> 00:10:57,720
And since r1 is equal to r2,
I'm just going to call that--
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call them both r.
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00:10:59,550 --> 00:11:06,690
And this is for
small delta theta.
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00:11:06,690 --> 00:11:10,140
So that tells us that
the vector just rotates.
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When I have a large vector,
and I add a small perpendicular
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00:11:14,070 --> 00:11:18,300
vector, the result is to
rotate the original vector
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without changing its length.
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00:11:19,920 --> 00:11:22,500
As long as the angle is
small or, equivalently, as
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00:11:22,500 --> 00:11:25,970
long as delta r is very small
compared to my original vector
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00:11:25,970 --> 00:11:29,670
length, I'll get
a pure rotation.
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00:11:29,670 --> 00:11:32,050
In addition, again
looking at this triangle,
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00:11:32,050 --> 00:11:39,340
notice that delta r is
equal to [AUDIO OUT], which
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00:11:39,340 --> 00:11:47,050
I can describe as r, times
the sine of delta theta.
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00:11:47,050 --> 00:11:49,810
And again, if the
angle is small,
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00:11:49,810 --> 00:11:51,820
if delta theta's a
small angle, then this
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00:11:51,820 --> 00:11:55,990
is well approximated by
r times the angle delta
194
00:11:55,990 --> 00:11:58,180
theta in radians.
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00:11:58,180 --> 00:12:06,920
And again, this is
for small delta theta.
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00:12:06,920 --> 00:12:11,530
So now if I divide
this last equation
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00:12:11,530 --> 00:12:16,870
by delta t, small time interval
in which this rotation is
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00:12:16,870 --> 00:12:24,420
happening, then I write this
as delta r vector divided
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00:12:24,420 --> 00:12:25,560
by delta t.
200
00:12:25,560 --> 00:12:27,850
And I take the
magnitude of that.
201
00:12:27,850 --> 00:12:37,530
That is equal to r delta
theta divided by delta t.
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00:12:37,530 --> 00:12:41,640
And if I go to the limit
as delta t gets small,
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00:12:41,640 --> 00:12:45,690
as delta t approaches
0, then I can write this
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00:12:45,690 --> 00:12:50,497
as the derivative of the
vector r, the time derivative,
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00:12:50,497 --> 00:12:52,080
I should say the
magnitude of the time
206
00:12:52,080 --> 00:12:53,730
derivative of vector r.
207
00:12:53,730 --> 00:13:00,630
And that's equal to the length
r times the time derivative
208
00:13:00,630 --> 00:13:04,350
of the angle d theta dt.
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00:13:04,350 --> 00:13:07,500
But d theta dt is just
the angular velocity.
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00:13:07,500 --> 00:13:11,760
So I could write that
as r times capital omega
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00:13:11,760 --> 00:13:13,320
for the angular velocity.
212
00:13:13,320 --> 00:13:16,140
And this is just
our familiar result
213
00:13:16,140 --> 00:13:21,780
that for circular motion, for
the rotation of a position
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00:13:21,780 --> 00:13:27,330
vector, the velocity
is equal to the radius
215
00:13:27,330 --> 00:13:34,044
of the circle times the angular
velocity of the rotation.
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00:13:34,044 --> 00:13:35,460
Another way of
thinking of that is
217
00:13:35,460 --> 00:13:39,780
that the magnitude of the
rate of change of the position
218
00:13:39,780 --> 00:13:42,030
vector, which we
call the velocity,
219
00:13:42,030 --> 00:13:44,520
is equal to the length
of the rotating vector
220
00:13:44,520 --> 00:13:49,680
r times the angular
velocity of the rotation.
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00:13:49,680 --> 00:13:52,740
But there's nothing special
about the position vector
222
00:13:52,740 --> 00:13:56,110
that I used in order
to do this analysis.
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00:13:56,110 --> 00:14:02,970
So if instead of a position
vector I considered any vector,
224
00:14:02,970 --> 00:14:11,780
so let's say this is my
rotational motion of a vector
225
00:14:11,780 --> 00:14:17,000
that I'll call A. So
that's A at time t.
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00:14:17,000 --> 00:14:25,790
And then at some later time,
that's A of t plus delta t.
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00:14:25,790 --> 00:14:33,260
This vector is my delta A. I'll
call this angle delta theta.
228
00:14:33,260 --> 00:14:37,070
And so the vector A is
rotating in that direction
229
00:14:37,070 --> 00:14:39,920
with an angular
velocity capital omega.
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00:14:39,920 --> 00:14:45,560
Now in the example I just did,
my vector A is actually r of t.
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00:14:45,560 --> 00:14:47,390
But so everywhere
where I have an r here,
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00:14:47,390 --> 00:14:51,860
I can just write an A. This
is now just an arbitrary
233
00:14:51,860 --> 00:14:56,120
vector in space that is rotating
at an angular velocity capital
234
00:14:56,120 --> 00:14:57,050
omega.
235
00:14:57,050 --> 00:15:00,050
And what we see, using
the same analysis,
236
00:15:00,050 --> 00:15:06,830
we would find that the magnitude
of the time rate of change
237
00:15:06,830 --> 00:15:10,670
of the vector, of
the rotating vector,
238
00:15:10,670 --> 00:15:13,760
is equal to the length of
the rotating vector, which
239
00:15:13,760 --> 00:15:20,450
is A, times d theta dt.
240
00:15:20,450 --> 00:15:22,700
Or in other words, the
length of the rotating
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00:15:22,700 --> 00:15:26,750
vector times the angular
velocity of the rotation.
242
00:15:26,750 --> 00:15:28,500
That's true for any vector.
243
00:15:28,500 --> 00:15:29,000
OK?
244
00:15:29,000 --> 00:15:31,330
This is a general
result. v equals
245
00:15:31,330 --> 00:15:36,920
r omega is just a special
case of this general rule
246
00:15:36,920 --> 00:15:40,010
where, in that case, my rotating
vector is a position vector.
247
00:15:40,010 --> 00:15:43,280
But this is true for any vector
that's rotating in space.
248
00:15:43,280 --> 00:15:46,610
In particular for a
rotating angular momentum--
249
00:15:52,430 --> 00:15:58,440
for a rotating angular
momentum vector,
250
00:15:58,440 --> 00:16:04,410
I have that the magnitude of the
time derivative of the rotating
251
00:16:04,410 --> 00:16:09,111
angular momentum vector is
just equal to the length
252
00:16:09,111 --> 00:16:11,610
of the angular momentum vector,
the magnitude of the angular
253
00:16:11,610 --> 00:16:16,570
momentum, times the angular
velocity of rotation.
254
00:16:16,570 --> 00:16:19,176
Now in addition, in the
particular case of a rotating
255
00:16:19,176 --> 00:16:21,550
angular momentum vector or of
any angular momentum vector
256
00:16:21,550 --> 00:16:27,160
rather, we know that the time
derivative of the angular
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00:16:27,160 --> 00:16:31,920
momentum vector is also
equal to the torque vector.
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00:16:31,920 --> 00:16:33,930
So I can set these
two things equal.
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00:16:33,930 --> 00:16:36,510
In the case of a rotating
angular momentum vector,
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00:16:36,510 --> 00:16:39,000
the magnitude of
the torque is given
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00:16:39,000 --> 00:16:43,260
by the magnitude of the
rotating angular momentum
262
00:16:43,260 --> 00:16:46,600
vector times the angular
speed of rotation.
263
00:16:46,600 --> 00:16:49,680
And that's just using
the general behavior
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00:16:49,680 --> 00:16:52,430
of a rotating vector in space.