Breaking down forces for free body diagrams

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.700
- [Instructor] Let's say
we have some type of hard,

00:00:02.700 --> 00:00:05.180
flat, frictionless
surface right over here.

00:00:05.180 --> 00:00:08.120
That's my drawing of a hard,
flat, frictionless surface.

00:00:08.120 --> 00:00:10.740
And on that, I have a block,

00:00:10.740 --> 00:00:13.870
and that block is not
accelerating in any direction.

00:00:13.870 --> 00:00:16.230
It is just sitting there.

00:00:16.230 --> 00:00:18.580
And let's say we know
the weight of that block.

00:00:18.580 --> 00:00:21.030
It is a 10-newton block.

00:00:21.030 --> 00:00:23.020
So my question to you is,

00:00:23.020 --> 00:00:25.570
pause this video, and think
about what are all of the forces

00:00:25.570 --> 00:00:28.173
that are acting on this block?

00:00:29.010 --> 00:00:30.560
All right, now let's work
through this together.

00:00:30.560 --> 00:00:32.300
And to do it, I'm going to draw

00:00:32.300 --> 00:00:36.420
what's known as a free-body diagram

00:00:36.420 --> 00:00:38.250
to think about all of the forces.

00:00:38.250 --> 00:00:41.180
And the reason why it's
called a free-body diagram

00:00:41.180 --> 00:00:43.230
is that we just focus on this one body.

00:00:43.230 --> 00:00:45.130
And we don't draw
everything else around it,

00:00:45.130 --> 00:00:47.160
and we just draw the forces acting on it.

00:00:47.160 --> 00:00:49.150
And there's actually two typical ways

00:00:49.150 --> 00:00:51.200
of drawing a free-body diagram.

00:00:51.200 --> 00:00:52.033
I'll do them both.

00:00:52.033 --> 00:00:53.360
So first, I could draw it this way.

00:00:53.360 --> 00:00:55.053
So this is my block here.

00:00:55.990 --> 00:00:58.830
Now, I told you that it weighs 10 newtons.

00:00:58.830 --> 00:01:00.070
The weight of an object,

00:01:00.070 --> 00:01:03.500
that's the force of gravity
acting on that object,

00:01:03.500 --> 00:01:04.990
and it would be downwards.

00:01:04.990 --> 00:01:08.220
So we have, from this 10
newtons right over here,

00:01:08.220 --> 00:01:11.560
we know that there is a downward force,

00:01:11.560 --> 00:01:12.730
the force of gravity

00:01:12.730 --> 00:01:16.230
acting on the mass of
this object of 10 newtons.

00:01:16.230 --> 00:01:17.630
It has a magnitude of 10 newtons,

00:01:17.630 --> 00:01:19.820
and the force is acting downwards.

00:01:19.820 --> 00:01:23.580
We could say this is the
magnitude of the force of gravity.

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And when you draw a free-body diagram,

00:01:25.390 --> 00:01:28.820
it's typical to show
your vectors originating

00:01:28.820 --> 00:01:30.900
out of the center of that,

00:01:30.900 --> 00:01:32.930
of that object in your drawing.

00:01:32.930 --> 00:01:33.870
Now, my question to you is,

00:01:33.870 --> 00:01:36.130
is that the only force acting on this?

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If you think it is, what
would happen to the object?

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Well, it would start

00:01:39.640 --> 00:01:42.400
or it would be accelerating downwards.

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But I just said that this is not

00:01:43.470 --> 00:01:44.860
accelerating in any direction.

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So there must be something
that counteracts this,

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and there is.

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There's the normal force

00:01:49.970 --> 00:01:52.510
of this surface acting on the block.

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That surface is what's keeping the block

00:01:54.570 --> 00:01:56.270
from accelerating downwards.

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And I will do that with
this vector right over here.

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So it's going to be going upwards,

00:02:00.440 --> 00:02:02.330
and it's going to have
the exact same magnitude,

00:02:02.330 --> 00:02:04.070
just in the opposite direction.

00:02:04.070 --> 00:02:08.120
So I could say the magnitude
of the normal force,

00:02:08.120 --> 00:02:10.340
the normal force is
magnitude right over here,

00:02:10.340 --> 00:02:12.290
is also going to be equal to 10 newtons,

00:02:12.290 --> 00:02:13.400
but it's going upwards.

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And we can see that these
two are going to net out,

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and so you have no net force

00:02:19.440 --> 00:02:21.480
acting in this vertical dimension.

00:02:21.480 --> 00:02:22.540
And I have no forces,

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I haven't thought about any or drawn any

00:02:24.320 --> 00:02:25.560
in the horizontal direction.

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And so that's why you have
no net force in total,

00:02:28.220 --> 00:02:31.090
and this thing isn't
going to be accelerated.

00:02:31.090 --> 00:02:32.760
Now, I mentioned that there's other ways

00:02:32.760 --> 00:02:34.440
to draw a free-body diagram.

00:02:34.440 --> 00:02:36.680
Another way that you
might see it is like this,

00:02:36.680 --> 00:02:38.163
where you see the body.

00:02:39.030 --> 00:02:41.210
And from the outside of the body,

00:02:41.210 --> 00:02:43.110
you see the vectors being drawn.

00:02:43.110 --> 00:02:46.503
So in this situation, you
have ten newtons downward,

00:02:47.890 --> 00:02:50.830
and you would have ten newtons upward.

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This is another way that you might see

00:02:53.830 --> 00:02:56.320
free-body diagrams drawn.

00:02:56.320 --> 00:02:57.720
Now, what I want to do is

00:02:57.720 --> 00:03:00.140
do something interesting to this block.

00:03:00.140 --> 00:03:01.740
Let me redraw it.

00:03:01.740 --> 00:03:05.050
So I have my surface here,

00:03:05.050 --> 00:03:09.460
my hard, frictionless
surface, and it's flat.

00:03:09.460 --> 00:03:11.760
And I still have my block here.

00:03:11.760 --> 00:03:14.320
It's my 10-newton block.

00:03:14.320 --> 00:03:16.840
But now I'm going to apply a force to it.

00:03:16.840 --> 00:03:19.840
I am going to apply a force

00:03:21.160 --> 00:03:22.823
that is in this direction.

00:03:23.690 --> 00:03:26.320
It's in this direction, and its magnitude,

00:03:26.320 --> 00:03:30.170
let's say its magnitude is 20 newtons.

00:03:30.170 --> 00:03:32.490
And just so that we know the direction,

00:03:32.490 --> 00:03:34.700
this angle right over here,

00:03:34.700 --> 00:03:37.560
let's say that that is 60 degrees.

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I'll say theta is equal to 60 degrees.

00:03:40.220 --> 00:03:41.790
The magnitude

00:03:41.790 --> 00:03:44.540
of this force is equal to

00:03:44.540 --> 00:03:45.880
20 newtons.

00:03:45.880 --> 00:03:49.800
So what would the free-body
diagram now look like?

00:03:49.800 --> 00:03:52.430
Well, it might be tempting
to just draw the force

00:03:52.430 --> 00:03:54.610
right on one of these free-body diagrams,

00:03:54.610 --> 00:03:56.600
something like that,

00:03:56.600 --> 00:04:00.090
something like that, and
that would not be inaccurate.

00:04:00.090 --> 00:04:01.730
But we would have to watch out

00:04:01.730 --> 00:04:04.100
because this force is acting at an angle.

00:04:04.100 --> 00:04:05.270
So if we were to break it up

00:04:05.270 --> 00:04:07.570
into its horizontal and
vertical components,

00:04:07.570 --> 00:04:09.910
some part of that force
is acting downward.

00:04:09.910 --> 00:04:12.090
And so you're actually going
to have a larger normal force

00:04:12.090 --> 00:04:13.330
to counteract that.

00:04:13.330 --> 00:04:14.720
And some other component is going

00:04:14.720 --> 00:04:16.660
to be working horizontally.

00:04:16.660 --> 00:04:19.900
And so what we want to do
is actually break things up.

00:04:19.900 --> 00:04:21.010
'Cause if you leave it in this angle,

00:04:21.010 --> 00:04:22.540
it gets very, very confusing.

00:04:22.540 --> 00:04:24.970
So what I want to do is I want to break up

00:04:24.970 --> 00:04:28.360
this new blue force into its horizontal

00:04:28.360 --> 00:04:29.940
and vertical components.

00:04:29.940 --> 00:04:31.870
And to do so, we just have to remember

00:04:31.870 --> 00:04:34.390
a little bit of our basic trigonometry.

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If I have a force like this,

00:04:36.900 --> 00:04:38.910
if I have a force like this

00:04:38.910 --> 00:04:42.530
and it is acting at an
angle theta right over here,

00:04:42.530 --> 00:04:43.880
with the horizontal,

00:04:43.880 --> 00:04:46.830
and I want to break it
up into its horizontal

00:04:47.860 --> 00:04:50.470
and its vertical components,

00:04:50.470 --> 00:04:52.600
and its vertical components,

00:04:52.600 --> 00:04:56.940
if the magnitude of the
hypotenuse is capital F,

00:04:56.940 --> 00:04:59.800
then the magnitude of the
adjacent side to this angle,

00:04:59.800 --> 00:05:01.490
this comes straight out of soh-cah-toa,

00:05:01.490 --> 00:05:03.310
from our right triangle trigonometry,

00:05:03.310 --> 00:05:05.830
it would be the magnitude
of our hypotenuse

00:05:05.830 --> 00:05:07.760
times the cosine

00:05:07.760 --> 00:05:09.860
of this angle.

00:05:09.860 --> 00:05:12.380
And the magnitude of
the vertical component,

00:05:12.380 --> 00:05:15.300
that would be the
magnitude of our hypotenuse

00:05:15.300 --> 00:05:17.250
times the sine

00:05:17.250 --> 00:05:18.270
of that angle.

00:05:18.270 --> 00:05:20.610
And you could think about
it the other way as well,

00:05:20.610 --> 00:05:22.853
if the force was like this,

00:05:24.380 --> 00:05:26.530
where it's just going in
the opposite direction.

00:05:26.530 --> 00:05:29.930
But once again, you have this angle theta.

00:05:29.930 --> 00:05:32.940
And now the components
would look like this,

00:05:32.940 --> 00:05:35.830
where the vertical component
would have the same magnitude,

00:05:35.830 --> 00:05:38.100
but now it would be pointing downwards.

00:05:38.100 --> 00:05:41.170
And the horizontal component
would have the same magnitude,

00:05:41.170 --> 00:05:44.020
but now it is pointing to the left.

00:05:44.020 --> 00:05:45.670
It is the same idea.

00:05:45.670 --> 00:05:47.730
If this force has magnitude F

00:05:47.730 --> 00:05:50.700
that's represented by the
hypotenuse of this triangle,

00:05:50.700 --> 00:05:54.600
then the magnitude of
our horizontal component

00:05:54.600 --> 00:05:57.730
is still going to be F cosine theta.

00:05:57.730 --> 00:05:59.780
The vector is not going
in the other direction.

00:05:59.780 --> 00:06:02.850
And the magnitude of our
vertical component here

00:06:02.850 --> 00:06:04.970
is going to be

00:06:04.970 --> 00:06:07.850
F sine theta.

00:06:07.850 --> 00:06:11.030
And so what about this scenario over here?

00:06:11.030 --> 00:06:12.160
Well, in this scenario,

00:06:12.160 --> 00:06:14.833
our vertical component
is gonna look like this,

00:06:15.900 --> 00:06:20.230
and our horizontal component
is going to look like this.

00:06:20.230 --> 00:06:23.430
And so what's the magnitude
of our horizontal component?

00:06:23.430 --> 00:06:27.300
Well, it's going to be the
magnitude of our hypotenuse

00:06:27.300 --> 00:06:30.440
times the cosine of the 60 degrees.

00:06:30.440 --> 00:06:33.200
So this is going to be 20 newtons

00:06:33.200 --> 00:06:36.200
times the cosine of 60 degrees.

00:06:36.200 --> 00:06:37.770
And it's really helpful,

00:06:37.770 --> 00:06:40.290
both trigonometry and physics classes,

00:06:40.290 --> 00:06:43.470
to know the values of
your sines, your cosines,

00:06:43.470 --> 00:06:45.750
and your tangents at
angles like zero degrees,

00:06:45.750 --> 00:06:49.900
30 degrees, 60 degrees,
90 degrees, 45 degrees.

00:06:49.900 --> 00:06:51.900
You could use a calculator
here, but it's useful to know

00:06:51.900 --> 00:06:54.670
that the cosine of 60 degrees is 1/2.

00:06:54.670 --> 00:06:56.200
So 20 times 1/2,

00:06:56.200 --> 00:07:00.710
this is going to be equal to 10 newtons.

00:07:00.710 --> 00:07:02.330
And if we want to know the magnitude

00:07:02.330 --> 00:07:04.230
of our vertical component,

00:07:04.230 --> 00:07:07.480
well, this is going to be 20 newtons

00:07:07.480 --> 00:07:09.480
times the sine

00:07:09.480 --> 00:07:10.860
of 60 degrees.

00:07:10.860 --> 00:07:12.350
Once again, this is useful to know.

00:07:12.350 --> 00:07:14.750
It is square root of three over two.

00:07:14.750 --> 00:07:17.010
And so 20 divided by two is 10.

00:07:17.010 --> 00:07:21.000
So this is going to be 10
square root of three newtons.

00:07:21.000 --> 00:07:22.760
And so we can use that information.

00:07:22.760 --> 00:07:25.040
We've broken up our original vector

00:07:25.040 --> 00:07:27.220
into two component vectors
that, if you took their sum,

00:07:27.220 --> 00:07:29.000
you would get your original one.

00:07:29.000 --> 00:07:31.500
But what's useful now is
that we've broken it up

00:07:31.500 --> 00:07:33.090
into vectors that are parallel

00:07:33.090 --> 00:07:35.080
or perpendicular to our surface,

00:07:35.080 --> 00:07:36.760
which will allow us to think about

00:07:36.760 --> 00:07:40.150
what nets against these things
that I already have in place.

00:07:40.150 --> 00:07:41.580
So let me draw that.

00:07:41.580 --> 00:07:44.930
So actually, I'll first draw
this type of free-body diagram.

00:07:44.930 --> 00:07:47.363
So there is my block.

00:07:48.250 --> 00:07:52.370
And I have the force of
gravity acting on it downwards.

00:07:52.370 --> 00:07:54.000
I will draw it right over here.

00:07:54.000 --> 00:07:55.530
So that's 10 newtons.

00:07:55.530 --> 00:07:57.680
That is the force of
gravity acting downwards.

00:07:57.680 --> 00:08:00.170
Now, is that the only thing
that I have acting downwards?

00:08:00.170 --> 00:08:02.320
No, I also have the vertical component

00:08:02.320 --> 00:08:03.720
of this applied force.

00:08:03.720 --> 00:08:05.350
And so this is going downwards

00:08:05.350 --> 00:08:07.640
10 square roots of three newtons.

00:08:07.640 --> 00:08:09.780
And these aren't drawn perfectly to scale,

00:08:09.780 --> 00:08:11.930
but hopefully you get the idea.

00:08:11.930 --> 00:08:16.640
So this is 10 square roots of
three newtons, just like that.

00:08:16.640 --> 00:08:18.410
And now what is our normal force,

00:08:18.410 --> 00:08:21.010
assuming that our surface
is able to not be compressed

00:08:21.010 --> 00:08:24.210
in any way, that it is a
hard, frictionless surface?

00:08:24.210 --> 00:08:25.500
Well, now our normal force is going

00:08:25.500 --> 00:08:27.420
to counteract both of these forces.

00:08:27.420 --> 00:08:29.690
Our normal force might
look something like this.

00:08:29.690 --> 00:08:31.800
Once again, I haven't drawn
it completely to scale.

00:08:31.800 --> 00:08:34.160
But this would be 10 plus

00:08:34.160 --> 00:08:38.220
10 square roots of three newtons.

00:08:38.220 --> 00:08:40.900
And what about now in
the horizontal direction?

00:08:40.900 --> 00:08:44.350
Well, I have this blue vector right here,

00:08:44.350 --> 00:08:46.430
and so that is going to the right

00:08:46.430 --> 00:08:50.050
with a magnitude of 10 newtons,

00:08:50.050 --> 00:08:51.380
10 newtons.

00:08:51.380 --> 00:08:53.350
And so now you can hopefully appreciate

00:08:53.350 --> 00:08:56.620
why a free-body diagram is
really, really, really useful.

00:08:56.620 --> 00:08:57.880
Just looking at this,

00:08:57.880 --> 00:09:01.240
I can predict what's going
to happen to my block.

00:09:01.240 --> 00:09:03.020
I would say, look, this upward force

00:09:03.020 --> 00:09:05.370
is completely netted out
by these downward forces,

00:09:05.370 --> 00:09:07.250
or these downward forces
are completely netted out

00:09:07.250 --> 00:09:08.490
by this upward force.

00:09:08.490 --> 00:09:10.750
And the only net force that I have

00:09:10.750 --> 00:09:12.750
is ten newtons to the right.

00:09:12.750 --> 00:09:14.270
And so that lets me know that,

00:09:14.270 --> 00:09:16.910
hey, since I have a
net force to the right,

00:09:16.910 --> 00:09:21.393
this block is going to
accelerate to the right.


Free body diagram with angled forces: worked example

WEBVTT
Kind: captions
Language: en

00:00:00.200 --> 00:00:01.590
- [Instructor] So what
we have depicted here.

00:00:01.590 --> 00:00:03.010
We have a block and let's say

00:00:03.010 --> 00:00:05.260
that this block is completely stationary

00:00:05.260 --> 00:00:07.720
and it's being pushed up against this

00:00:07.720 --> 00:00:09.480
non-frictionless wall.

00:00:09.480 --> 00:00:12.330
So this wall does have
friction with the block.

00:00:12.330 --> 00:00:14.130
It's being pushed by this force

00:00:14.130 --> 00:00:17.010
of magnitude F and its direction

00:00:17.010 --> 00:00:20.520
makes an angle of theta
with the horizontal.

00:00:20.520 --> 00:00:23.250
What I want you to do is pause this video

00:00:23.250 --> 00:00:25.700
and construct a free body diagram

00:00:25.700 --> 00:00:28.430
for this block and include the

00:00:28.430 --> 00:00:30.850
vertical and horizontal components

00:00:30.850 --> 00:00:33.460
of this force right over here,

00:00:33.460 --> 00:00:35.600
but also include other things.

00:00:35.600 --> 00:00:39.450
Include the force, include the force

00:00:39.450 --> 00:00:41.910
of gravity acting on this block.

00:00:41.910 --> 00:00:45.730
Include the force of friction

00:00:45.730 --> 00:00:48.560
acting on this block and include

00:00:48.560 --> 00:00:51.880
the normal force of the wall

00:00:51.880 --> 00:00:53.610
acting on the block as well.

00:00:53.610 --> 00:00:56.103
Pause the video and
try to have a go at it.

00:00:57.020 --> 00:00:58.540
So before I even start to draw

00:00:58.540 --> 00:01:01.130
the free body diagram, let's break down

00:01:01.130 --> 00:01:04.990
this force into its vertical
and horizontal components.

00:01:04.990 --> 00:01:06.550
So the first thing,

00:01:06.550 --> 00:01:09.230
let me do its vertical component.

00:01:09.230 --> 00:01:11.310
So it would look like that,

00:01:11.310 --> 00:01:13.810
and it's horizontal component

00:01:13.810 --> 00:01:17.023
would look like this.

00:01:17.870 --> 00:01:21.120
And so what's the magnitude
of the vertical component?

00:01:21.120 --> 00:01:24.130
Well it is opposite
this angle that we know.

00:01:24.130 --> 00:01:25.930
This is the angle that is theta

00:01:25.930 --> 00:01:28.410
and so this is going to be,

00:01:28.410 --> 00:01:30.030
the vertical component is going to be

00:01:30.030 --> 00:01:32.600
F sine of theta.

00:01:32.600 --> 00:01:33.990
We've seen that in previous videos

00:01:33.990 --> 00:01:34.990
and it comes straight out

00:01:34.990 --> 00:01:36.820
of right triangle trigonometry.

00:01:36.820 --> 00:01:39.530
Encourage you to review that
if this looks unfamiliar.

00:01:39.530 --> 00:01:42.990
And the magnitude of the
horizontal component,

00:01:42.990 --> 00:01:46.770
that is going to be F cosine theta.

00:01:46.770 --> 00:01:48.990
This side right over here is adjacent

00:01:48.990 --> 00:01:51.860
to the angle, SOH CAH TOA.

00:01:51.860 --> 00:01:53.110
And now with that out of the way

00:01:53.110 --> 00:01:56.380
we can draw our free body diagram.

00:01:56.380 --> 00:01:58.990
So let me draw that free body,

00:01:58.990 --> 00:02:01.090
let me draw that block and I'm really

00:02:01.090 --> 00:02:03.480
just gonna focus on the block only.

00:02:03.480 --> 00:02:05.720
And we know a couple of
things that are going on.

00:02:05.720 --> 00:02:08.940
We know that we have
this horizontal force,

00:02:08.940 --> 00:02:11.293
F cosine theta, so let me draw that.

00:02:12.140 --> 00:02:16.670
So the magnitude there is F cosine theta.

00:02:16.670 --> 00:02:19.330
We know we have this vertical force,

00:02:19.330 --> 00:02:21.573
F sine theta, so let me draw that.

00:02:22.760 --> 00:02:26.310
So this would be F and that one

00:02:26.310 --> 00:02:28.220
is actually a little bit shorter,

00:02:28.220 --> 00:02:29.480
it's obviously not drawn perfectly

00:02:29.480 --> 00:02:34.480
to scale, but this would be F sine theta.

00:02:34.790 --> 00:02:36.690
Now let's think about these characters

00:02:36.690 --> 00:02:37.770
right over here.

00:02:37.770 --> 00:02:40.780
We have the force of gravity

00:02:41.770 --> 00:02:43.900
and so that's going to be acting downward.

00:02:43.900 --> 00:02:45.993
So it would look something like this.

00:02:47.260 --> 00:02:51.670
So we have, and it would
have magnitude F sub g.

00:02:51.670 --> 00:02:53.080
I'm not drawing the arrow now because

00:02:53.080 --> 00:02:55.110
I'm just talking about
the magnitude of this vector.

00:02:55.110 --> 00:02:57.340
Here I'm referring to the entire vector,

00:02:57.340 --> 00:03:01.010
I'm referring to its
magnitude and direction.

00:03:01.010 --> 00:03:03.080
Now what about the force of friction?

00:03:03.080 --> 00:03:04.920
Let's assume that we're in a situation

00:03:04.920 --> 00:03:07.970
where the magnitude of
the vertical component

00:03:07.970 --> 00:03:12.075
of this applied force, F sine theta

00:03:12.075 --> 00:03:15.080
is less than the magnitude

00:03:15.080 --> 00:03:17.110
of the force of gravity.

00:03:17.110 --> 00:03:20.320
Well in that situation
if there was no friction,

00:03:20.320 --> 00:03:22.330
the block would start
accelerating downward,

00:03:22.330 --> 00:03:24.060
because you would have
a net force downward.

00:03:24.060 --> 00:03:25.360
We haven't talked about the forces

00:03:25.360 --> 00:03:26.760
to the left and right yet.

00:03:26.760 --> 00:03:29.720
But as we mentioned,
this thing is stationary

00:03:29.720 --> 00:03:31.500
and the force of friction is going

00:03:31.500 --> 00:03:34.470
to act against the direction of motion.

00:03:34.470 --> 00:03:37.410
And so in this situation,
the force of friction

00:03:37.410 --> 00:03:39.930
will act upwards, so we would have a force

00:03:39.930 --> 00:03:41.760
of friction just like that and its

00:03:41.760 --> 00:03:46.520
magnitude would be F sub lowercase f.

00:03:46.520 --> 00:03:48.460
And then last but not least what about

00:03:48.460 --> 00:03:50.000
the normal force?

00:03:50.000 --> 00:03:51.990
Well if this block is not accelerating

00:03:51.990 --> 00:03:53.540
in any direction that means

00:03:53.540 --> 00:03:55.440
that the normal force must perfectly

00:03:55.440 --> 00:03:57.960
counteract this force to the right

00:03:57.960 --> 00:04:00.100
which is the horizontal component

00:04:00.100 --> 00:04:01.630
of this applied force.

00:04:01.630 --> 00:04:03.420
And so our normal force is going

00:04:03.420 --> 00:04:07.640
to go to the left and
it would look like this.

00:04:07.640 --> 00:04:11.890
So it's magnitude is F sub capital N.

00:04:11.890 --> 00:04:12.740
So there you have it,

00:04:12.740 --> 00:04:14.640
we have drawn a free body diagram

00:04:14.640 --> 00:04:16.660
for this scenario right over here.

00:04:16.660 --> 00:04:19.210
If these two were equal, then you would

00:04:19.210 --> 00:04:20.470
have no force of friction

00:04:20.470 --> 00:04:22.470
or the force of friction would be zero.

00:04:22.470 --> 00:04:24.260
There would be nothing for
it to be counteracting.

00:04:24.260 --> 00:04:26.290
These two would perfectly cancel out.

00:04:26.290 --> 00:04:29.510
And if the magnitude of
the vertical component

00:04:29.510 --> 00:04:30.980
of the applied force were greater

00:04:30.980 --> 00:04:33.040
than the force of
gravity without friction,

00:04:33.040 --> 00:04:34.990
it would start to accelerate upwards

00:04:34.990 --> 00:04:36.370
and so the force of friction would

00:04:36.370 --> 00:04:39.510
act against that motion,

00:04:39.510 --> 00:04:42.250
but let's just go with this
scenario right over here

00:04:42.250 --> 00:04:43.191
and start to appreciate why

00:04:43.191 --> 00:04:45.880
free body diagrams are so useful.

00:04:45.880 --> 00:04:48.670
Because we can start to set up equations

00:04:48.670 --> 00:04:50.530
that relate these magnitudes.

00:04:50.530 --> 00:04:53.500
We can say look if this
box is not accelerating

00:04:53.500 --> 00:04:55.960
in any way, there's zero net forces

00:04:55.960 --> 00:04:58.180
in the horizontal and
the vertical directions,

00:04:58.180 --> 00:05:00.270
well then we could say that F sub N

00:05:00.270 --> 00:05:02.320
completely counteracts
F times cosine of theta.

00:05:02.320 --> 00:05:07.050
So we could say F sub N is equal to,

00:05:07.050 --> 00:05:11.090
is equal to F cosine theta,

00:05:11.090 --> 00:05:15.620
and we could also say that F sine theta,

00:05:15.620 --> 00:05:19.350
F sine theta plus the magnitude

00:05:19.350 --> 00:05:20.910
of the force of friction.

00:05:20.910 --> 00:05:22.880
Plus the magnitude of
the force of friction

00:05:22.880 --> 00:05:24.900
that these would completely counteract

00:05:24.900 --> 00:05:27.960
the magnitude of the force of gravity,

00:05:27.960 --> 00:05:30.270
because it's going in
the opposite direction.

00:05:30.270 --> 00:05:34.340
So these would be equal to F sub g.

00:05:34.340 --> 00:05:36.840
So once you set up equations like this,

00:05:36.840 --> 00:05:38.860
if you know all but
one of these variables,

00:05:38.860 --> 00:05:40.660
you can figure out the other ones

00:05:40.660 --> 00:05:42.713
which is very useful in physics.


Open primaries, closed primaries, and blanket primaries

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.070
- [Presenter] Let's talk
about primary elections,

00:00:03.070 --> 00:00:06.060
which are often known as primaries.

00:00:06.060 --> 00:00:07.700
One way to think about them is that

00:00:07.700 --> 00:00:09.610
they're just preliminary elections

00:00:09.610 --> 00:00:13.290
used to get down to a
fewer number of candidates.

00:00:13.290 --> 00:00:15.190
A very simple example would be

00:00:15.190 --> 00:00:18.940
let's say there's a congressional
seat in your district,

00:00:18.940 --> 00:00:22.920
and there's three folks who
are interested in that position

00:00:22.920 --> 00:00:25.560
who associate themselves
with the Democrats.

00:00:25.560 --> 00:00:28.920
And let's say there's also
three folks in your district

00:00:28.920 --> 00:00:31.130
who are interested in
running for that position

00:00:31.130 --> 00:00:33.870
who associate themselves
with the Republicans.

00:00:33.870 --> 00:00:35.740
There are some situations
where you can have

00:00:35.740 --> 00:00:37.820
just a big preliminary election

00:00:37.820 --> 00:00:39.310
where they all run against each other.

00:00:39.310 --> 00:00:40.940
We'll talk about that in a second.

00:00:40.940 --> 00:00:43.340
But, usually you have a situation

00:00:43.340 --> 00:00:44.900
where they try to think about, well,

00:00:44.900 --> 00:00:47.050
let's just pick one to
represent the Democrats,

00:00:47.050 --> 00:00:49.630
and let's just pick one to
represent the Republicans.

00:00:49.630 --> 00:00:52.760
And so that's where the
primary election would happen.

00:00:52.760 --> 00:00:56.250
So you would have a Democratic primary

00:00:56.250 --> 00:00:58.300
where these folks would
run against each other,

00:00:58.300 --> 00:01:01.660
and whoever gets the most
votes would then go on

00:01:01.660 --> 00:01:05.970
to represent the Democratic
Party in that election.

00:01:05.970 --> 00:01:07.770
So let's say that person wins,

00:01:07.770 --> 00:01:11.170
then they will go on into
the general election.

00:01:11.170 --> 00:01:14.460
Likewise, these three folks
would run against each other

00:01:14.460 --> 00:01:17.170
in the Republican primary,

00:01:17.170 --> 00:01:19.850
and whoever gets the most votes,

00:01:19.850 --> 00:01:22.330
let's say it's that
person right over there,

00:01:22.330 --> 00:01:24.930
they would go to the general election

00:01:24.930 --> 00:01:28.430
to go against likely the
Democrat right over here.

00:01:28.430 --> 00:01:31.840
So this is the person who
would go against the Democrat.

00:01:31.840 --> 00:01:36.480
And then they are going to
compete in the general election

00:01:36.480 --> 00:01:38.280
against each other, head to head,

00:01:38.280 --> 00:01:41.170
and then whoever gets that would
get the congressional seat.

00:01:41.170 --> 00:01:43.670
Now even in this type of primary,

00:01:43.670 --> 00:01:45.980
you could call this a partisan primary,

00:01:45.980 --> 00:01:47.600
there's an interesting question.

00:01:47.600 --> 00:01:52.150
Who votes in this
Democratic Party primary?

00:01:52.150 --> 00:01:53.770
Who votes in that?

00:01:53.770 --> 00:01:57.647
And, who votes in the
Republican Party primary?

00:01:57.647 --> 00:02:00.030
And there's actually two systems for this,

00:02:00.030 --> 00:02:02.310
or two general buckets,
and there's some nuances

00:02:02.310 --> 00:02:04.350
from region to region.

00:02:04.350 --> 00:02:07.700
But one way is that only
registered Democrats

00:02:07.700 --> 00:02:09.350
could vote in the Democratic primary,

00:02:09.350 --> 00:02:11.570
pick who represents their
party in the general,

00:02:11.570 --> 00:02:13.650
and only registered Republicans

00:02:13.650 --> 00:02:16.090
can vote in the Republican primary.

00:02:16.090 --> 00:02:18.750
That situation where only registered folks

00:02:18.750 --> 00:02:21.100
can vote in their respective primaries,

00:02:21.100 --> 00:02:23.640
that is called a closed primary.

00:02:23.640 --> 00:02:25.890
It's called closed because it's not open

00:02:25.890 --> 00:02:28.210
to just anyone who wants
to vote in the primary.

00:02:28.210 --> 00:02:29.810
Only the people who have already

00:02:29.810 --> 00:02:32.190
registered in that party can vote.

00:02:32.190 --> 00:02:34.040
Now you can imagine the opposite of that

00:02:34.040 --> 00:02:39.040
is the open primary, so open primary,

00:02:39.180 --> 00:02:41.660
where even though these
are partisan primaries,

00:02:41.660 --> 00:02:42.610
you are trying to figure out

00:02:42.610 --> 00:02:44.400
who's going to represent the Democrats,

00:02:44.400 --> 00:02:46.780
and who's going to
represent the Republicans,

00:02:46.780 --> 00:02:48.960
you can allow people who are either

00:02:48.960 --> 00:02:50.220
not affiliated with a party

00:02:50.220 --> 00:02:52.400
or even people who are
affiliated with the other party

00:02:52.400 --> 00:02:54.840
to vote in your primary.

00:02:54.840 --> 00:02:57.370
So, for example, in
this Democratic primary,

00:02:57.370 --> 00:02:58.660
there could be, for some reason,

00:02:58.660 --> 00:03:01.290
folks who feel passionate
about this candidate

00:03:01.290 --> 00:03:03.190
who are actually registered Republicans

00:03:03.190 --> 00:03:05.180
but who they wanna vote in this primary.

00:03:05.180 --> 00:03:06.680
Or you could have people
who are unaffiliated

00:03:06.680 --> 00:03:08.270
who would vote in this primary

00:03:08.270 --> 00:03:11.100
if we are dealing with an open primary.

00:03:11.100 --> 00:03:12.830
Now one interesting question is,

00:03:12.830 --> 00:03:15.430
is how could the dynamics change

00:03:15.430 --> 00:03:19.413
if we're dealing with a open
versus a closed primary?

00:03:20.450 --> 00:03:23.270
Well, political scientists
like to think a lot about this,

00:03:23.270 --> 00:03:25.010
and even states like to think about this,

00:03:25.010 --> 00:03:26.460
in terms of what is appropriate?

00:03:26.460 --> 00:03:28.040
What is most representative?

00:03:28.040 --> 00:03:30.570
What gives the people of that state

00:03:30.570 --> 00:03:33.730
the fairest voice in their elections?

00:03:33.730 --> 00:03:36.660
Now people who like closed
primaries might say, well look,

00:03:36.660 --> 00:03:37.970
we're trying to figure out who represents

00:03:37.970 --> 00:03:39.820
the Democratic Party; it
should only be Democrats.

00:03:39.820 --> 00:03:40.850
We're trying to figure out who represents

00:03:40.850 --> 00:03:43.960
the Republican Party; it
should only be Republicans.

00:03:43.960 --> 00:03:46.150
Now criticism of closed
primaries might be,

00:03:46.150 --> 00:03:49.330
well, if you limit who gets
to vote in the primary,

00:03:49.330 --> 00:03:53.390
then these candidates are only
going to cater to the base.

00:03:53.390 --> 00:03:55.170
They're only going to
cater to the interests

00:03:55.170 --> 00:03:57.270
of people in their
party, and in particular,

00:03:57.270 --> 00:03:59.330
they might cater to people who are

00:03:59.330 --> 00:04:01.520
at the extremes of their party.

00:04:01.520 --> 00:04:05.900
For example, let's say we
have a situation like this.

00:04:05.900 --> 00:04:09.290
Let's say that this is the
population right over here,

00:04:09.290 --> 00:04:13.200
and people at the right end,
these are conservatives.

00:04:13.200 --> 00:04:14.113
Conservatives.

00:04:15.000 --> 00:04:16.253
Conservative.

00:04:17.110 --> 00:04:20.890
And these folks right
over here are liberal.

00:04:20.890 --> 00:04:23.650
And if you were to look at
the population as a whole,

00:04:23.650 --> 00:04:26.010
let's say the distribution
in the population.

00:04:26.010 --> 00:04:27.360
Let's say it looks something like this.

00:04:27.360 --> 00:04:31.460
You have some people here
who are quite liberal,

00:04:31.460 --> 00:04:34.820
and then you have actually
the bulk of the people

00:04:34.820 --> 00:04:37.010
who are some place in between,

00:04:37.010 --> 00:04:39.550
and then you have another bump of folks

00:04:39.550 --> 00:04:42.520
who are actually quite conservative.

00:04:42.520 --> 00:04:44.140
So you would guess that, hey, look,

00:04:44.140 --> 00:04:47.190
a view that's someplace in
between, maybe a moderate view,

00:04:47.190 --> 00:04:50.210
would actually maybe be most
representative of the people.

00:04:50.210 --> 00:04:53.410
But if you have closed
primaries, what might happen?

00:04:53.410 --> 00:04:54.700
Well, the Democratic primary,

00:04:54.700 --> 00:04:56.680
the people who would be eligible to vote,

00:04:56.680 --> 00:04:59.828
would be these people right over here.

00:04:59.828 --> 00:05:01.220
It would be these people, the people

00:05:01.220 --> 00:05:03.690
who have registered as Democrats.

00:05:03.690 --> 00:05:07.990
And in the Republican
primary, well, if it's closed,

00:05:07.990 --> 00:05:09.870
only the registered
Republicans would vote,

00:05:09.870 --> 00:05:11.860
maybe these people right over here.

00:05:11.860 --> 00:05:12.870
And these people in the middle,

00:05:12.870 --> 00:05:17.140
these would be the
unaffiliated-with-a-party people.

00:05:17.140 --> 00:05:20.330
And so, if you have
different candidates here,

00:05:20.330 --> 00:05:22.900
let's say that this is candidate one,

00:05:22.900 --> 00:05:24.470
where they're here on the spectrum.

00:05:24.470 --> 00:05:26.990
This is candidate two, which
is here on the spectrum,

00:05:26.990 --> 00:05:29.570
and candidate three is
here on the spectrum.

00:05:29.570 --> 00:05:31.960
You could imagine that
for this population,

00:05:31.960 --> 00:05:36.410
they're all going to say things
that speak to this group.

00:05:36.410 --> 00:05:38.710
And in particular, they might actually,

00:05:38.710 --> 00:05:41.520
the person who wins, might
actually be very successful

00:05:41.520 --> 00:05:45.270
of getting the people at the
extreme left right over here.

00:05:45.270 --> 00:05:47.640
And so that person maybe is a person

00:05:47.640 --> 00:05:50.010
who goes off to the general election.

00:05:50.010 --> 00:05:52.310
Likewise, the same
thing might be happening

00:05:52.310 --> 00:05:54.110
on the Republican side.

00:05:54.110 --> 00:05:55.290
You have the different candidates.

00:05:55.290 --> 00:05:58.090
Maybe their personal views
put 'em right over here.

00:05:58.090 --> 00:06:01.140
And it's this person who
is furthest to the right

00:06:01.140 --> 00:06:05.630
who's really able to appeal
to the folks right over here

00:06:05.630 --> 00:06:09.600
who ends up winning and
represents the Republican Party.

00:06:09.600 --> 00:06:12.070
Well then what happens
in the general election?

00:06:12.070 --> 00:06:15.290
The entire population
has to pick between folks

00:06:15.290 --> 00:06:19.540
that actually are more
indicative of either extreme.

00:06:19.540 --> 00:06:21.690
That maybe the closed primary system,

00:06:21.690 --> 00:06:23.580
it wouldn't have been
so good for a candidate

00:06:23.580 --> 00:06:25.950
who is reasonably moderate
on the Democratic side.

00:06:25.950 --> 00:06:27.250
Maybe someone is there,

00:06:27.250 --> 00:06:29.550
or a candidate who is right over here.

00:06:29.550 --> 00:06:30.670
Or maybe there's a candidate

00:06:30.670 --> 00:06:32.840
whose views are right in the middle.

00:06:32.840 --> 00:06:34.970
First of all, if they
want to represent a party,

00:06:34.970 --> 00:06:38.220
they would have to moot
pick one of these parties.

00:06:38.220 --> 00:06:41.160
And then to have a shot, they
would have to represent views

00:06:41.160 --> 00:06:43.180
that would be to the left or to the right

00:06:43.180 --> 00:06:45.170
of where they truly are,
and maybe they wouldn't have

00:06:45.170 --> 00:06:46.860
as good of a shot as the people

00:06:46.860 --> 00:06:48.810
who actually are quite liberal

00:06:48.810 --> 00:06:51.540
or the folks who are actually
are quite conservative.

00:06:51.540 --> 00:06:53.900
So that's a criticism
of the closed primary.

00:06:53.900 --> 00:06:56.720
An open primary might make
this a little bit better,

00:06:56.720 --> 00:06:59.560
because you have these
registered Democrats here.

00:06:59.560 --> 00:07:01.650
But folks from here might decide, hey,

00:07:01.650 --> 00:07:03.010
I wanna go vote there as well.

00:07:03.010 --> 00:07:04.550
So they might moderate things.

00:07:04.550 --> 00:07:06.040
Or folks from here might say, hey,

00:07:06.040 --> 00:07:08.260
I wanna go vote here as well.

00:07:08.260 --> 00:07:09.840
Now critics of an open primary would say,

00:07:09.840 --> 00:07:11.040
well those aren't the people who

00:07:11.040 --> 00:07:13.020
really are registered in that party.

00:07:13.020 --> 00:07:15.530
And you could also have
a phenomenon of rating,

00:07:15.530 --> 00:07:18.770
where people maybe on the Republican side

00:07:18.770 --> 00:07:20.440
could say, look, you know what?

00:07:20.440 --> 00:07:21.800
I know who's gonna win here.

00:07:21.800 --> 00:07:24.120
The Republican primary
doesn't need my vote.

00:07:24.120 --> 00:07:25.290
I'm gonna go and try to pick

00:07:25.290 --> 00:07:27.660
the weakest candidate on the other side

00:07:27.660 --> 00:07:29.230
so that in the general election,

00:07:29.230 --> 00:07:31.820
my candidate has the best shot.

00:07:31.820 --> 00:07:34.510
Now third type of primary is known

00:07:34.510 --> 00:07:37.630
as a blanket primary,

00:07:37.630 --> 00:07:41.040
often a nonpartisan blanket primary.

00:07:41.040 --> 00:07:43.270
In a nonpartisan blanket primary,

00:07:43.270 --> 00:07:47.610
instead of going through
this partisan process,

00:07:47.610 --> 00:07:49.630
all the candidates here would go

00:07:49.630 --> 00:07:52.380
into one preliminary election.

00:07:52.380 --> 00:07:54.250
And in that preliminary election,

00:07:54.250 --> 00:07:56.980
the top two candidates would then

00:07:56.980 --> 00:07:58.543
go on to the general election.

00:08:00.690 --> 00:08:03.710
So two candidates are going to move on

00:08:03.710 --> 00:08:07.040
in a blanket primary, a
nonpartisan blanket primary,

00:08:07.040 --> 00:08:09.610
and what's interesting there
is you can actually get

00:08:09.610 --> 00:08:12.350
two candidates from the same party going.

00:08:12.350 --> 00:08:14.240
Proponents of a blanket primary say, hey,

00:08:14.240 --> 00:08:15.500
this is the fairest of it all.

00:08:15.500 --> 00:08:17.090
Instead of making things partisan,

00:08:17.090 --> 00:08:19.470
just let everyone run
against everyone else,

00:08:19.470 --> 00:08:23.430
and the top two will then
get to the general election.

00:08:23.430 --> 00:08:25.840
Now you'll see all three of
these in the United States

00:08:25.840 --> 00:08:28.830
at congressional elections
or even statewide elections.

00:08:28.830 --> 00:08:31.590
But if we're thinking about
presidential elections,

00:08:31.590 --> 00:08:33.740
we're thinking mainly about these two.

00:08:33.740 --> 00:08:35.450
And when we're talking
about presidential election,

00:08:35.450 --> 00:08:37.440
these are part of primary season

00:08:37.440 --> 00:08:40.860
where it's thinking
about how many delegates

00:08:40.860 --> 00:08:44.330
candidates would get during
the national convention

00:08:44.330 --> 00:08:46.900
where they actually pick their candidate.

00:08:46.900 --> 00:08:48.260
I'll leave you there.

00:08:48.260 --> 00:08:49.720
It's an interesting way to think

00:08:49.720 --> 00:08:51.590
about the different ways of voting,

00:08:51.590 --> 00:08:54.226
especially voting in a preliminary way

00:08:54.226 --> 00:08:56.500
to get down to a fewer
number of candidates.

00:08:56.500 --> 00:08:58.943
I'll let you think about
which one you prefer.


The presidential incumbency advantage

WEBVTT
Kind: captions
Language: en

00:00:00.561 --> 00:00:03.874
- What we're going to do
in this video is talk about

00:00:03.874 --> 00:00:05.874
the incumbent advantage.

00:00:06.844 --> 00:00:09.043
And, this is the idea
that the person who is

00:00:09.043 --> 00:00:11.411
already in power, the incumbent,

00:00:11.411 --> 00:00:13.598
has an advantage in elections.

00:00:13.598 --> 00:00:15.589
And, in particular,
we're going to focus on

00:00:15.589 --> 00:00:19.113
Presidential elections,
although this idea is pretty

00:00:19.113 --> 00:00:22.271
common across different offices.

00:00:22.271 --> 00:00:25.079
Now, it turns out that if
you go back over, roughly,

00:00:25.079 --> 00:00:28.621
the last 100 years, in
Presidential elections that had

00:00:28.621 --> 00:00:31.682
and incumbent, that had a
sitting President running

00:00:31.682 --> 00:00:34.895
for a second term or maybe
a third, or a forth term

00:00:34.895 --> 00:00:39.603
in the case of FDR, you have
more than 80% of the time,

00:00:39.603 --> 00:00:42.267
it goes to the incumbent.

00:00:42.267 --> 00:00:44.016
And this right over
here is a picture of the

00:00:44.016 --> 00:00:48.020
Presidential Debates from
C-SPAN and this in 2012

00:00:48.020 --> 00:00:51.141
between President Barack
Obama and Mitt Romney.

00:00:51.141 --> 00:00:53.952
And President Obama, of
course, was the incumbent

00:00:53.952 --> 00:00:55.803
and he went on to win.

00:00:55.803 --> 00:01:00.045
And, once again, not unusual,
this happened 80% of the time.

00:01:00.045 --> 00:01:03.424
The last time that a challenger
was able to unseat an

00:01:03.424 --> 00:01:08.174
incumbent was in 1992 when
you have President Clinton was

00:01:09.665 --> 00:01:13.550
able to beat President George H. W. Bush,

00:01:13.550 --> 00:01:15.286
George W. Bush's father.

00:01:15.286 --> 00:01:18.619
George H. W. Bush and the time before ...

00:01:18.619 --> 00:01:21.818
and this is 26 years ago,
26 years before the time

00:01:21.818 --> 00:01:23.654
of the making of this video.

00:01:23.654 --> 00:01:26.544
And, to go even before that,
the last time before 1992

00:01:26.544 --> 00:01:30.791
would have been 1980, when
Reagan is able to beat Carter.

00:01:30.791 --> 00:01:33.756
So, this is a very unusual
thing for a challenger

00:01:33.756 --> 00:01:35.913
to be able to beat an incumbent.

00:01:35.913 --> 00:01:37.670
Now one thing that I would like you to do

00:01:37.670 --> 00:01:38.972
is pause this video.

00:01:38.972 --> 00:01:43.972
Think about why incumbents
might have such an advantage.

00:01:44.190 --> 00:01:47.553
Okay, so there's many
scholarly papers written about

00:01:47.553 --> 00:01:50.184
the idea of the incumbent
advantage, both at the

00:01:50.184 --> 00:01:52.449
Presidential level and at other levels.

00:01:52.449 --> 00:01:55.192
But we can come up with
some ideas as to why

00:01:55.192 --> 00:01:57.288
the incumbent has the advantage.

00:01:57.288 --> 00:02:00.893
One is people already imagine
that individual as President

00:02:00.893 --> 00:02:03.642
because they are the President.

00:02:03.642 --> 00:02:07.307
On top of that, because
they are already President,

00:02:07.307 --> 00:02:10.689
they've had many years of being
in the national spotlight.

00:02:10.689 --> 00:02:12.729
And even during the
Presidential campaign, they are

00:02:12.729 --> 00:02:15.373
still the President and so
they have that bully pulpit

00:02:15.373 --> 00:02:19.153
that President Roosevelt,
Theodore Roosevelt talked about.

00:02:19.153 --> 00:02:23.016
Bully pulpit, where they
can get people's attention

00:02:23.016 --> 00:02:24.554
when they want to.

00:02:24.554 --> 00:02:26.763
Even though the challenger
might be well-funded, and

00:02:26.763 --> 00:02:28.493
the press is definitely
going to pay attention to the

00:02:28.493 --> 00:02:30.311
challenger and give them a lot of voice.

00:02:30.311 --> 00:02:33.601
Because the President is
the President, they have the

00:02:33.601 --> 00:02:36.390
State of the Union Address,
they can call Special Sessions

00:02:36.390 --> 00:02:39.125
of Congress, they have
regular press briefings,

00:02:39.125 --> 00:02:43.231
and so they have a much
larger voice even compared to

00:02:43.231 --> 00:02:45.564
a very prominent challenger.

00:02:46.517 --> 00:02:50.192
On top of that, there might
be some notion of experience

00:02:50.192 --> 00:02:53.223
and this idea of experience
can go in many directions.

00:02:53.223 --> 00:02:56.630
No matter which candidate
has more life experience,

00:02:56.630 --> 00:03:01.014
which one is older, one could
argue that no one has ever

00:03:01.014 --> 00:03:02.647
really ever been the President

00:03:02.647 --> 00:03:04.330
unless they are the President.

00:03:04.330 --> 00:03:07.599
And, so someone who has been
in that job for four years

00:03:07.599 --> 00:03:10.345
has been able to build
some experiences that maybe

00:03:10.345 --> 00:03:13.095
they'll be able to build
on for their second term.

00:03:13.095 --> 00:03:15.219
While, the other person might have to,

00:03:15.219 --> 00:03:17.124
no matter what they've done
in the rest of their life,

00:03:17.124 --> 00:03:20.154
being the President is
fairly new experience.

00:03:20.154 --> 00:03:23.661
The other idea is that they
are experienced campaigners.

00:03:23.661 --> 00:03:26.287
That they have already run
a successful Presidential

00:03:26.287 --> 00:03:29.825
campaign and this is
their second time around,

00:03:29.825 --> 00:03:32.647
which could help in winning that campaign.

00:03:32.647 --> 00:03:34.867
Another idea that might
help an incumbent...

00:03:34.867 --> 00:03:37.297
and once again, this is
me thinking about it,

00:03:37.297 --> 00:03:38.921
I encourage you to think
about it and see if you

00:03:38.921 --> 00:03:40.796
buy these or could come up with others...

00:03:40.796 --> 00:03:43.833
is maybe there is a sense of patriotism.

00:03:43.833 --> 00:03:46.558
That people are used to
thinking of the incumbent as

00:03:46.558 --> 00:03:48.954
their President for the last several years

00:03:48.954 --> 00:03:51.883
and so they feel some
sense of loyalty to them.

00:03:51.883 --> 00:03:54.248
This could even be
supercharged if maybe it's in

00:03:54.248 --> 00:03:57.734
wartime or if the country
is facing some type of

00:03:57.734 --> 00:04:00.371
foreign policy crisis that
they want to stand behind

00:04:00.371 --> 00:04:03.249
the person during that time of crisis.

00:04:03.249 --> 00:04:04.555
They don't want to change
who is in charge in

00:04:04.555 --> 00:04:07.362
the middle of a war or
something like that.

00:04:07.362 --> 00:04:08.940
So, I will leave you there.

00:04:08.940 --> 00:04:12.723
These are just some ideas
why we see this phenomenon

00:04:12.723 --> 00:04:14.949
of an incumbent advantage.

00:04:14.949 --> 00:04:18.364
I encourage you to think
about why this is happening.


Example: Intersection of sine and cosine

WEBVTT
Kind: captions
Language: en

00:00:01.010 --> 00:00:02.370
- [Instructor] We're
asked at how many points

00:00:02.370 --> 00:00:04.590
do the graph of y equals sine of theta

00:00:04.590 --> 00:00:06.590
and y equal cosine theta

00:00:06.590 --> 00:00:09.640
intersect for theta
between zero and two pi?

00:00:09.640 --> 00:00:11.780
And it's zero is less
than or equal to theta

00:00:11.780 --> 00:00:13.280
which is less than or equal to two pi,

00:00:13.280 --> 00:00:15.710
so we're gonna include zero and two pi

00:00:15.710 --> 00:00:19.200
in the possible values for theta.

00:00:19.200 --> 00:00:20.380
So to do this,

00:00:20.380 --> 00:00:23.490
I've set up a little chart
for theta, cosine theta,

00:00:23.490 --> 00:00:24.380
and sine theta,

00:00:24.380 --> 00:00:26.630
and we can use this to,
and the unit circle,

00:00:26.630 --> 00:00:29.280
to hopefully quickly graph what the graphs

00:00:29.280 --> 00:00:32.460
of y equals sine theta and
y equals cosine theta are

00:00:32.460 --> 00:00:35.820
and then we can think about
how many times they intersect

00:00:35.820 --> 00:00:38.670
and maybe where they actually intersect.

00:00:38.670 --> 00:00:40.140
So let's get started.

00:00:40.140 --> 00:00:41.210
So first of all just to be clear,

00:00:41.210 --> 00:00:42.160
this is a unit circle.

00:00:42.160 --> 00:00:44.720
This is the x-axis, this is the y-axis.

00:00:44.720 --> 00:00:47.040
Over here we're going to
graph these two graphs.

00:00:47.040 --> 00:00:48.810
So this is going to be the y-axis,

00:00:48.810 --> 00:00:51.880
and it's going to be a
function of theta, not x,

00:00:51.880 --> 00:00:53.900
on the horizontal axis.

00:00:53.900 --> 00:00:56.090
So first let's think about what happens

00:00:56.090 --> 00:00:58.830
when theta is equal to zero.

00:00:58.830 --> 00:01:01.060
So when theta is equal to zero,

00:01:01.060 --> 00:01:02.820
you're at this point right over here,

00:01:02.820 --> 00:01:03.720
let me do it in a different color,

00:01:03.720 --> 00:01:06.810
you're at this point right
over here on the unit circle,

00:01:06.810 --> 00:01:09.530
and what coordinate is that?

00:01:09.530 --> 00:01:11.260
Well that's the point

00:01:11.260 --> 00:01:14.320
one comma zero.

00:01:14.320 --> 00:01:17.010
And so based on that,
what is cosine of theta

00:01:17.010 --> 00:01:19.440
when theta is equal to zero?

00:01:19.440 --> 00:01:21.730
Well cosine of theta is one

00:01:21.730 --> 00:01:24.740
and sine of theta is going to be zero.

00:01:24.740 --> 00:01:26.840
This is the x-axis at
the point of intersection

00:01:26.840 --> 00:01:28.690
with the unit circle, this is sorry,

00:01:28.690 --> 00:01:31.430
this is the x-coordinate at
the point of intersection

00:01:31.430 --> 00:01:33.610
with the unit circle,
this is the y-coordinate.

00:01:33.610 --> 00:01:34.880
Let's keep going.

00:01:34.880 --> 00:01:36.180
What about pi over two?

00:01:36.180 --> 00:01:37.740
So pi over two,

00:01:37.740 --> 00:01:39.640
we're right over here.

00:01:39.640 --> 00:01:41.053
What is that coordinate?

00:01:42.090 --> 00:01:45.750
Well that's now x is zero, y is one.

00:01:45.750 --> 00:01:49.110
So based on that, cosine of theta is zero,

00:01:49.110 --> 00:01:51.560
and what is sine of theta?

00:01:51.560 --> 00:01:53.060
Well that's going to be one.

00:01:53.060 --> 00:01:55.230
It's the y-coordinate right over here.

00:01:55.230 --> 00:01:56.920
Now let's go all the way to pi.

00:01:56.920 --> 00:01:58.800
Now let's go to all the way to pi.

00:01:58.800 --> 00:02:00.790
We're at this point in the unit circle.

00:02:00.790 --> 00:02:02.013
What is the coordinate?

00:02:02.920 --> 00:02:05.940
Well this is negative one comma zero,

00:02:05.940 --> 00:02:08.290
so what is cosine of theta?

00:02:08.290 --> 00:02:11.110
Well it's the x-coordinate
here, which is negative one.

00:02:11.110 --> 00:02:13.830
And sine of theta's going
to be the y-coordinate,

00:02:13.830 --> 00:02:16.170
which is zero.

00:02:16.170 --> 00:02:17.660
Now let's keep going.

00:02:17.660 --> 00:02:20.800
Now we're down here at three pi over two.

00:02:20.800 --> 00:02:23.890
If we go all the way around
to three pi over two,

00:02:23.890 --> 00:02:26.440
what is this coordinate?

00:02:26.440 --> 00:02:29.400
Well this is zero negative one.

00:02:29.400 --> 00:02:32.090
Cosine of theta is the x-coordinate here,

00:02:32.090 --> 00:02:35.230
so cosine of theta is going to be zero,

00:02:35.230 --> 00:02:37.950
and what is sine of theta going to be?

00:02:37.950 --> 00:02:40.830
Well it's going to negative one.

00:02:40.830 --> 00:02:43.150
And then finally we go back to two pi,

00:02:43.150 --> 00:02:45.510
which is making a full
revolution around the circle.

00:02:45.510 --> 00:02:47.170
We went all the way around,

00:02:47.170 --> 00:02:49.390
and we're back to this
point right over here.

00:02:49.390 --> 00:02:51.410
So the coordinate is the exact same thing

00:02:51.410 --> 00:02:54.063
as when the angle equaled radio,

00:02:54.063 --> 00:02:56.230
(laughing) equals zero radians,

00:02:56.230 --> 00:02:58.470
and so what is cosine of theta?

00:02:58.470 --> 00:02:59.660
Well that's one,

00:02:59.660 --> 00:03:03.120
and sine of theta is zero.

00:03:03.120 --> 00:03:05.380
And from this we can make
a rough sketch of the graph

00:03:05.380 --> 00:03:07.490
and think about where
they might intersect.

00:03:07.490 --> 00:03:09.750
So first let's do cosine of theta.

00:03:09.750 --> 00:03:11.310
When theta is zero,

00:03:11.310 --> 00:03:13.450
and let me draw, let me mark this off,

00:03:13.450 --> 00:03:15.957
so this is going to be
when y is equal to one,

00:03:15.957 --> 00:03:20.060
and this is when y is
equal to negative one.

00:03:20.060 --> 00:03:23.010
So y equals cosine of theta,

00:03:23.010 --> 00:03:24.747
I'm gonna graph, let's
see theta equals zero,

00:03:24.747 --> 00:03:26.750
cosine of theta equals one.

00:03:26.750 --> 00:03:29.200
So cosine of theta is equal to one.

00:03:29.200 --> 00:03:31.180
When theta's equal to pi over two,

00:03:31.180 --> 00:03:33.670
cosine of theta is zero.

00:03:33.670 --> 00:03:35.742
When theta's equal to pi,

00:03:35.742 --> 00:03:39.040
cosine of theta is negative one.

00:03:39.040 --> 00:03:41.940
When theta is equal to three pi over two,

00:03:41.940 --> 00:03:43.910
cosine of theta is equal to zero.

00:03:43.910 --> 00:03:45.200
That's this right over here.

00:03:45.200 --> 00:03:47.051
And then finally when theta's two pi,

00:03:47.051 --> 00:03:49.680
cosine of theta is one again,

00:03:49.680 --> 00:03:51.670
is one again.

00:03:51.670 --> 00:03:54.070
And the curve will look something,

00:03:54.070 --> 00:03:55.980
will look something like this,

00:03:55.980 --> 00:03:58.190
my best attempt to draw it.

00:03:58.190 --> 00:04:01.080
Make it a nice smooth curve.

00:04:01.080 --> 00:04:03.130
So it's going to look something,

00:04:03.130 --> 00:04:05.587
something, something like this.

00:04:05.587 --> 00:04:07.760
The look of these curves
should look somewhat familiar

00:04:07.760 --> 00:04:09.980
at this point, so it should
look something like this.

00:04:09.980 --> 00:04:14.580
So this is the graph of y
is equal to cosine of theta.

00:04:14.580 --> 00:04:16.970
Now let's do the same
thing for sine theta.

00:04:16.970 --> 00:04:18.160
When theta's equal to zero,

00:04:18.160 --> 00:04:21.080
sine theta is zero.

00:04:21.080 --> 00:04:24.803
When theta is pi over
two, sine of theta is one.

00:04:25.660 --> 00:04:27.910
When theta is equal to pi,

00:04:27.910 --> 00:04:30.630
sine of theta is zero.

00:04:30.630 --> 00:04:32.410
When theta's equal to three pi over two,

00:04:32.410 --> 00:04:34.530
sine of theta is negative one,

00:04:34.530 --> 00:04:36.210
is negative one.

00:04:36.210 --> 00:04:37.610
When theta's equal to two pi,

00:04:37.610 --> 00:04:40.690
sine of theta is equal to zero.

00:04:40.690 --> 00:04:43.638
And so the graph of sine of
theta's gonna look something,

00:04:43.638 --> 00:04:46.790
something like,

00:04:46.790 --> 00:04:50.540
something like this, my
best attempt at drawing it,

00:04:50.540 --> 00:04:53.790
is going to look something like this.

00:04:53.790 --> 00:04:54.860
So just visually,

00:04:54.860 --> 00:04:56.620
we can think about the question.

00:04:56.620 --> 00:04:59.590
At how many points do the
graphs of y equals sine of theta

00:04:59.590 --> 00:05:01.215
and y equal cosine of theta intersect

00:05:01.215 --> 00:05:03.260
for this range for theta,

00:05:03.260 --> 00:05:06.570
for theta being between zero and two pi,

00:05:06.570 --> 00:05:08.493
including those two points?

00:05:09.840 --> 00:05:11.190
Well you just look at this graph,

00:05:11.190 --> 00:05:12.920
you see there's two
points of intersection,

00:05:12.920 --> 00:05:15.681
this point right over here and this point

00:05:15.681 --> 00:05:17.830
right over here, just over the,

00:05:17.830 --> 00:05:19.480
between zero and two pi.

00:05:19.480 --> 00:05:20.580
These are cyclical graphs.

00:05:20.580 --> 00:05:22.270
If we kept going, they
would keep intersecting

00:05:22.270 --> 00:05:24.530
with each other, but
just over this two pi,

00:05:24.530 --> 00:05:27.370
just over this two pi range for theta,

00:05:27.370 --> 00:05:31.290
you would get two points of intersection.

00:05:31.290 --> 00:05:32.270
Now let's think about what they are

00:05:32.270 --> 00:05:33.190
because they look,

00:05:33.190 --> 00:05:36.160
they look to be pretty close between,

00:05:36.160 --> 00:05:38.650
between right between zero and pi over two

00:05:38.650 --> 00:05:42.070
and right between pi
and three pi over two.

00:05:42.070 --> 00:05:43.900
So let's look at our unit
circle if we can figure out

00:05:43.900 --> 00:05:44.733
what those values are.

00:05:44.733 --> 00:05:48.850
Let's it looks like, it looks
like this is at pi over four.

00:05:48.850 --> 00:05:50.080
So let's verify that.

00:05:50.080 --> 00:05:54.130
So let's think about what these
values are at pi over four.

00:05:54.130 --> 00:05:56.490
So pi over four is that angle

00:05:56.490 --> 00:05:58.300
or that's the terminal side of it,

00:05:58.300 --> 00:05:59.460
so this is pi over four.

00:05:59.460 --> 00:06:01.130
Pi over four is the exact same thing

00:06:01.130 --> 00:06:03.990
as a 45 degree angle.

00:06:03.990 --> 00:06:06.350
So let's do pi over four right over here.

00:06:06.350 --> 00:06:08.540
so we have to figure
out what this point is,

00:06:08.540 --> 00:06:10.500
what the coordinates are.

00:06:10.500 --> 00:06:12.190
So let's make this,

00:06:12.190 --> 00:06:14.840
let's make this a right triangle.

00:06:14.840 --> 00:06:16.050
It's a right triangle.

00:06:16.050 --> 00:06:18.210
And so what do we know
about this right triangle?

00:06:18.210 --> 00:06:19.870
And I'm gonna draw it right over here

00:06:19.870 --> 00:06:20.703
to make it a little clearer.

00:06:20.703 --> 00:06:23.190
This is a very typical
type of right triangle

00:06:23.190 --> 00:06:27.110
so it's good to get some
familiarity with it.

00:06:27.110 --> 00:06:30.310
So let me draw my best attempt, all right.

00:06:30.310 --> 00:06:31.660
So we know it's a right triangle.

00:06:31.660 --> 00:06:34.210
We know that this is 45 degrees.

00:06:34.210 --> 00:06:36.600
What is the length of the hypotenuse?

00:06:36.600 --> 00:06:38.810
Well this is a unit
circle, has radius one,

00:06:38.810 --> 00:06:41.670
so the length of the
hypotenuse here is one.

00:06:41.670 --> 00:06:44.783
And what do we know about
this angle right over here?

00:06:45.780 --> 00:06:50.500
Well we know that it
too must be 45 degrees

00:06:50.500 --> 00:06:53.160
because all of these angles
have to add up to 180.

00:06:53.160 --> 00:06:55.620
And since these two angles are the same,

00:06:55.620 --> 00:06:58.850
we know that these two sides
are going to be the same.

00:06:58.850 --> 00:07:01.390
And then we could use
the Pythagorean theorem

00:07:01.390 --> 00:07:03.340
to think about the length of those sides.

00:07:03.340 --> 00:07:05.170
So using the Pythagorean theorem,

00:07:05.170 --> 00:07:07.110
knowing that these two sides are equal,

00:07:07.110 --> 00:07:09.573
what do we get for the
length of those sides?

00:07:10.690 --> 00:07:13.070
Well let's call these,
if this has length a,

00:07:13.070 --> 00:07:15.000
well then this also has length a,

00:07:15.000 --> 00:07:16.720
and we can use the Pythagorean theorem

00:07:16.720 --> 00:07:20.070
and we could say a squared plus a squared

00:07:20.070 --> 00:07:22.760
is equal to the hypotenuse
squared, is equal to one.

00:07:22.760 --> 00:07:26.090
Or two a squared is equal to one.

00:07:26.090 --> 00:07:27.953
a squared is equal to 1/2.

00:07:28.987 --> 00:07:30.790
Take the principle root of both sides,

00:07:30.790 --> 00:07:32.470
a is equal to the square root of 1/2,

00:07:32.470 --> 00:07:33.800
which is the square root
of one which is one,

00:07:33.800 --> 00:07:35.690
over the square root of two.

00:07:35.690 --> 00:07:37.800
We can rationalize the denominator here

00:07:37.800 --> 00:07:39.610
by multiplying by square root of two

00:07:39.610 --> 00:07:41.880
over square root of two,

00:07:41.880 --> 00:07:43.600
which gives us a is equal to,

00:07:43.600 --> 00:07:45.580
in the numerator square root of two,

00:07:45.580 --> 00:07:46.980
and in the denominator square root of two

00:07:46.980 --> 00:07:49.430
times square root of two is two.

00:07:49.430 --> 00:07:52.130
So this length is square
root of two over two

00:07:52.130 --> 00:07:54.020
and this length is the same thing.

00:07:54.020 --> 00:07:55.760
So this length right over here

00:07:55.760 --> 00:07:57.760
is square root of two over two

00:07:57.760 --> 00:07:59.620
and this height right over here is also

00:07:59.620 --> 00:08:01.840
square root of two over two.

00:08:01.840 --> 00:08:02.730
So based on that,

00:08:02.730 --> 00:08:05.940
what is this coordinate point?

00:08:05.940 --> 00:08:10.230
Well it's square root of
two over two to the right

00:08:10.230 --> 00:08:11.170
in the positive direction,

00:08:11.170 --> 00:08:13.570
so x is equal to square
root of two over two,

00:08:13.570 --> 00:08:16.910
and y is square root of two over two

00:08:16.910 --> 00:08:19.070
in the upwards direction,
in the vertical direction,

00:08:19.070 --> 00:08:20.100
the positive vertical direction,

00:08:20.100 --> 00:08:22.240
so it's also square root of two over two.

00:08:22.240 --> 00:08:24.350
Cosine of theta is just the,

00:08:24.350 --> 00:08:25.420
is just the x-coordinate,

00:08:25.420 --> 00:08:27.440
so it's square root of two over two.

00:08:27.440 --> 00:08:29.127
Sine of theta's just the y-coordinate.

00:08:29.127 --> 00:08:31.830
So you see immediately that they do,

00:08:31.830 --> 00:08:33.660
they are indeed equal at that point.

00:08:33.660 --> 00:08:35.990
So at this point they are both equal to

00:08:35.990 --> 00:08:37.550
square root of two,

00:08:37.550 --> 00:08:41.600
they're both equal to
square root of two over two.

00:08:41.600 --> 00:08:43.470
Now what about this point right over here

00:08:43.470 --> 00:08:48.470
which looks right in between
pi and three pi over two?

00:08:48.640 --> 00:08:49.530
So that's going to be,

00:08:49.530 --> 00:08:52.000
so this is pi, this is three pi over two,

00:08:52.000 --> 00:08:53.531
it is right,

00:08:53.531 --> 00:08:56.090
it is right over here.

00:08:56.090 --> 00:08:58.590
So it's another pi over four plus pi,

00:08:58.590 --> 00:09:01.290
so pi plus pi over four

00:09:02.378 --> 00:09:04.920
is the same thing as four pi
over four plus pi over four.

00:09:04.920 --> 00:09:08.270
So this is the angle five pi over four.

00:09:08.270 --> 00:09:11.390
So this is five pi over four.

00:09:11.390 --> 00:09:14.490
So this is equal to five pi over four.

00:09:14.490 --> 00:09:17.040
So that's what we're trying to figure out.

00:09:17.040 --> 00:09:18.750
What are the value of these functions

00:09:18.750 --> 00:09:22.490
at theta equal five pi over four?

00:09:22.490 --> 00:09:24.120
Well there's multiple
ways to think about it.

00:09:24.120 --> 00:09:26.060
You could even use a little
bit of geometry to say,

00:09:26.060 --> 00:09:28.270
well if this is a 45 degree angle,

00:09:28.270 --> 00:09:32.440
then this right over here
is also a 45 degree angle.

00:09:32.440 --> 00:09:34.090
You could say that the reference angle

00:09:34.090 --> 00:09:36.770
in terms of degrees is 45 degrees.

00:09:36.770 --> 00:09:38.330
And we could do a very similar thing.

00:09:38.330 --> 00:09:41.040
We can draw a right triangle.

00:09:41.040 --> 00:09:43.490
We know the hypotenuse is one.

00:09:43.490 --> 00:09:45.729
We know that if this is a right angle,

00:09:45.729 --> 00:09:47.190
this is 45 degrees.

00:09:47.190 --> 00:09:50.620
If that's 45 degrees then
this is also 45 degrees,

00:09:50.620 --> 00:09:54.080
and we have a triangle
that's very similar,

00:09:54.080 --> 00:09:56.550
it's actually, they're
actually congruent triangles.

00:09:56.550 --> 00:09:59.780
So hypotenuse is one, 45-45-90,

00:09:59.780 --> 00:10:01.850
we then know that the length of this side

00:10:01.850 --> 00:10:03.380
is square root of two over two

00:10:03.380 --> 00:10:05.230
and the length of this side is
square root of two over two,

00:10:05.230 --> 00:10:08.150
the exact same logic we used over here.

00:10:08.150 --> 00:10:09.900
So based on that,

00:10:09.900 --> 00:10:11.150
based on that,

00:10:11.150 --> 00:10:14.940
what is the coordinate of that point?

00:10:14.940 --> 00:10:17.720
Well let's think about the x value.

00:10:17.720 --> 00:10:21.260
It's square root of two over
two in the negative direction.

00:10:21.260 --> 00:10:22.690
We have to go square root of two over two

00:10:22.690 --> 00:10:25.200
to the left of the origin,

00:10:25.200 --> 00:10:27.900
so it's negative square
root of two over two.

00:10:27.900 --> 00:10:30.810
This point right over,
this point on the a-xis

00:10:30.810 --> 00:10:33.030
is negative square root of two over two.

00:10:33.030 --> 00:10:34.250
What about the y value?

00:10:34.250 --> 00:10:36.813
Well we had to go square
root of two over two down,

00:10:36.813 --> 00:10:39.930
in the downward direction from the origin,

00:10:39.930 --> 00:10:43.580
so it's also negative
square root of two over two.

00:10:43.580 --> 00:10:46.844
So sine of, cosine of theta's
negative square root of two

00:10:46.844 --> 00:10:49.210
over two and sine of theta's also negative

00:10:49.210 --> 00:10:51.480
square root of two over two.

00:10:51.480 --> 00:10:54.360
And so we see that we do
indeed have the same value

00:10:54.360 --> 00:10:57.290
for cosine of theta and
sine of theta right there.

00:10:57.290 --> 00:10:59.050
They're both equal to,

00:10:59.050 --> 00:11:01.290
they're both at that point equal to

00:11:01.290 --> 00:11:05.813
the negative, the negative
square root of two over two.


Finding the 100th term in a sequence

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:03.070
- [Instructor] We are
asked what is the value

00:00:03.070 --> 00:00:06.100
of the 100th term in this sequence,

00:00:06.100 --> 00:00:08.440
and the first term is 15, then nine,

00:00:08.440 --> 00:00:09.730
then three, then negative three.

00:00:09.730 --> 00:00:11.910
So let's write it like this in a table.

00:00:11.910 --> 00:00:13.670
So if we have the term,

00:00:13.670 --> 00:00:16.040
just so we have things straight,

00:00:16.040 --> 00:00:18.180
and then we have the value,

00:00:18.180 --> 00:00:20.890
and then we have the value of the term.

00:00:20.890 --> 00:00:23.360
I'll do a nice little table here.

00:00:23.360 --> 00:00:26.870
So our first term we saw is 15.

00:00:26.870 --> 00:00:30.830
Our second term is nine.

00:00:30.830 --> 00:00:33.440
Our third term is three.

00:00:33.440 --> 00:00:34.980
I'm just really copying this down,

00:00:34.980 --> 00:00:37.770
but I'm making sure we associate
it with the right term.

00:00:37.770 --> 00:00:39.890
And then our fourth term,

00:00:39.890 --> 00:00:42.600
our fourth term is negative three.

00:00:42.600 --> 00:00:43.570
And they wanna ask,

00:00:43.570 --> 00:00:46.820
they want us to figure
out what the 100th term

00:00:46.820 --> 00:00:47.800
of this sequence is going to be.

00:00:47.800 --> 00:00:49.120
So let's see what's happening here,

00:00:49.120 --> 00:00:52.010
if we can discern some type of pattern.

00:00:52.010 --> 00:00:53.940
So we went from the first
term to the second term,

00:00:53.940 --> 00:00:54.880
what happened?

00:00:54.880 --> 00:00:57.780
15 to nine, looks like
we went down by six.

00:00:57.780 --> 00:00:59.880
It's always good to
think about just how much

00:00:59.880 --> 00:01:01.010
the numbers changed by.

00:01:01.010 --> 00:01:03.020
That's always the
simplest type of pattern.

00:01:03.020 --> 00:01:06.150
So we went down by six, we subtracted six.

00:01:06.150 --> 00:01:07.920
Then to go from nine to three,

00:01:07.920 --> 00:01:09.994
well we subtracted six again.

00:01:09.994 --> 00:01:12.770
We subtracted six again.

00:01:12.770 --> 00:01:15.079
And then to go from
three to negative three,

00:01:15.079 --> 00:01:18.480
well we, we subtracted six again.

00:01:18.480 --> 00:01:21.840
We subtracted six again.

00:01:21.840 --> 00:01:24.920
So it looks like every term,

00:01:24.920 --> 00:01:26.660
you subtract six.

00:01:26.660 --> 00:01:29.280
So the second term is going to be six less

00:01:29.280 --> 00:01:30.113
than the first term.

00:01:30.113 --> 00:01:31.867
The third term is going to be

00:01:31.867 --> 00:01:34.940
12 minus from the first term,

00:01:34.940 --> 00:01:37.870
or six subtracted twice.

00:01:37.870 --> 00:01:38.830
So in the third term,

00:01:38.830 --> 00:01:41.160
you subtract a six twice.

00:01:41.160 --> 00:01:42.550
In the fourth term,

00:01:42.550 --> 00:01:45.850
you subtract six three times.

00:01:45.850 --> 00:01:48.100
So whatever term you're looking at,

00:01:48.100 --> 00:01:51.680
you subtract six one less
than that many times.

00:01:51.680 --> 00:01:53.600
Let me write this down just so,

00:01:53.600 --> 00:01:55.630
notice when your first term,

00:01:55.630 --> 00:01:58.310
you have 15 and you don't
subtract six at all,

00:01:58.310 --> 00:02:01.590
or you could say you
subtract six zero times.

00:02:01.590 --> 00:02:05.090
So you could say this is 15 minus

00:02:06.220 --> 00:02:09.240
six times or let me
write it better this way,

00:02:09.240 --> 00:02:12.950
minus zero times six.

00:02:12.950 --> 00:02:15.060
That's what that first
term is right there.

00:02:15.060 --> 00:02:16.620
What's the second term?

00:02:16.620 --> 00:02:18.440
This is 15,

00:02:18.440 --> 00:02:20.760
it's just we just subtracted six once,

00:02:20.760 --> 00:02:25.070
or you could say minus one times six.

00:02:25.070 --> 00:02:26.930
Or you could say plus
one times negative six,

00:02:26.930 --> 00:02:30.770
either way, we're
subtracting the six once.

00:02:30.770 --> 00:02:32.420
Now what's happening here?

00:02:32.420 --> 00:02:34.300
This is 15,

00:02:34.300 --> 00:02:36.080
this is 15

00:02:36.080 --> 00:02:38.490
minus two times negative six,

00:02:38.490 --> 00:02:41.364
or sorry minus two times six,

00:02:41.364 --> 00:02:43.960
minus two times six.

00:02:43.960 --> 00:02:46.470
We're subtracting a six twice.

00:02:46.470 --> 00:02:48.060
What's the fourth term?

00:02:48.060 --> 00:02:50.050
This is 15 minus,

00:02:50.050 --> 00:02:53.680
we're subtracting the six
three times from the 15,

00:02:53.680 --> 00:02:56.980
so minus three times six.

00:02:56.980 --> 00:02:59.520
So if you see the pattern here,

00:02:59.520 --> 00:03:02.230
when our term, when we
have our fourth term,

00:03:02.230 --> 00:03:05.030
we have the term minus one right there,

00:03:05.030 --> 00:03:06.560
the fourth term we have a three,

00:03:06.560 --> 00:03:08.741
the third term we have a two,

00:03:08.741 --> 00:03:11.900
the second term we have a one.

00:03:11.900 --> 00:03:14.340
So if we had the nth term,

00:03:14.340 --> 00:03:16.410
if we just had the nth term here,

00:03:16.410 --> 00:03:17.470
what's this going to be?

00:03:17.470 --> 00:03:18.970
It's going to be 15

00:03:19.840 --> 00:03:23.430
minus, you see it's going to
be n minus one right here,

00:03:23.430 --> 00:03:26.090
right when n is four,
n minus one is three.

00:03:26.090 --> 00:03:29.200
When n is three, n minus one is two.

00:03:29.200 --> 00:03:32.250
When n is two, n minus one is one.

00:03:32.250 --> 00:03:36.080
When n is one, n minus one is zero.

00:03:36.080 --> 00:03:37.300
So we're going to have,

00:03:37.300 --> 00:03:39.510
this term right here is n minus one,

00:03:39.510 --> 00:03:44.410
so minus n minus one times six.

00:03:44.410 --> 00:03:48.300
So if you wanna figure out the
100th term of this sequence,

00:03:48.300 --> 00:03:50.130
I didn't even have to write
it in this general term,

00:03:50.130 --> 00:03:51.530
you could just look at this pattern.

00:03:51.530 --> 00:03:53.620
It's going to be, and I'll do it in pink,

00:03:53.620 --> 00:03:55.300
the 100th term in our sequence,

00:03:55.300 --> 00:03:57.740
I'll continue our table
down, is gonna be what?

00:03:57.740 --> 00:04:00.680
It's going to be 15 minus

00:04:00.680 --> 00:04:04.570
100 minus one, which is 99,

00:04:04.570 --> 00:04:06.730
times six, right?

00:04:06.730 --> 00:04:08.050
I just followed the pattern.

00:04:08.050 --> 00:04:10.550
One, you had a zero here,
two, you had a one here,

00:04:10.550 --> 00:04:11.830
three, you had a two here,

00:04:11.830 --> 00:04:13.970
100, you're gonna have a 99 here.

00:04:13.970 --> 00:04:16.150
So let's just calculate what this is.

00:04:16.150 --> 00:04:18.150
What's 99 times six?

00:04:18.150 --> 00:04:21.470
So 99 times six,

00:04:21.470 --> 00:04:22.770
actually you could do this in your head.

00:04:22.770 --> 00:04:25.110
You could say that's going to be six less

00:04:25.110 --> 00:04:28.940
than 100 times six, which is 600,

00:04:28.940 --> 00:04:31.550
and six less is 594.

00:04:31.550 --> 00:04:32.810
But if you didn't wanna do it that way,

00:04:32.810 --> 00:04:34.400
you just do it the old-fashioned way.

00:04:34.400 --> 00:04:39.330
Six times nine is 54, carry the five.

00:04:39.330 --> 00:04:42.291
Nine times six or six times nine is 54.

00:04:42.291 --> 00:04:46.053
54 plus five is 594.

00:04:46.053 --> 00:04:47.940
So this right here

00:04:48.790 --> 00:04:51.660
is 594,

00:04:51.660 --> 00:04:53.320
and then to figure out what 15,

00:04:53.320 --> 00:04:54.940
so we wanna figure out,

00:04:54.940 --> 00:04:59.113
we wanna figure out what 15 minus 594 is,

00:05:00.170 --> 00:05:02.620
and this can sometimes be confusing,

00:05:02.620 --> 00:05:05.090
but the way I always
process this in my head

00:05:05.090 --> 00:05:07.540
is I say that this is the exact same thing

00:05:07.540 --> 00:05:09.270
as the negative

00:05:09.270 --> 00:05:14.000
of 594 minus 15.

00:05:14.000 --> 00:05:15.270
And if you don't believe me,

00:05:15.270 --> 00:05:17.010
distribute out this negative sign.

00:05:17.010 --> 00:05:20.670
Negative one times 594 is negative 594,

00:05:20.670 --> 00:05:24.010
negative one times
negative 15 is positive 15.

00:05:24.010 --> 00:05:25.990
So these two statements are equivalent.

00:05:25.990 --> 00:05:29.240
This is much easier for
my brain to understand.

00:05:29.240 --> 00:05:31.620
So what's 594 minus 15?

00:05:31.620 --> 00:05:33.230
You should do it, we
could do this in our head.

00:05:33.230 --> 00:05:37.900
594 minus 14 would be 580,

00:05:37.900 --> 00:05:42.900
and then 580 minus one more would be 579.

00:05:42.940 --> 00:05:45.920
So that right there is 579,

00:05:45.920 --> 00:05:49.000
and then we have this negative
sign sitting out there.

00:05:49.000 --> 00:05:51.250
So our the 100th term in our sequence

00:05:51.250 --> 00:05:53.963
will be negative 579.


Projectile motion graphs

WEBVTT
Kind: captions
Language: en

00:00:00.340 --> 00:00:01.730
- [Instructor] So in
each of these pictures

00:00:01.730 --> 00:00:02.920
we have a different scenario.

00:00:02.920 --> 00:00:06.840
We have someone standing at
the edge of a cliff on Earth,

00:00:06.840 --> 00:00:08.550
and in this first scenario,

00:00:08.550 --> 00:00:12.300
they are launching a
projectile up into the air.

00:00:12.300 --> 00:00:14.980
In this one they're just
throwing it straight out.

00:00:14.980 --> 00:00:17.640
They're not throwing it up or
down but just straight out.

00:00:17.640 --> 00:00:19.723
And here they're throwing the projectile

00:00:19.723 --> 00:00:22.140
at an angle downwards.

00:00:22.140 --> 00:00:23.680
And so what we're going
to do in this video

00:00:23.680 --> 00:00:27.080
is think about for each of
these initial velocity vectors,

00:00:27.080 --> 00:00:29.930
what would the acceleration versus time,

00:00:29.930 --> 00:00:31.530
the velocity versus time,

00:00:31.530 --> 00:00:34.930
and the position versus
time graphs look like

00:00:34.930 --> 00:00:38.840
in both the y and the x directions.

00:00:38.840 --> 00:00:40.780
So I encourage you to pause this video

00:00:40.780 --> 00:00:42.880
and think about it on your own
or even take out some paper

00:00:42.880 --> 00:00:45.930
and try to solve it
before I work through it.

00:00:45.930 --> 00:00:48.210
So let's first think about acceleration

00:00:48.210 --> 00:00:50.200
in the vertical dimension,

00:00:50.200 --> 00:00:53.020
acceleration in the y direction.

00:00:53.020 --> 00:00:54.550
We're assuming we're on
Earth and we're going

00:00:54.550 --> 00:00:55.890
to ignore air resistance.

00:00:55.890 --> 00:00:58.440
We can assume we're in some
type of a laboratory vacuum

00:00:58.440 --> 00:01:00.350
and this person had maybe
an astronaut suit on

00:01:00.350 --> 00:01:02.120
even though they're on Earth.

00:01:02.120 --> 00:01:05.120
What would be the acceleration
in the vertical direction?

00:01:05.120 --> 00:01:08.830
Well the acceleration due to
gravity will be downwards,

00:01:08.830 --> 00:01:10.670
and it's going to be constant.

00:01:10.670 --> 00:01:12.960
We're going to assume
constant acceleration.

00:01:12.960 --> 00:01:17.470
So the acceleration is
going to look like this.

00:01:17.470 --> 00:01:19.540
And if the magnitude of the acceleration

00:01:19.540 --> 00:01:20.670
due to gravity is g,

00:01:20.670 --> 00:01:23.070
we could call this negative g to show

00:01:23.070 --> 00:01:25.470
that it is a downward acceleration.

00:01:25.470 --> 00:01:27.570
Once the projectile is let loose,

00:01:27.570 --> 00:01:29.800
that's the way it's
going to be accelerated.

00:01:29.800 --> 00:01:31.870
Now what about in the x direction?

00:01:31.870 --> 00:01:34.010
Well if we assume no air resistance,

00:01:34.010 --> 00:01:36.260
then there's not going
to be any acceleration

00:01:36.260 --> 00:01:38.130
or deceleration in the x direction.

00:01:38.130 --> 00:01:39.560
So it's just going to be,

00:01:39.560 --> 00:01:41.420
it's just going to stay right at zero

00:01:41.420 --> 00:01:43.210
and it's not going to change.

00:01:43.210 --> 00:01:45.550
And what I've just drawn
here is going to be true

00:01:45.550 --> 00:01:48.000
for all three of these scenarios

00:01:48.000 --> 00:01:50.800
because the direction
with which you throw it,

00:01:50.800 --> 00:01:53.930
that doesn't somehow affect
the acceleration due to gravity

00:01:53.930 --> 00:01:57.830
once the ball is actually
out of your hands.

00:01:57.830 --> 00:02:00.070
So now let's think about velocity.

00:02:00.070 --> 00:02:03.320
So what is going to be the
velocity in the y direction

00:02:03.320 --> 00:02:05.173
for this first scenario?

00:02:06.090 --> 00:02:08.780
Well we could take our
initial velocity vector

00:02:08.780 --> 00:02:10.350
that has this velocity at an angle

00:02:10.350 --> 00:02:13.240
and break it up into
its y and x components.

00:02:13.240 --> 00:02:15.543
So this would be its y component.

00:02:16.798 --> 00:02:19.020
We just take the top part of
this vector right over here,

00:02:19.020 --> 00:02:20.930
the head of it, and go to the left,

00:02:20.930 --> 00:02:23.580
and so that would be the
magnitude of its y component,

00:02:23.580 --> 00:02:27.510
and then this would be the
magnitude of its x component.

00:02:27.510 --> 00:02:30.410
So the y component, it starts positive,

00:02:30.410 --> 00:02:31.610
so it's like that,

00:02:31.610 --> 00:02:34.720
but remember our acceleration
is a constant negative.

00:02:34.720 --> 00:02:38.000
So our velocity is going to
decrease at a constant rate.

00:02:38.000 --> 00:02:40.670
So our velocity in this first scenario

00:02:40.670 --> 00:02:42.960
is going to look something,

00:02:42.960 --> 00:02:46.600
is going to look something like that.

00:02:46.600 --> 00:02:49.060
Now what about the velocity
in the x direction?

00:02:49.060 --> 00:02:51.440
We see that it starts positive,

00:02:51.440 --> 00:02:53.130
so it's going to start positive,

00:02:53.130 --> 00:02:55.820
and if we're in a world
with no air resistance,

00:02:55.820 --> 00:02:57.610
well then it's just
going to stay positive.

00:02:57.610 --> 00:03:00.010
Notice we have zero acceleration,

00:03:00.010 --> 00:03:03.270
so our velocity is just
going to stay positive.

00:03:03.270 --> 00:03:05.140
One of the things to really keep in mind

00:03:05.140 --> 00:03:07.540
when we start doing
two-dimensional projectile motion

00:03:07.540 --> 00:03:09.100
like we're doing right over here

00:03:09.100 --> 00:03:13.140
is once you break down your
vectors into x and y components,

00:03:13.140 --> 00:03:15.640
you can treat them
completely independently.

00:03:15.640 --> 00:03:17.520
That something will
decelerate in the y direction,

00:03:17.520 --> 00:03:19.330
but it doesn't mean that
it's going to decelerate

00:03:19.330 --> 00:03:21.130
in the x direction.

00:03:21.130 --> 00:03:23.450
Now what would the velocities look like

00:03:23.450 --> 00:03:25.990
for this blue scenario?

00:03:25.990 --> 00:03:28.690
Well our velocity in our y direction,

00:03:28.690 --> 00:03:31.610
we start off with no
velocity in our y direction

00:03:31.610 --> 00:03:33.940
so it's going to be right over here.

00:03:33.940 --> 00:03:36.380
But then we are going to
be accelerated downward,

00:03:36.380 --> 00:03:39.150
so our velocity is going
to get more and more

00:03:39.150 --> 00:03:42.690
and more negative as time passes.

00:03:42.690 --> 00:03:45.750
And notice the slope on
these two lines are the same

00:03:45.750 --> 00:03:48.490
because the rate of
acceleration is the same,

00:03:48.490 --> 00:03:51.310
even though you had a
different starting point.

00:03:51.310 --> 00:03:53.860
Now what about the velocity
in the x direction here?

00:03:53.860 --> 00:03:55.770
It looks like this x initial velocity

00:03:55.770 --> 00:03:57.240
is a little bit more than this one,

00:03:57.240 --> 00:03:58.890
so maybe it's a little bit higher,

00:03:58.890 --> 00:04:01.353
but it stays constant once again.

00:04:02.260 --> 00:04:04.610
Now let's look at this third scenario.

00:04:04.610 --> 00:04:06.090
In this third scenario,

00:04:06.090 --> 00:04:09.940
what is our y velocity,
our initial y velocity?

00:04:09.940 --> 00:04:12.040
Well it would look something like that.

00:04:12.040 --> 00:04:15.540
And our initial x velocity
would look something like that.

00:04:15.540 --> 00:04:18.160
If we were to break things
down into their components.

00:04:18.160 --> 00:04:21.260
So our y velocity is starting negative,

00:04:21.260 --> 00:04:22.550
is starting negative,

00:04:22.550 --> 00:04:24.950
and then it's just going to
get more and more negative

00:04:24.950 --> 00:04:28.010
once the individual lets go of the ball.

00:04:28.010 --> 00:04:30.840
Because you have that
constant acceleration,

00:04:30.840 --> 00:04:32.250
that negative acceleration,

00:04:32.250 --> 00:04:34.420
so it's gonna look something like that.

00:04:34.420 --> 00:04:36.660
And what about in the x direction?

00:04:36.660 --> 00:04:38.930
Well looks like in the x
direction right over here

00:04:38.930 --> 00:04:40.350
is very similar to that one,

00:04:40.350 --> 00:04:42.440
so it might look something like this.

00:04:42.440 --> 00:04:44.570
I'll draw it slightly higher
just so you can see it,

00:04:44.570 --> 00:04:47.340
but once again the velocity
x direction stays the same

00:04:47.340 --> 00:04:49.450
because in all three scenarios,

00:04:49.450 --> 00:04:52.923
you have zero acceleration
in the x direction.

00:04:53.830 --> 00:04:57.210
Now last but not least
let's think about position.

00:04:57.210 --> 00:04:59.600
So they all start in the exact same place

00:04:59.600 --> 00:05:01.640
at both the x and y dimension,

00:05:01.640 --> 00:05:04.440
but as we see, they all have
different initial velocities,

00:05:04.440 --> 00:05:06.190
at least in the y dimension.

00:05:06.190 --> 00:05:08.500
So let's start with
the salmon colored one.

00:05:08.500 --> 00:05:09.870
So the salmon colored one,

00:05:09.870 --> 00:05:14.370
it starts off with a some
type of positive y position,

00:05:14.370 --> 00:05:17.710
maybe based on the height of
where the individual's hand is.

00:05:17.710 --> 00:05:19.310
And then what's going to happen?

00:05:19.310 --> 00:05:22.500
Well it's going to have
positive but decreasing velocity

00:05:22.500 --> 00:05:23.680
up until this point.

00:05:23.680 --> 00:05:26.170
At this point its velocity is zero.

00:05:26.170 --> 00:05:28.380
So its position is going to go up

00:05:28.380 --> 00:05:31.780
but at ever decreasing
rates until you get right

00:05:31.780 --> 00:05:33.680
to that point right over there,

00:05:33.680 --> 00:05:36.820
and then we see the velocity
starts becoming more

00:05:36.820 --> 00:05:40.010
and more and more and more negative.

00:05:40.010 --> 00:05:42.600
So it would look something like,

00:05:42.600 --> 00:05:45.510
something like that.

00:05:45.510 --> 00:05:48.340
Now what would be the x position

00:05:48.340 --> 00:05:51.050
of this first scenario?

00:05:51.050 --> 00:05:54.620
Well if we make this position
right over here zero,

00:05:54.620 --> 00:05:58.870
then we would start our x
position would start over here,

00:05:58.870 --> 00:06:02.650
and since we have a constant
positive x velocity,

00:06:02.650 --> 00:06:06.690
our x position would just
increase at a constant rate.

00:06:06.690 --> 00:06:09.050
It would do something like that.

00:06:09.050 --> 00:06:11.200
Now what about this blue scenario?

00:06:11.200 --> 00:06:12.430
Well this blue scenario,

00:06:12.430 --> 00:06:15.920
we are starting in the exact same place

00:06:15.920 --> 00:06:17.610
as in our pink scenario,

00:06:17.610 --> 00:06:20.730
and then our initial y velocity is zero,

00:06:20.730 --> 00:06:22.700
and then it just gets
more and more and more

00:06:22.700 --> 00:06:23.810
and more negative.

00:06:23.810 --> 00:06:25.520
So it would look something,

00:06:25.520 --> 00:06:28.403
it would look something like this.

00:06:29.350 --> 00:06:32.760
Now what about the x position?

00:06:32.760 --> 00:06:34.220
Well our x position,

00:06:34.220 --> 00:06:36.340
we had a slightly higher velocity,

00:06:36.340 --> 00:06:38.120
at least the way that I drew it over here,

00:06:38.120 --> 00:06:41.410
so we our x position would
increase at a constant rate

00:06:41.410 --> 00:06:43.330
and it would be a slightly
higher constant rate.

00:06:43.330 --> 00:06:45.080
So it would have a slightly higher slope

00:06:45.080 --> 00:06:47.190
than we saw for the pink one.

00:06:47.190 --> 00:06:49.090
Now the yellow scenario,

00:06:49.090 --> 00:06:51.700
once again we're starting
in the exact same place,

00:06:51.700 --> 00:06:54.947
and here we're already starting
with a negative velocity

00:06:54.947 --> 00:06:57.410
and it's only gonna get more
and more and more negative.

00:06:57.410 --> 00:06:59.380
So it's just gonna do something like this.

00:06:59.380 --> 00:07:01.480
It's gonna get more and
more and more negative.

00:07:01.480 --> 00:07:02.420
It's a little bit hard to see,

00:07:02.420 --> 00:07:03.800
but it would do something like that.

00:07:03.800 --> 00:07:06.792
And if the in the x direction,

00:07:06.792 --> 00:07:11.490
our velocity is roughly the
same as the blue scenario,

00:07:11.490 --> 00:07:13.430
then our x position over
time for the yellow one

00:07:13.430 --> 00:07:16.360
is gonna look pretty pretty similar.

00:07:16.360 --> 00:07:18.840
So this is just a way to visualize

00:07:18.840 --> 00:07:21.250
how things would behave
in terms of position,

00:07:21.250 --> 00:07:25.530
velocity, and acceleration
in the y and x directions

00:07:25.530 --> 00:07:26.990
and to appreciate, one,

00:07:26.990 --> 00:07:29.000
how to draw and visualize these graphs

00:07:29.000 --> 00:07:30.690
and conceptualize them,

00:07:30.690 --> 00:07:32.760
but also to appreciate that you can treat,

00:07:32.760 --> 00:07:35.890
once you break your initial
velocity vectors down,

00:07:35.890 --> 00:07:38.120
you can treat the different dimensions,

00:07:38.120 --> 00:07:40.963
the x and the y dimensions, independently.