Newton's Laws Examples (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:01.620
Welcome back.

00:00:01.620 --> 00:00:05.710
And I will now do that same
problem in a much easier way.

00:00:05.710 --> 00:00:07.400
Let me clear it a little bit.

00:00:10.300 --> 00:00:12.810
So the original problem we said,
you know, we apply some

00:00:12.810 --> 00:00:19.330
force to m sub 0 and
that gives some

00:00:19.330 --> 00:00:21.340
acceleration, a sub 0.

00:00:21.340 --> 00:00:25.090
And then we said when we apply
the same force to a

00:00:25.090 --> 00:00:32.310
combination of m sub 0 and
m sub 1, we get 1/5 the

00:00:32.310 --> 00:00:33.560
acceleration.

00:00:35.780 --> 00:00:37.900
And we worked it through
with all the variables.

00:00:37.900 --> 00:00:39.390
What I'll show you is you can
actually do this type of

00:00:39.390 --> 00:00:41.680
problem just by substituting
numbers.

00:00:41.680 --> 00:00:43.610
This is kind of quick and dirty,
but it's good to do a

00:00:43.610 --> 00:00:44.740
reality check.

00:00:44.740 --> 00:00:47.720
And oftentimes, you can solve
the problem without having to

00:00:47.720 --> 00:00:49.360
go through all the
variable mess.

00:00:49.360 --> 00:00:50.930
So I can just pick some
numbers here.

00:00:50.930 --> 00:00:56.060
So I could say, well what if F
sub 0 is equal to 10 Newtons,

00:00:56.060 --> 00:01:02.070
m sub 0 is equal to-- I don't
know-- 2 kilograms. Than a sub

00:01:02.070 --> 00:01:06.320
0 is equal to-- well, it would
be 10 divided by 2.

00:01:06.320 --> 00:01:08.200
Because force is equal to
mass times acceleration.

00:01:08.200 --> 00:01:12.940
So it'd be 5 meters per
second squared.

00:01:12.940 --> 00:01:18.810
And then in this case, this
would be 10 Newtons.

00:01:18.810 --> 00:01:20.250
1/5 a sub 0?

00:01:20.250 --> 00:01:21.540
Well that will be 1/5 this.

00:01:21.540 --> 00:01:24.940
So it would be 1 meters
per second squared.

00:01:24.940 --> 00:01:28.030
And then we could solve for
what the new mass is.

00:01:28.030 --> 00:01:28.860
How would we do that?

00:01:28.860 --> 00:01:33.030
Well we have force, which is
10 Newtons, is equal to the

00:01:33.030 --> 00:01:34.120
sum of the masses.

00:01:34.120 --> 00:01:37.600
So m1 plus m sub 0.

00:01:37.600 --> 00:01:42.060
But m sub 0 we already learned
is 2 kilograms. Times the

00:01:42.060 --> 00:01:42.810
acceleration.

00:01:42.810 --> 00:01:46.110
Times 1/5 a sub 0, which is 1
meter per second squared.

00:01:48.910 --> 00:01:53.650
So then we have-- this 1, we
could ignore it essentially.

00:01:53.650 --> 00:01:57.200
So then we essentially have that
10-- and since all of our

00:01:57.200 --> 00:01:59.490
units are right, we can kind of
drop the units because we

00:01:59.490 --> 00:02:00.530
know they work out.

00:02:00.530 --> 00:02:04.220
10 is equal to m1 plus 2.

00:02:04.220 --> 00:02:07.550
So you get m1 is equal to 8.

00:02:07.550 --> 00:02:10.550
And then once again, if we want
to know the ratio of m

00:02:10.550 --> 00:02:14.290
sub 0 to m sub 1, we can just
substitute the numbers.

00:02:14.290 --> 00:02:18.440
m sub 0 is 2 kilograms. m
sub 1 is 8 kilograms. So

00:02:18.440 --> 00:02:19.970
the ratio is 1:4.

00:02:19.970 --> 00:02:22.450
You probably find that
a little bit easier.

00:02:22.450 --> 00:02:23.700
Let's do another problem.

00:02:28.570 --> 00:02:29.610
Whoops.

00:02:29.610 --> 00:02:30.570
Invert colors.

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OK.

00:02:33.650 --> 00:02:38.230
This next problem I think
you'll find interesting.

00:02:38.230 --> 00:02:42.860
So let's say I have a sky diver
and he's in his sky

00:02:42.860 --> 00:02:47.820
diver position, falling
towards the ground.

00:02:47.820 --> 00:02:51.120
And let's say he weighs 70
kilograms. So his mass is

00:02:51.120 --> 00:02:57.340
equal to 70 kilograms. Let's say
that the terminal velocity

00:02:57.340 --> 00:02:59.080
is 120 miles per hour.

00:02:59.080 --> 00:03:03.330
So he's moving downward at 120
miles per hour, which is

00:03:03.330 --> 00:03:04.260
actually accurate.

00:03:04.260 --> 00:03:07.450
I've gone sky diving.

00:03:07.450 --> 00:03:09.560
And if we convert that-- you
could convert that for fun

00:03:09.560 --> 00:03:10.690
into the metric system.

00:03:10.690 --> 00:03:12.270
But I'll do that for you.

00:03:12.270 --> 00:03:14.660
But it's good to know just so
you have a sense of how fast

00:03:14.660 --> 00:03:18.140
you fall when you sky dive
before the parachute opens.

00:03:18.140 --> 00:03:22.910
This translates to about
53.6 meters per second.

00:03:22.910 --> 00:03:24.790
I'm reading this from a problem
from a website at the

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University of Oregon.

00:03:27.090 --> 00:03:31.030
But anyway, they are asking
us, what force does air

00:03:31.030 --> 00:03:33.870
resistance exert on
the sky diver?

00:03:33.870 --> 00:03:35.350
So let's be clear
a couple things.

00:03:35.350 --> 00:03:37.960
This 120 miles per hour,
this is the sky

00:03:37.960 --> 00:03:40.500
diver's terminal velocity.

00:03:40.500 --> 00:03:42.880
And if you're not familiar with
what terminal velocity

00:03:42.880 --> 00:03:46.010
is, I will now explain
it to you.

00:03:46.010 --> 00:03:50.610
So when you fall from a plane,
you have a bunch of wind

00:03:50.610 --> 00:03:51.510
pushing on you.

00:03:51.510 --> 00:03:54.100
You have a lot of
wind resistance.

00:03:54.100 --> 00:03:56.480
It causes friction; it slows you
down as you can imagine.

00:03:56.480 --> 00:03:57.900
I mean that's how a
parachute works.

00:04:02.610 --> 00:04:04.600
It creates a lot more resistance
from the wind and

00:04:04.600 --> 00:04:05.600
then you slow down.

00:04:05.600 --> 00:04:11.760
So the terminal velocity is the
velocity at which you no

00:04:11.760 --> 00:04:13.230
longer go faster than.

00:04:13.230 --> 00:04:17.420
So it's the velocity at which
you stop accelerating or it's

00:04:17.420 --> 00:04:20.690
the velocity you reach and you
don't go any faster than that.

00:04:20.690 --> 00:04:23.940
It's basically based on
your wind resistance.

00:04:23.940 --> 00:04:26.560
So at the terminal velocity
your acceleration is 0.

00:04:29.580 --> 00:04:34.180
So what we know is, is that
the force of the air-- we

00:04:34.180 --> 00:04:36.720
could call that the air force.

00:04:36.720 --> 00:04:41.460
So we know that the force of
the air is exactly equal to

00:04:41.460 --> 00:04:42.710
the force of gravity.

00:04:46.260 --> 00:04:47.620
And how do we know that?

00:04:47.620 --> 00:04:50.370
Because the guy's not
accelerating.

00:04:50.370 --> 00:04:51.660
It's his terminal velocity.

00:04:51.660 --> 00:04:54.070
He's at a very high speed.

00:04:54.070 --> 00:04:56.060
He had accelerated all the
way to this point.

00:04:56.060 --> 00:04:59.160
But the more he accelerated
and the faster he got, the

00:04:59.160 --> 00:05:03.250
more resistance the wind
provided up to a point where

00:05:03.250 --> 00:05:05.810
the wind provided so much
resistance that he stopped

00:05:05.810 --> 00:05:06.790
going any faster.

00:05:06.790 --> 00:05:08.010
And that's the terminal
velocity.

00:05:08.010 --> 00:05:11.680
So at terminal velocity, the
force of the air is equal to

00:05:11.680 --> 00:05:14.120
the force of gravity.

00:05:14.120 --> 00:05:15.640
What's the force of gravity?

00:05:15.640 --> 00:05:18.250
Well the force of gravity is
just the guy's weight.

00:05:18.250 --> 00:05:23.490
So the force of gravity is equal
to the guy's mass, 70

00:05:23.490 --> 00:05:25.240
kilograms. And we have
our units right.

00:05:25.240 --> 00:05:29.580
70 kilograms. Times the
acceleration of gravity.

00:05:29.580 --> 00:05:32.940
Well the acceleration of gravity
is 9.8 roughly meters

00:05:32.940 --> 00:05:34.580
per second squared.

00:05:34.580 --> 00:05:36.940
We could use a calculator
to calculate this.

00:05:40.730 --> 00:05:41.540
I feel cheap now.

00:05:41.540 --> 00:05:43.790
I could have done it
by hand anyway.

00:05:43.790 --> 00:05:45.840
686.

00:05:45.840 --> 00:05:52.760
So it equals 686 Newtons.

00:05:52.760 --> 00:05:55.140
The second part of the
question-- and this is

00:05:55.140 --> 00:05:56.650
interesting.

00:05:56.650 --> 00:06:00.060
If a sky diver pulls in their
arm and aims their body

00:06:00.060 --> 00:06:04.010
downward, so now the sky diver
looks more like this and he

00:06:04.010 --> 00:06:06.500
pulled in his arms and he
aimed his body down.

00:06:06.500 --> 00:06:09.770
So he's diving, really.

00:06:09.770 --> 00:06:13.810
The terminal velocity can be
increased to about 180 miles

00:06:13.810 --> 00:06:17.070
per hour or 80.5 meters
per second.

00:06:17.070 --> 00:06:17.680
They give us this.

00:06:17.680 --> 00:06:19.310
We could've figured
it out though.

00:06:19.310 --> 00:06:22.400
So now he's going
a lot faster.

00:06:22.400 --> 00:06:25.850
Roughly 50% faster than he
was, or maybe-- well, 30%

00:06:25.850 --> 00:06:27.830
faster than he was
going before.

00:06:27.830 --> 00:06:30.040
He's going a lot faster
and why is that?

00:06:30.040 --> 00:06:33.960
Because he's more
aerodynamic now.

00:06:33.960 --> 00:06:36.760
We'll do more on pressure later,
but I want you to get

00:06:36.760 --> 00:06:40.110
the intuition that when you're
laying flat there's just a lot

00:06:40.110 --> 00:06:41.470
of wind pressing against
your body.

00:06:41.470 --> 00:06:44.560
You have a lot of surface area
exposed to the wind.

00:06:44.560 --> 00:06:46.970
But when you're diving in this
situation, like the sky diver

00:06:46.970 --> 00:06:49.660
is, he has a lot less
exposed to the wind.

00:06:49.660 --> 00:06:50.450
Really just his head.

00:06:50.450 --> 00:06:53.420
His head is breaking the
wind and nothing else.

00:06:53.420 --> 00:07:00.760
And that's why it takes a lot
more speed for the force of

00:07:00.760 --> 00:07:04.420
the wind resistance to match
the force of gravity.

00:07:04.420 --> 00:07:07.270
So the question is asking, if
the sky diver pulls in their

00:07:07.270 --> 00:07:10.310
arms and aims their body
downward, the terminal

00:07:10.310 --> 00:07:14.510
velocity can be increased to
about 80.5 meters per second

00:07:14.510 --> 00:07:15.880
or 180 miles per hour.

00:07:15.880 --> 00:07:17.140
So he's going very fast.

00:07:17.140 --> 00:07:22.720
What force does air resistance
now exert on the sky diver?

00:07:22.720 --> 00:07:24.460
And I'll let you think about
that for a second.

00:07:24.460 --> 00:07:27.580
Maybe you want to pause it
and think of it yourself.

00:07:27.580 --> 00:07:30.120
And now that you've unpaused it,
I'll tell you that this is

00:07:30.120 --> 00:07:31.290
a trick question.

00:07:31.290 --> 00:07:33.810
Because once again, the sky
diver has reached a new

00:07:33.810 --> 00:07:35.270
terminal velocity.

00:07:35.270 --> 00:07:39.500
By definition, at the terminal
velocity, the sky diver is no

00:07:39.500 --> 00:07:40.960
longer accelerating.

00:07:40.960 --> 00:07:43.940
The sky diver is not going any
faster because the wind

00:07:43.940 --> 00:07:46.713
resistance is so strong that
it completely matches the

00:07:46.713 --> 00:07:47.920
force of gravity.

00:07:47.920 --> 00:07:50.950
So once again, the wind
resistance, the force of the

00:07:50.950 --> 00:07:53.430
air or the force of the
wind, is equal to

00:07:53.430 --> 00:07:55.250
the force of gravity.

00:07:55.250 --> 00:07:56.620
And what is the force
of gravity?

00:07:56.620 --> 00:07:57.770
Well that's his weight.

00:07:57.770 --> 00:07:58.890
And we already figured
that out.

00:07:58.890 --> 00:08:01.720
That was 686 Newtons.

00:08:01.720 --> 00:08:02.640
And I know what you're
thinking.

00:08:02.640 --> 00:08:05.560
You're saying, Sal, this
doesn't make sense.

00:08:05.560 --> 00:08:11.070
He's now going so much faster,
doesn't the air exert more

00:08:11.070 --> 00:08:12.410
force on him?

00:08:12.410 --> 00:08:13.810
Well no.

00:08:13.810 --> 00:08:16.640
The air is exerting
the same force.

00:08:16.640 --> 00:08:18.990
If he was going that same speed,
but he was flattened

00:08:18.990 --> 00:08:21.940
out, I would agree with you.

00:08:21.940 --> 00:08:24.680
The air would be exerting
more force on him.

00:08:24.680 --> 00:08:27.820
But what's happened now is that
he's once again reached a

00:08:27.820 --> 00:08:29.970
state where his acceleration
is 0.

00:08:29.970 --> 00:08:32.870
It's at a higher velocity
and I want you

00:08:32.870 --> 00:08:33.980
think a lot about this.

00:08:33.980 --> 00:08:36.250
He's at a much higher
velocity now.

00:08:38.950 --> 00:08:41.330
On his head, for example,
there's a lot

00:08:41.330 --> 00:08:42.360
more wind going by.

00:08:42.360 --> 00:08:46.200
But it's pressing on a
smaller surface area.

00:08:46.200 --> 00:08:48.520
And I'm not going to go too
much into detail of

00:08:48.520 --> 00:08:49.560
pressure right now.

00:08:49.560 --> 00:08:51.730
But I want you to get
that intuition.

00:08:51.730 --> 00:08:56.360
So although the wind is going a
lot faster, it's going a lot

00:08:56.360 --> 00:08:59.460
faster on a smaller area.

00:08:59.460 --> 00:09:02.390
And its actual force is
the exact same thing.

00:09:02.390 --> 00:09:04.850
And we know that because he's
not accelerating anymore.

00:09:04.850 --> 00:09:07.060
Because he's at his
terminal velocity.

00:09:07.060 --> 00:09:08.070
So think about that a bit.

00:09:08.070 --> 00:09:10.000
It's a bit of a trick question,
but I think it gives

00:09:10.000 --> 00:09:12.890
you a good intuition on what
acceleration means, what

00:09:12.890 --> 00:09:15.370
terminal velocity means, and
it'll start to give you a

00:09:15.370 --> 00:09:18.430
little bit of an intuition
on even wind

00:09:18.430 --> 00:09:19.820
resistance, on pressure.

00:09:19.820 --> 00:09:21.860
I'll see you in the
next video.


Newton's Laws Problems (part 1)

WEBVTT
Kind: captions
Language: en

00:00:00.890 --> 00:00:01.820
Welcome back.

00:00:01.820 --> 00:00:04.340
Now that we've hopefully,
learned a little bit about

00:00:04.340 --> 00:00:07.670
Newton's law, let's apply them
to solve some problems.

00:00:07.670 --> 00:00:10.630
Let's say I have a-- I don't
know-- some kind of vehicle, a

00:00:10.630 --> 00:00:12.390
car, a motorcycle
or something.

00:00:12.390 --> 00:00:18.150
And let's say its mass is 500
grams. And let's say that I

00:00:18.150 --> 00:00:23.150
can accelerate this vehicle at
an acceleration of-- I don't

00:00:23.150 --> 00:00:33.410
know-- 3 centimeters
per second squared.

00:00:33.410 --> 00:00:37.500
What's the force that I need
to apply to this mass to

00:00:37.500 --> 00:00:39.210
accelerate it at this speed?

00:00:39.210 --> 00:00:41.180
And we want the answer
in Newton's.

00:00:41.180 --> 00:00:42.976
So what's the force?

00:00:42.976 --> 00:00:45.130
So we're just going to use
Newton's second law and

00:00:45.130 --> 00:00:48.820
Newton's second law tells us
force is equal to mass times

00:00:48.820 --> 00:00:49.810
acceleration.

00:00:49.810 --> 00:00:53.410
So you might want to, or you
might be tempted just to

00:00:53.410 --> 00:00:56.550
multiply mass times acceleration
and you'd get

00:00:56.550 --> 00:01:04.530
force is equal to-- let's see,
500 grams times 3 centimeters

00:01:04.530 --> 00:01:05.700
per second squared.

00:01:05.700 --> 00:01:12.440
And you would get force
is equal to 1,500 gram

00:01:12.440 --> 00:01:16.320
centimeters per second
squared.

00:01:16.320 --> 00:01:18.880
And if you did this,
you would be right.

00:01:18.880 --> 00:01:21.250
Although your answer would not
be in Newtons and now you

00:01:21.250 --> 00:01:23.670
would have to somehow,
try to convert this

00:01:23.670 --> 00:01:25.990
set of units to Newtons.

00:01:25.990 --> 00:01:27.020
And what are Newtons?

00:01:27.020 --> 00:01:30.850
Well we learned when we did
Newton's laws that a Newton,

00:01:30.850 --> 00:01:35.360
so 1 Newton is equal
to 1 kilogram

00:01:35.360 --> 00:01:37.900
meter per second squared.

00:01:37.900 --> 00:01:40.250
So somehow we have to convert
the gram to kilograms and we

00:01:40.250 --> 00:01:42.260
have to convert the centimeters
to meters.

00:01:42.260 --> 00:01:45.050
We could do it after the fact
here, or what I find it easier

00:01:45.050 --> 00:01:48.150
to do is actually to convert the
mass and the acceleration

00:01:48.150 --> 00:01:51.200
units first and then just
do the F equals ma.

00:01:51.200 --> 00:01:55.360
So what's 500 grams
in kilograms?

00:01:55.360 --> 00:02:00.940
Well 500 grams is half-- well,
a kilogram as a thousand

00:02:00.940 --> 00:02:04.010
grams. So 500 is going to
be half a kilogram.

00:02:07.440 --> 00:02:11.430
1 kilogram is a thousand grams,
so 0.5 kilograms is 500

00:02:11.430 --> 00:02:13.040
grams.

00:02:13.040 --> 00:02:18.000
Similarly, 3 centimeters
is how many meters?

00:02:18.000 --> 00:02:24.370
Well, 1 meter is 300-- sorry.

00:02:24.370 --> 00:02:25.510
I think I'm dehydrated.

00:02:25.510 --> 00:02:28.150
1 meter is a hundred
centimeters, right?

00:02:28.150 --> 00:02:36.280
So 3 centimeters is 0.03 meters
per second squared.

00:02:36.280 --> 00:02:38.530
Hopefully this make
sense to you.

00:02:38.530 --> 00:02:40.740
3 centimeters is 0.03 meters.

00:02:40.740 --> 00:02:41.870
And now we already.

00:02:41.870 --> 00:02:44.770
We have our mass in kilograms
and we have our acceleration

00:02:44.770 --> 00:02:46.130
in meters per second squared.

00:02:46.130 --> 00:02:49.130
And if this is confusing, you
should watch the unit videos

00:02:49.130 --> 00:02:50.710
because this is all I'm doing;
I'm just doing unit

00:02:50.710 --> 00:02:51.660
conversion.

00:02:51.660 --> 00:02:53.880
So let's go back to force
equals mass times

00:02:53.880 --> 00:02:55.280
acceleration.

00:02:55.280 --> 00:03:04.850
So the force was equal to
0.5 kilograms times the

00:03:04.850 --> 00:03:12.970
acceleration, which is 0.03
meters per second squared.

00:03:12.970 --> 00:03:13.700
And this equals--

00:03:13.700 --> 00:03:15.490
What's 0.5 times 0.03?

00:03:15.490 --> 00:03:18.400
I'll do it down here just
because multiplying decimals

00:03:18.400 --> 00:03:21.370
seems to be a problem for a
lot of people, including

00:03:21.370 --> 00:03:22.980
myself, many times.

00:03:22.980 --> 00:03:26.000
So what you do, you just
multiply the numbers.

00:03:26.000 --> 00:03:29.510
5 times 03 or 5 times 3 is 15.

00:03:29.510 --> 00:03:31.640
And then, how many points do
we have behind the decimal?

00:03:31.640 --> 00:03:33.210
How many digits behind
the decimal?

00:03:33.210 --> 00:03:33.740
Let's see.

00:03:33.740 --> 00:03:35.130
We have 1, 2, 3.

00:03:35.130 --> 00:03:37.660
So 1, 2, 3.

00:03:37.660 --> 00:03:40.240
We have to add the 0 because
we need three spaces behind

00:03:40.240 --> 00:03:41.730
the decimal point.

00:03:41.730 --> 00:03:50.190
So we get the force is equal to
0.015 kilogram meters per

00:03:50.190 --> 00:03:52.640
second squared.

00:03:52.640 --> 00:03:54.400
And this is a Newton.

00:03:54.400 --> 00:04:01.066
So the force is equal
to 0.015 Newtons.

00:04:01.066 --> 00:04:03.470
Let's do another problem.

00:04:03.470 --> 00:04:05.280
And this one's going to be-- and
actually, I think you'll

00:04:05.280 --> 00:04:09.600
find most of the difficult
Newton's laws problems or

00:04:09.600 --> 00:04:12.120
force problems, they're just
some combination of making

00:04:12.120 --> 00:04:15.120
sure you get the units right
when we're talking about in

00:04:15.120 --> 00:04:16.680
one dimension.

00:04:16.680 --> 00:04:20.959
The difficult part is usually
getting the units right or

00:04:20.959 --> 00:04:22.710
just the math, just
the algebra.

00:04:22.710 --> 00:04:24.610
So if you have trouble with this
it's usually because you

00:04:24.610 --> 00:04:26.890
have to just brush up a little
bit on the algebra.

00:04:26.890 --> 00:04:29.100
The physics itself is just
force equals mass times

00:04:29.100 --> 00:04:32.130
acceleration as we will
see in this problem.

00:04:32.130 --> 00:04:35.950
So let's say that when I apply
some force, some particular

00:04:35.950 --> 00:04:40.700
force, I use this
little 0 here.

00:04:40.700 --> 00:04:43.360
So I call that force F sub 0.

00:04:43.360 --> 00:04:44.820
So this means a particular
force.

00:04:44.820 --> 00:04:46.010
This is some value.

00:04:46.010 --> 00:04:50.600
When I apply that force to some
mass, let's call that m

00:04:50.600 --> 00:04:54.070
sub 0, I get some
acceleration.

00:04:54.070 --> 00:04:56.810
I get acceleration a sub 0.

00:04:56.810 --> 00:04:57.720
We could've put numbers here.

00:04:57.720 --> 00:05:02.810
We could've said, well, if I
apply a force of 10 Newtons to

00:05:02.810 --> 00:05:07.940
a mass of-- I don't know--
to a mass of let's say 2

00:05:07.940 --> 00:05:14.330
kilograms, I have an
acceleration of 5 meters per

00:05:14.330 --> 00:05:15.170
second squared.

00:05:15.170 --> 00:05:16.890
But I'm just doing this because
this could be any

00:05:16.890 --> 00:05:17.790
relationship.

00:05:17.790 --> 00:05:20.380
And let's say the problem
tells us that when I put

00:05:20.380 --> 00:05:23.680
another mass with this first
mass, so let's say that, you

00:05:23.680 --> 00:05:26.400
know-- let me draw
this diagram.

00:05:26.400 --> 00:05:29.480
So here's my mass, m sub 0.

00:05:29.480 --> 00:05:34.610
When I apply a force of f sub 0
to it, I get an acceleration

00:05:34.610 --> 00:05:36.250
of a sub 0.

00:05:36.250 --> 00:05:38.390
Now the problem tell us when I
add another mass-- so let's

00:05:38.390 --> 00:05:39.880
says I stack it up and
we're like in an ice

00:05:39.880 --> 00:05:41.670
skating ring or something.

00:05:41.670 --> 00:05:44.196
And I stack another mass
up here, and let's

00:05:44.196 --> 00:05:46.990
call this mass m1.

00:05:46.990 --> 00:05:49.890
When I stack another mass on
here-- so let me redraw it

00:05:49.890 --> 00:05:51.250
actually down here.

00:05:51.250 --> 00:05:53.420
Because it's a different case.

00:05:53.420 --> 00:05:57.720
And I apply the same force,
and now I have

00:05:57.720 --> 00:05:59.020
this new mass on here.

00:05:59.020 --> 00:06:01.926
I'll do it in red.

00:06:01.926 --> 00:06:03.920
This is m1.

00:06:03.920 --> 00:06:08.540
The problem tells us that my new
acceleration is 1/5 of the

00:06:08.540 --> 00:06:09.650
original acceleration.

00:06:09.650 --> 00:06:12.070
So it's 1/5 of whatever
this was.

00:06:12.070 --> 00:06:16.390
So it's 1/5 a sub 0.

00:06:16.390 --> 00:06:21.620
So the question is, what
is the ratio of m

00:06:21.620 --> 00:06:23.610
sub 0 to m sub 1?

00:06:23.610 --> 00:06:28.590
So m sub 0 to m sub 1
is equal to what?

00:06:28.590 --> 00:06:30.990
And I'm going to keep it in
abstract variables just to

00:06:30.990 --> 00:06:31.710
confuse you.

00:06:31.710 --> 00:06:33.200
And then I'll show you that you
can actually substitute

00:06:33.200 --> 00:06:35.720
numbers and the problem becomes
a little easier.

00:06:35.720 --> 00:06:38.210
And you might want to pause it
and try it for yourself.

00:06:38.210 --> 00:06:39.670
So let's work it through.

00:06:39.670 --> 00:06:41.590
So we know we have
this relationship

00:06:41.590 --> 00:06:43.250
to start off with.

00:06:43.250 --> 00:06:48.760
And just for simplicity, let's
write what m sub 0 is in terms

00:06:48.760 --> 00:06:49.720
of F and a.

00:06:49.720 --> 00:06:54.890
So we just divide both sides by
a sub 0 and you get F sub 0

00:06:54.890 --> 00:06:58.250
divided by a sub 0 is
equal to m sub 0.

00:06:58.250 --> 00:06:58.580
Good.

00:06:58.580 --> 00:07:00.970
So let's just put that
aside for a second.

00:07:00.970 --> 00:07:04.340
And let's do that same
relationship here with this.

00:07:04.340 --> 00:07:09.160
So here, this relationship tells
us that F sub 0 is equal

00:07:09.160 --> 00:07:19.130
to m1 plus m0 times this
new acceleration, which

00:07:19.130 --> 00:07:20.660
is 1/5 a sub 0.

00:07:24.570 --> 00:07:28.390
And so, if we divide both sides
by this term right here,

00:07:28.390 --> 00:07:32.820
we get-- dividing by 1/5
is the same thing as

00:07:32.820 --> 00:07:34.260
multiplying by 5.

00:07:34.260 --> 00:07:44.060
So you get 5 F sub 0 over a sub
0 is equal to m1 plus m0.

00:07:44.060 --> 00:07:47.370
I just divided both sides
by this term right here.

00:07:47.370 --> 00:07:48.910
Well, what's F sub 0?

00:07:48.910 --> 00:07:49.490
What's this?

00:07:49.490 --> 00:07:50.340
Let me switch colors again.

00:07:50.340 --> 00:07:53.910
What's F sub 0 divided
by a sub 0?

00:07:53.910 --> 00:07:54.570
Well it's here's.

00:07:54.570 --> 00:07:55.860
It's what we solved for
in the beginning.

00:07:55.860 --> 00:07:57.770
We just got it from
this relationship.

00:07:57.770 --> 00:07:59.310
So we could substitute.

00:07:59.310 --> 00:08:02.780
5 times F sub 0 divided
by a sub 0 is the same

00:08:02.780 --> 00:08:05.425
thing as 5 times m0.

00:08:05.425 --> 00:08:06.790
Draw a line here.

00:08:06.790 --> 00:08:07.930
So we have a new relationship.

00:08:07.930 --> 00:08:13.330
5m0 is equal to m1 plus m0.

00:08:13.330 --> 00:08:15.960
All I did is I substituted
this for

00:08:15.960 --> 00:08:17.980
this, or this for that.

00:08:17.980 --> 00:08:20.150
And I used this relationship,
which we got in the beginning

00:08:20.150 --> 00:08:21.370
to do that.

00:08:21.370 --> 00:08:22.830
And now what do we have?

00:08:22.830 --> 00:08:25.660
We have 5 m sub 0 is equal
to m1 plus m0.

00:08:25.660 --> 00:08:28.460
We could subtract m0
from both sides.

00:08:28.460 --> 00:08:32.270
You get 4 m0 is equal to m1.

00:08:35.520 --> 00:08:39.460
You could divide both sides by
m0 and you got 4 is equal to

00:08:39.460 --> 00:08:42.730
m1 over m0.

00:08:42.730 --> 00:08:44.760
And you could invert this
relationship and you can get

00:08:44.760 --> 00:08:49.210
m0 over m sub 1 is
equal to 1/4.

00:08:49.210 --> 00:08:54.060
So what we learned is the ratio
of the old mass to the

00:08:54.060 --> 00:08:56.400
new mass is 1 to 4.

00:08:56.400 --> 00:08:57.410
And that's a problem.

00:08:57.410 --> 00:08:59.750
And actually, I will leave it
for you as an exercise to

00:08:59.750 --> 00:09:01.410
figure out-- to just
do the same

00:09:01.410 --> 00:09:04.500
problem using the numbers.

00:09:04.500 --> 00:09:06.650
I will do that actually, in the
next video just to show

00:09:06.650 --> 00:09:07.750
you that that actually
would've been a

00:09:07.750 --> 00:09:08.860
simpler way to do it.

00:09:08.860 --> 00:09:12.160
But it's good to get used to
this just so you can solve

00:09:12.160 --> 00:09:15.710
things in general terms. I'll
see you in the next video.


Newton's Second Law of Motion

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:01.550
Welcome back.

00:00:01.550 --> 00:00:03.460
We're now ready for Newton's
second law.

00:00:03.460 --> 00:00:05.970
And Newton's second law can
simply be stated-- and you've

00:00:05.970 --> 00:00:12.310
probably seen this before as
force is equal to mass times

00:00:12.310 --> 00:00:13.100
acceleration.

00:00:13.100 --> 00:00:18.480
This is probably, if not the
most famous formula in all of

00:00:18.480 --> 00:00:20.730
time or all of physics,
it's up there.

00:00:20.730 --> 00:00:23.190
It's probably up there with
E equals mc squared.

00:00:23.190 --> 00:00:24.810
But that one's a little
bit more complicated.

00:00:24.810 --> 00:00:26.030
So what does this tell us?

00:00:26.030 --> 00:00:29.550
This tells us that the force,
the net force upon an object,

00:00:29.550 --> 00:00:33.520
is equal to the object's mass
times its acceleration.

00:00:33.520 --> 00:00:35.490
So let's stay in the metric
system because most of what

00:00:35.490 --> 00:00:37.660
you'll do in physic class is in
the metric system, and that

00:00:37.660 --> 00:00:40.160
tends to be because the metric
system makes more sense.

00:00:40.160 --> 00:00:42.390
So let's say that I have
a 1 kilogram object.

00:00:47.330 --> 00:00:50.430
So its mass is 1 kilogram.

00:00:50.430 --> 00:00:53.450
And it's being pulled down
at-- let's say its

00:00:53.450 --> 00:00:54.850
acceleration.

00:00:54.850 --> 00:01:01.040
It's being accelerated downward
at 9.8 meters per

00:01:01.040 --> 00:01:02.290
second squared.

00:01:05.209 --> 00:01:06.850
These kind of units should be
familiar with you from all the

00:01:06.850 --> 00:01:08.380
projectile motion problems.

00:01:08.380 --> 00:01:11.380
So the force applied on that
object in order to get this

00:01:11.380 --> 00:01:16.820
type of acceleration would be--
you just multiply mass

00:01:16.820 --> 00:01:17.570
times acceleration.

00:01:17.570 --> 00:01:25.610
The force would have had to be
9.8 kilogram times the meter.

00:01:25.610 --> 00:01:26.250
kilogram.

00:01:26.250 --> 00:01:29.540
times meter over
second square.

00:01:29.540 --> 00:01:31.240
That's the force applied
on the object.

00:01:31.240 --> 00:01:33.950
And you're saying, sal,
this is very messy.

00:01:33.950 --> 00:01:36.340
I don't like writing kilogram
meters per second squared.

00:01:36.340 --> 00:01:39.315
And you are in luck because
there is a unit and that unit

00:01:39.315 --> 00:01:40.450
is the Newton.

00:01:40.450 --> 00:01:46.490
1 Newton is equal to 1 kilogram

00:01:46.490 --> 00:01:49.400
meter per second squared.

00:01:49.400 --> 00:01:57.740
So if I'm pulling down on an
object at 9.8 Newtons, that's

00:01:57.740 --> 00:01:58.440
just this, right?

00:01:58.440 --> 00:02:00.340
This is 1 Newton.

00:02:00.340 --> 00:02:05.050
If I'm pulling down at 9.8
Newtons on an object that is 1

00:02:05.050 --> 00:02:12.230
kilogram, its acceleration is
going to be 9.8 meters per

00:02:12.230 --> 00:02:14.340
second squared down.

00:02:14.340 --> 00:02:17.200
And notice I said the word down,
but I didn't write it

00:02:17.200 --> 00:02:19.300
anywhere in the formula.

00:02:19.300 --> 00:02:22.690
And I guess we can imply that
both force and acceleration

00:02:22.690 --> 00:02:28.700
have direction by writing
this in the formula.

00:02:28.700 --> 00:02:32.620
That force is a vector and
acceleration is a vector.

00:02:32.620 --> 00:02:36.500
And so we could have written
9.8 Newtons-- I don't know.

00:02:36.500 --> 00:02:38.250
You'll never see this
convention.

00:02:38.250 --> 00:02:43.990
We could say Newtons down is
equal to 1 kilogram times 9.8

00:02:43.990 --> 00:02:47.340
meters per second down.

00:02:47.340 --> 00:02:49.010
So what can we do with
this formula?

00:02:49.010 --> 00:02:54.300
Well we can solve problems.
So let's say

00:02:54.300 --> 00:02:55.490
that I have an object.

00:02:55.490 --> 00:02:58.350
So my object weighs--
not weighs.

00:02:58.350 --> 00:03:00.400
The mass of my object.

00:03:00.400 --> 00:03:03.190
And I'll differentiate between
weight and mass in a second.

00:03:03.190 --> 00:03:07.110
Let's say the mass of some
object is-- I don't know-- 50

00:03:07.110 --> 00:03:12.085
kilograms. That's how much a
normal person might weigh or a

00:03:12.085 --> 00:03:13.480
light person.

00:03:13.480 --> 00:03:19.300
Mass weighs 50 kilograms. And
let's say we're in an inertial

00:03:19.300 --> 00:03:20.100
frame of reference.

00:03:20.100 --> 00:03:23.390
We're in deep space, so we don't
have all these other--

00:03:23.390 --> 00:03:26.130
the force of wind and
the force of gravity

00:03:26.130 --> 00:03:27.380
acting on us, et cetera.

00:03:31.600 --> 00:03:34.260
My force, let's say I apply
it to the right.

00:03:34.260 --> 00:03:36.530
So we know that force
is a vector.

00:03:36.530 --> 00:03:43.686
Let's say I apply a force of--
I don't know-- 100 Newtons.

00:03:43.686 --> 00:03:45.400
And let's say I apply
it to the right.

00:03:49.600 --> 00:03:56.820
So this is the object, 50
kilograms. And I'm applying a

00:03:56.820 --> 00:04:02.220
force to the right
of 100 Newtons.

00:04:02.220 --> 00:04:05.060
So what's going to happen
to this object?

00:04:05.060 --> 00:04:07.410
Well, let's use the formula.

00:04:07.410 --> 00:04:12.640
Force is equal to mass
times acceleration.

00:04:12.640 --> 00:04:14.790
The force is 100 Newtons.

00:04:14.790 --> 00:04:18.730
100 Newtons is equal
to the mass.

00:04:18.730 --> 00:04:24.260
The mass is 50 kilograms.
50 kilograms times the

00:04:24.260 --> 00:04:26.700
acceleration.

00:04:26.700 --> 00:04:33.830
So we can divide both sides by
50 and you get 100 Newtons

00:04:33.830 --> 00:04:40.580
over 50 kilograms is equal
to the acceleration.

00:04:40.580 --> 00:04:42.930
And it's 100 Newtons
to the right.

00:04:42.930 --> 00:04:43.950
I'll use this little
arrow here.

00:04:43.950 --> 00:04:46.160
That's not a traditional
convention, but that's how we

00:04:46.160 --> 00:04:47.220
know it's to the right.

00:04:47.220 --> 00:04:48.770
So it's 100 divided by 50.

00:04:48.770 --> 00:04:51.150
So it's 2.

00:04:51.150 --> 00:04:56.010
We get this weird units here,
Newtons per kilogram is equal

00:04:56.010 --> 00:04:58.250
to the acceleration
to the right.

00:04:58.250 --> 00:04:59.590
This is also going to be to
the right because the

00:04:59.590 --> 00:05:02.380
direction of the force is going
to be the same as the

00:05:02.380 --> 00:05:04.540
direction of the acceleration.

00:05:04.540 --> 00:05:06.400
So what is this, 2 Newtons
per kilogram?

00:05:06.400 --> 00:05:09.210
Well, if you remember-- well you
could just guess that the

00:05:09.210 --> 00:05:10.730
unit of acceleration is meters
per second squared.

00:05:10.730 --> 00:05:12.590
But let's show that this
simplifies to that.

00:05:12.590 --> 00:05:15.060
So we said earlier that-- let
me just switch colors.

00:05:15.060 --> 00:05:22.270
That a Newton is kilogram meter
per second squared.

00:05:22.270 --> 00:05:27.470
And we're taking this Newton
over this kilogram over

00:05:27.470 --> 00:05:29.140
kilogram, right?

00:05:29.140 --> 00:05:31.680
So that will cancel out with
that and you get meters per

00:05:31.680 --> 00:05:32.850
second squared.

00:05:32.850 --> 00:05:35.900
And you wouldn't have to do this
on a test. Essentially,

00:05:35.900 --> 00:05:37.690
if you did everything right, you
would know that the unit

00:05:37.690 --> 00:05:40.000
acceleration is meters
per second squared.

00:05:40.000 --> 00:05:42.170
So you would have the
acceleration-- I'm just

00:05:42.170 --> 00:05:46.620
switching the two sides--
is equal to 2

00:05:46.620 --> 00:05:50.000
meters per second squared.

00:05:50.000 --> 00:05:51.250
And it'll be to the right.

00:05:54.600 --> 00:05:55.630
So that's useful.

00:05:55.630 --> 00:06:00.720
We just figured out based on how
hard I push something, how

00:06:00.720 --> 00:06:03.760
fast it's going to accelerate
while I push it.

00:06:03.760 --> 00:06:04.990
And you could use the
same formula to

00:06:04.990 --> 00:06:05.850
figure out other things.

00:06:05.850 --> 00:06:11.820
Let's say I know that an object
is accelerating-- let's

00:06:11.820 --> 00:06:16.430
say my acceleration is
3 meters per second

00:06:16.430 --> 00:06:18.600
squared to the right.

00:06:18.600 --> 00:06:20.910
Let's say to the left, just
to switch things.

00:06:20.910 --> 00:06:26.880
And let's say that I know the
force being applied on it is--

00:06:26.880 --> 00:06:31.950
I don't know-- 30 Newtons
to the left.

00:06:31.950 --> 00:06:33.520
And I want to figure
out the mass.

00:06:33.520 --> 00:06:34.520
Well you use the same thing.

00:06:34.520 --> 00:06:39.740
You say force, 30 Newtons to
the left is equal to mass

00:06:39.740 --> 00:06:41.440
times acceleration.

00:06:41.440 --> 00:06:45.530
Times 3 meters per second
squared to the left.

00:06:45.530 --> 00:06:49.660
Divide both sides by the 3
meters per second and you get

00:06:49.660 --> 00:06:54.370
30 Newtons over 3 meters
per second squared is

00:06:54.370 --> 00:06:56.030
equal to the mass.

00:06:56.030 --> 00:06:59.520
30 divided by 3 is 10.

00:06:59.520 --> 00:07:02.050
You can figure out that Newtons
is kilogram meters per

00:07:02.050 --> 00:07:02.750
second squared.

00:07:02.750 --> 00:07:05.370
So you're just left with
10 kilograms is

00:07:05.370 --> 00:07:07.580
equal to the mass.

00:07:07.580 --> 00:07:10.590
It's very important that if you
see a problem where the

00:07:10.590 --> 00:07:14.060
answer's given in-- I don't
know-- kilometers per second

00:07:14.060 --> 00:07:17.410
squared or you know, instead of
giving it in kilograms it's

00:07:17.410 --> 00:07:20.110
giving it in grams or decagrams,
you should convert

00:07:20.110 --> 00:07:24.360
back to kilograms or meters just
so you make sure you're

00:07:24.360 --> 00:07:25.210
using the right units.

00:07:25.210 --> 00:07:27.180
And that tends to be frankly,
I think, the

00:07:27.180 --> 00:07:28.120
hardest thing for people.

00:07:28.120 --> 00:07:32.750
And we'll do all of that when
we tackle harder problems.

00:07:32.750 --> 00:07:35.020
I think now is a good time back
to actually differentiate

00:07:35.020 --> 00:07:37.220
between mass and weight.

00:07:37.220 --> 00:07:39.340
And you've probably thought the
two were interchangeable,

00:07:39.340 --> 00:07:40.710
but they're not.

00:07:40.710 --> 00:07:46.100
Mass is how much of an
object there is.

00:07:46.100 --> 00:07:48.440
You can almost view it as how
much of the stuff there is or

00:07:48.440 --> 00:07:50.300
you can almost it view it--
how many atoms there are.

00:07:50.300 --> 00:07:51.435
But even atoms have mass.

00:07:51.435 --> 00:07:54.050
So just how much
stuff there is.

00:07:54.050 --> 00:07:57.530
And another way to view mass is,
how much does the object

00:07:57.530 --> 00:07:58.880
resist change?

00:07:58.880 --> 00:08:02.250
And that actually falls
out of F equals ma.

00:08:02.250 --> 00:08:07.260
Because if our mass is bigger,
it's going to take a lot more

00:08:07.260 --> 00:08:09.220
force to make it accelerate
a certain amount.

00:08:09.220 --> 00:08:11.760
If the mass is smaller it'll
take less force.

00:08:11.760 --> 00:08:15.340
So mass can be viewed as how
much stuff there is, of an

00:08:15.340 --> 00:08:16.400
object there is.

00:08:16.400 --> 00:08:20.440
Or you can view it as how hard
is it to change what that

00:08:20.440 --> 00:08:21.040
object is doing.

00:08:21.040 --> 00:08:24.480
If it's stationary, how hard
is it to accelerate it?

00:08:24.480 --> 00:08:27.190
If it's moving, how hard
is it to maybe stop it?

00:08:27.190 --> 00:08:28.930
Which would essentially
be decelerating.

00:08:28.930 --> 00:08:31.400
How hard is it to accelerate
an object?

00:08:31.400 --> 00:08:35.600
Weight is actually how much
is-- what is the force of

00:08:35.600 --> 00:08:37.650
earth upon an object?

00:08:37.650 --> 00:08:39.840
So you're weight would actually
change if you go from

00:08:39.840 --> 00:08:42.600
one planet to another because
the force of gravity changes.

00:08:42.600 --> 00:08:46.760
So your weight is 1/6 on the
moon as it is on earth because

00:08:46.760 --> 00:08:48.310
the pull of gravity is 1/6.

00:08:48.310 --> 00:08:49.450
But your mass doesn't change.

00:08:49.450 --> 00:08:51.810
There's still the same amount
of Sal on earth as

00:08:51.810 --> 00:08:54.400
there is on the moon.

00:08:54.400 --> 00:08:57.550
So your weight really-- when you
ask someone in Europe and

00:08:57.550 --> 00:08:59.950
they say hey, you know, I weigh
50 kilograms. You should

00:08:59.950 --> 00:09:03.190
say, no, you don't weigh 50
kilograms. You weigh whatever

00:09:03.190 --> 00:09:04.870
50 times 9.8 is.

00:09:04.870 --> 00:09:08.890
That's like 400 something--
you weigh

00:09:08.890 --> 00:09:11.860
490 Newtons or something.

00:09:11.860 --> 00:09:14.450
This is mass.

00:09:14.450 --> 00:09:17.400
And it's interesting because in
the English system, and all

00:09:17.400 --> 00:09:18.860
of us Americans, we use
the English system.

00:09:18.860 --> 00:09:21.790
When we say that we weigh 10
pounds, we're actually using

00:09:21.790 --> 00:09:25.680
the correct terminology
because pounds

00:09:25.680 --> 00:09:28.730
are a unit of force.

00:09:28.730 --> 00:09:32.390
We're saying, if I weigh-- and
I do weigh about 150 pounds.

00:09:32.390 --> 00:09:33.940
That means the earth is this
pulling on me with

00:09:33.940 --> 00:09:36.280
150 pounds of force.

00:09:36.280 --> 00:09:39.770
And actually, turns out that
my mass is measured in the

00:09:39.770 --> 00:09:41.880
unit called a slug, which
we might discuss later.

00:09:41.880 --> 00:09:43.485
Actually, we'll do some problems
where we do it in the

00:09:43.485 --> 00:09:45.620
metric system and the
English system.

00:09:45.620 --> 00:09:47.930
And I'll see you in the
next presentation.


Newton's Third Law of Motion

WEBVTT
Kind: captions
Language: en

00:00:00.860 --> 00:00:02.710
Now we're ready for Newton's
third law.

00:00:02.710 --> 00:00:05.040
And Newton's third law, in some
ways, I think is the most

00:00:05.040 --> 00:00:09.230
fun because it's-- at least it
was to me, the least intuitive

00:00:09.230 --> 00:00:10.020
of all the laws.

00:00:10.020 --> 00:00:12.570
But once you really kind of
understand it a lot of things

00:00:12.570 --> 00:00:13.290
start to make sense.

00:00:13.290 --> 00:00:16.970
So Newton's third law
essentially says-- actually, I

00:00:16.970 --> 00:00:19.100
can tell you kind of what
you might have heard.

00:00:19.100 --> 00:00:21.160
A lot of people say every
action has an equal and

00:00:21.160 --> 00:00:22.690
opposite reaction.

00:00:22.690 --> 00:00:26.080
Another way of thinking about
it is, if there's an object

00:00:26.080 --> 00:00:28.460
and it exerts a force
on another object.

00:00:28.460 --> 00:00:34.150
So let's say I have a-- oh,
let me think of something.

00:00:34.150 --> 00:00:41.460
Let's say that I have a fist. So
let me draw the fist. So I

00:00:41.460 --> 00:00:45.660
have a fist and it is punching
someone's face.

00:00:45.660 --> 00:00:48.110
So this is a face, and
they're not happy.

00:00:51.030 --> 00:00:56.830
And let's say this fist is
punching the face with a force

00:00:56.830 --> 00:01:00.840
of-- I don't know--
10 Newtons.

00:01:00.840 --> 00:01:01.700
Let's not make it so violent.

00:01:01.700 --> 00:01:04.860
Let's say that this is a hand
massaging the face.

00:01:04.860 --> 00:01:09.940
It's pressing upon the face with
a force of 10 Newtons.

00:01:09.940 --> 00:01:14.210
So what Newton's third law tells
us is that the face is

00:01:14.210 --> 00:01:17.380
also, I guess, we could
say punching the hand.

00:01:17.380 --> 00:01:19.280
Or-- well no, we're not
using the violent.

00:01:19.280 --> 00:01:23.280
The face is also pressing upon
the hand with a force of 10

00:01:23.280 --> 00:01:28.280
Newtons in the opposite
direction.

00:01:28.280 --> 00:01:30.360
I guess you would say
along the same line.

00:01:30.360 --> 00:01:32.080
So does that make any sense?

00:01:32.080 --> 00:01:36.540
Because it seems like the hand
is doing something to the face

00:01:36.540 --> 00:01:37.830
and not the other way around.

00:01:37.830 --> 00:01:42.420
But if you think about it, when
you press on someone's

00:01:42.420 --> 00:01:46.330
face, their face might kind of
press in a little bit, but you

00:01:46.330 --> 00:01:50.530
also feel something on your
fist or on your hand.

00:01:50.530 --> 00:01:53.190
I mean think of it, maybe a
better example would be

00:01:53.190 --> 00:01:57.950
instead of someone's face,
imagine if it was a tree that

00:01:57.950 --> 00:02:01.700
you've decided to massage
or, I guess, punch.

00:02:01.700 --> 00:02:05.510
So here is the tree.

00:02:05.510 --> 00:02:06.790
That is the tree.

00:02:06.790 --> 00:02:08.880
And the same thing
would happen.

00:02:08.880 --> 00:02:11.970
If you were to press upon the
tree with-- if you were to

00:02:11.970 --> 00:02:15.350
punch the tree essentially,
the tree is essentially

00:02:15.350 --> 00:02:17.540
punching back with the
exact same force.

00:02:17.540 --> 00:02:20.650
And here it makes sense because
your hand will hurt.

00:02:20.650 --> 00:02:24.020
And maybe in this case, the
face will hurt because the

00:02:24.020 --> 00:02:26.900
face kind of gives way while
your fist doesn't.

00:02:26.900 --> 00:02:29.750
But here the tree's not giving
way and your fist will.

00:02:29.750 --> 00:02:35.170
Another way to think about it
is if-- well, and this is

00:02:35.170 --> 00:02:36.840
probably the least intuitive.

00:02:36.840 --> 00:02:42.380
If I have the earth and
here am I standing

00:02:42.380 --> 00:02:43.630
on top of the earth.

00:02:45.940 --> 00:02:48.330
So we already figured out that
the pull of the earth or the

00:02:48.330 --> 00:02:51.940
force of gravity upon me,
it's pulling down

00:02:51.940 --> 00:02:54.820
upon me at 150 pounds.

00:02:54.820 --> 00:02:55.580
That's the force.

00:02:55.580 --> 00:02:56.710
And you know, we could
say the Newton.

00:02:56.710 --> 00:02:59.120
But pounds is a unit of
force, it's weight.

00:02:59.120 --> 00:03:02.710
But also, Newton's third law
tells us that I'm actually, at

00:03:02.710 --> 00:03:06.870
the same moment, pulling
on the earth with a

00:03:06.870 --> 00:03:10.620
force of 150 pounds.

00:03:10.620 --> 00:03:15.890
And this might not make a lot
of sense to you, but you can

00:03:15.890 --> 00:03:17.210
think about it this way.

00:03:17.210 --> 00:03:20.360
When I'm stepping on-- let's
say I'm stepping on a soft

00:03:20.360 --> 00:03:23.010
surface, like sand
or something.

00:03:23.010 --> 00:03:26.440
My feet will compress a little
bit, but so does the sand.

00:03:26.440 --> 00:03:29.460
Well, depending on which
one's softer.

00:03:29.460 --> 00:03:32.360
And another way to think about
it also is, if me and the

00:03:32.360 --> 00:03:35.440
earth are both in deep space.

00:03:35.440 --> 00:03:38.990
And I am, you could say, falling
towards the earth

00:03:38.990 --> 00:03:40.780
because the earth is
pulling on me.

00:03:40.780 --> 00:03:43.140
How do we know that the earth
isn't falling towards me?

00:03:43.140 --> 00:03:44.850
I mean it's kind of arbitrary.

00:03:44.850 --> 00:03:45.960
There's no frame of reference.

00:03:45.960 --> 00:03:47.100
We're both in deep space.

00:03:47.100 --> 00:03:48.360
There's nothing else
to look at.

00:03:48.360 --> 00:03:51.070
We're essentially, falling
towards each other.

00:03:51.070 --> 00:03:53.200
I'm not necessarily falling to
the earth, the earth's not

00:03:53.200 --> 00:03:54.580
necessarily falling to me, we're
just falling towards

00:03:54.580 --> 00:03:55.310
each other.

00:03:55.310 --> 00:03:57.740
And that's another way of
thinking about this.

00:03:57.740 --> 00:03:59.870
So you could think about every
example where a force applies

00:03:59.870 --> 00:04:00.500
to something.

00:04:00.500 --> 00:04:03.390
And if you really think about
it, the force is going the

00:04:03.390 --> 00:04:04.150
other way as well.

00:04:04.150 --> 00:04:09.600
For example, if I were to take
I bat to this tree and swing

00:04:09.600 --> 00:04:10.700
on it really hard.

00:04:10.700 --> 00:04:13.490
So I were to swing on this tree
really hard, I have a

00:04:13.490 --> 00:04:16.040
good chance of breaking
that bat.

00:04:16.040 --> 00:04:18.839
Even though you would have
thought, hey, that bat is

00:04:18.839 --> 00:04:20.690
applying the force
to the tree.

00:04:20.690 --> 00:04:22.240
But why is the bat breaking?

00:04:22.240 --> 00:04:25.700
Because the tree is applying
an equal force to the bat.

00:04:25.700 --> 00:04:27.780
And actually, if I did it
perfectly, if I had like--

00:04:27.780 --> 00:04:33.540
let's say I had two bats
or two swords.

00:04:33.540 --> 00:04:37.080
For some reason I think I'm
going a little too violent

00:04:37.080 --> 00:04:37.700
with these example.

00:04:37.700 --> 00:04:41.250
But I guess you know we're
talking about forces.

00:04:41.250 --> 00:04:44.990
So maybe violence is
justified here.

00:04:44.990 --> 00:04:46.810
But let's say I have two swords
that are completely

00:04:46.810 --> 00:04:48.060
identical hitting each other.

00:04:51.260 --> 00:04:53.540
And let's say I keep increasing
the force at which

00:04:53.540 --> 00:04:56.110
they're kind of going in
opposite directions.

00:04:56.110 --> 00:04:58.600
At some point, they're
going to break.

00:04:58.600 --> 00:05:01.640
If I just keep increasing the
force on-- you know, one guy

00:05:01.640 --> 00:05:03.490
is swinging in one direction,
one guy is swinging in the

00:05:03.490 --> 00:05:04.590
exact opposite direction.

00:05:04.590 --> 00:05:08.180
And the force just keeps
increasing, at some point,

00:05:08.180 --> 00:05:09.720
they're going to break.

00:05:09.720 --> 00:05:14.480
And you could say that well--
this guy says well, I was the

00:05:14.480 --> 00:05:17.650
only guy swinging because
this guy was stationery.

00:05:17.650 --> 00:05:18.880
And the other guy will say,
well, I was really guy

00:05:18.880 --> 00:05:21.530
swinging because this
guy was stationary.

00:05:21.530 --> 00:05:23.140
Not one of them is going to
break, they're both going to

00:05:23.140 --> 00:05:28.230
break because even though this
mauve sword was pushing on

00:05:28.230 --> 00:05:32.020
this blue sword with some force,
the blue sword was

00:05:32.020 --> 00:05:34.850
essentially pushing back with
the exact same force.

00:05:34.850 --> 00:05:36.780
So these are completely
identical swords.

00:05:36.780 --> 00:05:38.320
At some point, they're
going to break.

00:05:38.320 --> 00:05:39.780
Another way we could think about
it, one of the swords

00:05:39.780 --> 00:05:41.410
could have just been held.

00:05:41.410 --> 00:05:42.390
You know, stationary.

00:05:42.390 --> 00:05:44.620
It could've been held stationary
by somebody and

00:05:44.620 --> 00:05:47.530
this other sword that-- if you
were to press down on it, kept

00:05:47.530 --> 00:05:51.040
increasing the force with which
you press, at some point

00:05:51.040 --> 00:05:52.780
they both would break.

00:05:52.780 --> 00:05:53.780
Because they're identical.

00:05:53.780 --> 00:05:55.570
If one was harder than the
other, than that one would

00:05:55.570 --> 00:05:56.920
stick around.

00:05:56.920 --> 00:05:58.130
Hopefully that gives you
a little intuition.

00:05:58.130 --> 00:06:01.940
I mean we could do a bunch
of more examples.

00:06:01.940 --> 00:06:02.690
I'm trying to think.

00:06:02.690 --> 00:06:05.460
Oh, let me think
of another one.

00:06:05.460 --> 00:06:06.250
A less violent one.

00:06:06.250 --> 00:06:09.770
Let's say we're in deep space
again and I have a-- I don't

00:06:09.770 --> 00:06:11.330
know-- I have a basketball.

00:06:11.330 --> 00:06:15.100
Let me do it in orange.

00:06:15.100 --> 00:06:19.260
I have a basketball and
it weighs 1 kilogram.

00:06:21.930 --> 00:06:32.360
And let's say that I weigh 50
kilograms. So let's say I

00:06:32.360 --> 00:06:38.120
push-- so my hand pushes on this
ball with a force of-- I

00:06:38.120 --> 00:06:40.200
don't know-- let's say I push
on that ball with a force of

00:06:40.200 --> 00:06:41.582
10 Newtons.

00:06:41.582 --> 00:06:44.930
10 Newtons to the right.

00:06:44.930 --> 00:06:48.220
What Newton's third law tells
me is that essentially, that

00:06:48.220 --> 00:06:51.530
basketball is going to push on
my hand with an equal and

00:06:51.530 --> 00:06:52.240
opposite force.

00:06:52.240 --> 00:06:55.830
So it's going to push on me with
a force of 10 Newtons.

00:06:55.830 --> 00:06:57.200
So what's going to happen?

00:06:57.200 --> 00:06:58.700
So we're touching.

00:06:58.700 --> 00:07:00.850
I'm pushing on 10 Newtons on
the basketball and we're in

00:07:00.850 --> 00:07:01.570
deep space.

00:07:01.570 --> 00:07:04.310
There's no gravity from random
stars, et cetera.

00:07:04.310 --> 00:07:06.200
And then the basketball's going
to push on me with the

00:07:06.200 --> 00:07:09.040
force of 10 Newtons
simultaneously.

00:07:09.040 --> 00:07:12.290
We know F equals ma.

00:07:12.290 --> 00:07:17.090
So the basketball, so 10 Newtons
is equal to 1 kilogram

00:07:17.090 --> 00:07:18.030
times acceleration.

00:07:18.030 --> 00:07:21.700
So acceleration is going to be
10 meters per second squared

00:07:21.700 --> 00:07:22.460
to the right.

00:07:22.460 --> 00:07:24.580
So as long as we're touching,
the basketball's going to

00:07:24.580 --> 00:07:29.620
accelerate at 10 meters per
second squared to the right.

00:07:29.620 --> 00:07:33.410
And simultaneously, I'm going
to accelerate at a certain

00:07:33.410 --> 00:07:34.460
acceleration to the left.

00:07:34.460 --> 00:07:36.880
And what's that going to be?

00:07:36.880 --> 00:07:41.710
50 kilograms. We know that the
force to the left is also

00:07:41.710 --> 00:07:42.700
going to be 10 Newtons.

00:07:42.700 --> 00:07:46.470
That equals 50 kilograms
times acceleration.

00:07:46.470 --> 00:07:52.530
So here, the acceleration is
going to be 1/5 meters per

00:07:52.530 --> 00:07:54.700
second squared.

00:07:54.700 --> 00:07:57.470
So we're both in deep space
floating around and I push on

00:07:57.470 --> 00:08:01.710
this 1 kilogram basketball with
a force of 10 Newtons.

00:08:01.710 --> 00:08:05.350
As long as I'm pushing on it,
it's going to accelerate at 10

00:08:05.350 --> 00:08:07.950
meters per second squared.

00:08:07.950 --> 00:08:11.950
But simultaneously, while I'm
pushing on it, it's exerting

00:08:11.950 --> 00:08:14.630
an equal and opposite force
on me of 10 Newtons.

00:08:14.630 --> 00:08:18.980
So I'm going to actually move
back a little bit at a slower

00:08:18.980 --> 00:08:19.500
acceleration.

00:08:19.500 --> 00:08:21.690
That's just because
I have more mass.

00:08:21.690 --> 00:08:24.090
At 1/5 meters per
second squared.

00:08:24.090 --> 00:08:26.190
Another example you could
think of is if

00:08:26.190 --> 00:08:27.520
someone shoots a gun.

00:08:27.520 --> 00:08:29.100
There's that-- I forgot
the term because

00:08:29.100 --> 00:08:29.950
I don't shoot guns.

00:08:29.950 --> 00:08:34.929
But your shoulder jerks back
as I've seen in the movies

00:08:34.929 --> 00:08:36.500
when a bullet is shot.

00:08:36.500 --> 00:08:39.090
That's because the gun is
exerting a force on that

00:08:39.090 --> 00:08:42.429
bullet and the bullet is
exerting an equal and opposite

00:08:42.429 --> 00:08:45.360
force on the gun, which
kind of pushes

00:08:45.360 --> 00:08:46.450
back on your shoulder.

00:08:46.450 --> 00:08:49.940
And the reason why the bullet
just goes a lot, lot faster

00:08:49.940 --> 00:08:53.060
forward than you and the gun go
backwards is because your

00:08:53.060 --> 00:08:57.770
mass is much, much, much
larger than the bullet.

00:08:57.770 --> 00:08:59.450
Hopefully that gives you a
little bit of intuition on

00:08:59.450 --> 00:09:00.580
Newton's third law.

00:09:00.580 --> 00:09:02.150
And this is kind of
non-intuitive.

00:09:02.150 --> 00:09:05.320
So look around you in the world,
look at all the forces

00:09:05.320 --> 00:09:07.860
that are being applied, and I
want you to think about when

00:09:07.860 --> 00:09:11.050
one force is being applied in
one direction, why does it

00:09:11.050 --> 00:09:13.785
make sense that another force,
an equal and opposite force,

00:09:13.785 --> 00:09:16.590
is being applied in the exact
opposite direction?

00:09:16.590 --> 00:09:19.000
I'll see you all in
the next video.


Newton's First Law of Motion

WEBVTT
Kind: captions
Language: en

00:00:00.680 --> 00:00:01.730
Good morning.

00:00:01.730 --> 00:00:02.900
It's February already.

00:00:02.900 --> 00:00:05.760
I'm back from my hiatus.

00:00:05.760 --> 00:00:08.020
I was so burned out doing all
those SAT problems. But now

00:00:08.020 --> 00:00:10.840
I'm ready and I will start
doing some physics.

00:00:10.840 --> 00:00:12.950
So we had done a bunch of
projectile motion, what

00:00:12.950 --> 00:00:14.310
happens you throw something
in the air or

00:00:14.310 --> 00:00:15.270
drop it from a cliff.

00:00:15.270 --> 00:00:17.990
But now I want to introduce you
to is how do you actually

00:00:17.990 --> 00:00:20.080
affect the acceleration
of an object?

00:00:20.080 --> 00:00:23.560
And to do that I'm going to
introduce you to Newton's

00:00:23.560 --> 00:00:24.810
three laws.

00:00:27.520 --> 00:00:30.660
To some degree what we were
doing before was derivative of

00:00:30.660 --> 00:00:31.550
what I'm going to do now.

00:00:31.550 --> 00:00:34.720
But this is kind of the backbone
of classical physics.

00:00:34.720 --> 00:00:36.510
So Newton's three laws.

00:00:36.510 --> 00:00:38.950
And you've probably heard
of these before.

00:00:38.950 --> 00:00:41.820
Newtow's three laws.

00:00:41.820 --> 00:00:44.270
Sometimes they're called
Newton's Laws of Motion.

00:00:44.270 --> 00:00:46.160
I've actually looked this up on
the web just to make sure

00:00:46.160 --> 00:00:49.520
and see if there's any correct
way of writing it, but every

00:00:49.520 --> 00:00:51.470
website seems to have
a different

00:00:51.470 --> 00:00:53.430
paraphrase of the laws.

00:00:53.430 --> 00:00:55.480
But hopefully, I can give
you an intuitive sense

00:00:55.480 --> 00:00:56.310
of what they are.

00:00:56.310 --> 00:01:08.010
So the first law is an object
at rest. An object at rest

00:01:08.010 --> 00:01:21.250
tends to stay at rest. And
an object in motion

00:01:21.250 --> 00:01:22.430
tends to stay in motion.

00:01:22.430 --> 00:01:25.520
This is what I learned when I
was a kid and now when I look

00:01:25.520 --> 00:01:27.110
at Wikipedia and things, there
are some paraphrases.

00:01:27.110 --> 00:01:28.720
And we'll go over those
paraphrases because I think

00:01:28.720 --> 00:01:29.740
they're instructive.

00:01:29.740 --> 00:01:31.910
Stay in motion.

00:01:31.910 --> 00:01:33.690
And you might say, Sal,
this is obvious.

00:01:33.690 --> 00:01:35.300
Why does Newton get
so much credit

00:01:35.300 --> 00:01:36.130
for stating the obvious?

00:01:36.130 --> 00:01:40.020
Obviously, if I look at my sofa
for example, it is an

00:01:40.020 --> 00:01:44.030
object at rest and if I keep
staring at it, it tends to

00:01:44.030 --> 00:01:49.590
stay at rest. Likewise, when
I look at a car crossing an

00:01:49.590 --> 00:01:52.170
intersection-- that's not a red
light, that's crossing an

00:01:52.170 --> 00:01:54.520
intersection, it's an
object in motion.

00:01:54.520 --> 00:01:59.040
And then, I don't know-- 10
seconds later, it's still

00:01:59.040 --> 00:02:01.480
staying in motion and of course,
it will stay in motion

00:02:01.480 --> 00:02:03.120
unless you press the
brakes or whatever.

00:02:03.120 --> 00:02:05.620
So you might say, well
Sal, this is the most

00:02:05.620 --> 00:02:06.670
obvious thing ever.

00:02:06.670 --> 00:02:09.380
This doesn't even need
to be written down.

00:02:09.380 --> 00:02:12.840
But let's say you were Newton
and you came to me-- it was in

00:02:12.840 --> 00:02:14.060
the 17th century.

00:02:14.060 --> 00:02:15.790
And you said, Sal, I have
these new laws.

00:02:15.790 --> 00:02:18.630
And the first is an object at
rest tends to stay at rest,

00:02:18.630 --> 00:02:20.860
and an object in motion tends
to stay in motion.

00:02:20.860 --> 00:02:23.290
And I would say Newton, I can
already disprove your law.

00:02:23.290 --> 00:02:27.140
Let's say I have an apple and
I'm holding it up at let's say

00:02:27.140 --> 00:02:31.270
my-- I'm holding it up with
my arm, so it's roughly my

00:02:31.270 --> 00:02:32.720
shoulder level.

00:02:32.720 --> 00:02:33.800
So I'm holding an apple.

00:02:33.800 --> 00:02:35.690
This is an apple.

00:02:35.690 --> 00:02:37.530
Looks like a heart,
but it's an apple.

00:02:37.530 --> 00:02:41.610
So I'm holding it with my hand,
I'm drawing my hand.

00:02:41.610 --> 00:02:43.170
I don't know if that makes sense
to you, but I'm holding

00:02:43.170 --> 00:02:44.010
it with my hand.

00:02:44.010 --> 00:02:46.760
And what happens when I
let go of that apple?

00:02:46.760 --> 00:02:49.890
So while I'm holding it with
my hand it's an object at

00:02:49.890 --> 00:02:51.180
rest, right?

00:02:51.180 --> 00:02:52.490
But then when I let
go, what happens?

00:02:52.490 --> 00:02:53.740
It falls.

00:02:53.740 --> 00:02:54.840
Falls to the ground.

00:02:54.840 --> 00:02:57.990
So I'll say, Newton, I just
disproved your first law.

00:02:57.990 --> 00:03:00.170
Because this was an object at
rest. And I did nothing to it.

00:03:00.170 --> 00:03:00.980
I just let go.

00:03:00.980 --> 00:03:03.680
I didn't apply, I didn't push
it, I didn't pull it.

00:03:03.680 --> 00:03:04.400
I didn't throw it.

00:03:04.400 --> 00:03:05.480
I didn't do anything.

00:03:05.480 --> 00:03:09.610
And when I let go it just
fell to the ground.

00:03:09.610 --> 00:03:12.100
It started moving without me
doing anything, even though it

00:03:12.100 --> 00:03:13.410
was an object at rest.

00:03:13.410 --> 00:03:15.610
And then Newton will say, oh,
well that's because there's a

00:03:15.610 --> 00:03:16.660
thing called gravity.

00:03:16.660 --> 00:03:17.670
And it's the force of gravity.

00:03:17.670 --> 00:03:20.690
And I would say, Newton, you
need to start to learn to not

00:03:20.690 --> 00:03:22.480
make up things.

00:03:22.480 --> 00:03:26.350
Just because you're law doesn't
make sense, you don't

00:03:26.350 --> 00:03:28.960
need to make up artificial
forces in the universe.

00:03:28.960 --> 00:03:31.410
But anyway, he would
end up being right.

00:03:31.410 --> 00:03:34.700
And the way to think about this,
if I did this exact same

00:03:34.700 --> 00:03:41.090
experiment while I was in space
and let's just say-- I

00:03:41.090 --> 00:03:43.260
was going to say orbit because
it would look like that, but

00:03:43.260 --> 00:03:47.190
even orbit is kind of a-- you're
still kind of falling

00:03:47.190 --> 00:03:49.000
towards the earth, it's just
you're moving-- well, I won't

00:03:49.000 --> 00:03:49.520
go into that.

00:03:49.520 --> 00:03:51.060
I'll go into orbit
at another time.

00:03:51.060 --> 00:03:53.850
But let's say we were just in
deep space and me and the

00:03:53.850 --> 00:03:57.870
apple were just floating
around in space.

00:03:57.870 --> 00:03:59.100
Maybe we're stationary.

00:03:59.100 --> 00:03:59.690
It's hard to say.

00:03:59.690 --> 00:04:01.620
We're floating with
respect to what?

00:04:01.620 --> 00:04:05.530
And then, if we're in space and
I let go of this apple,

00:04:05.530 --> 00:04:07.090
what happens to the apple?

00:04:07.090 --> 00:04:07.590
Nothing.

00:04:07.590 --> 00:04:08.690
It's not going to
fall anywhere.

00:04:08.690 --> 00:04:09.950
It's not going to move.

00:04:09.950 --> 00:04:12.260
And so whenever you think about
Newton's laws-- and

00:04:12.260 --> 00:04:13.440
that's why this is so amazing.

00:04:13.440 --> 00:04:15.330
He didn't know about space.

00:04:15.330 --> 00:04:18.570
He's living in this planet that
everything tends to fall

00:04:18.570 --> 00:04:20.850
and things start moving for no
reason because of whatever,

00:04:20.850 --> 00:04:23.470
gravity, and the wind
and whatever else.

00:04:23.470 --> 00:04:26.580
And he actually theorized that
there could be a place where

00:04:26.580 --> 00:04:30.030
there's no forces acting on
objects where if I were to let

00:04:30.030 --> 00:04:33.260
go of this apple, it would
just stay where it is.

00:04:33.260 --> 00:04:35.060
And similarly, the
object in motion

00:04:35.060 --> 00:04:35.970
tends to stay in motion.

00:04:35.970 --> 00:04:37.820
And there again I would've
told Newton, well, that

00:04:37.820 --> 00:04:38.850
doesn't make sense.

00:04:38.850 --> 00:04:41.920
If I were to-- I don't know.

00:04:41.920 --> 00:04:47.470
If I were to push a-- well, I
don't know if they had bowling

00:04:47.470 --> 00:04:48.200
balls back then.

00:04:48.200 --> 00:04:51.730
But if I were to roll a bowling
ball down a-- well

00:04:51.730 --> 00:04:55.870
let's say up a hill-- At some
point that bowling ball's

00:04:55.870 --> 00:04:57.120
going to slow down.

00:04:59.250 --> 00:05:00.960
If I rolled it up a hill,
at some point it's just

00:05:00.960 --> 00:05:01.590
going to slow down.

00:05:01.590 --> 00:05:03.570
And maybe if I got it right it
would just stop at the top if

00:05:03.570 --> 00:05:05.500
I did it perfectly.

00:05:05.500 --> 00:05:07.170
And I could say, look, this
was an object in motion.

00:05:07.170 --> 00:05:09.750
At some point it stops or it
actually turns back around.

00:05:09.750 --> 00:05:12.990
Or even if I were to roll it
this way, at some point it's

00:05:12.990 --> 00:05:13.860
just going to stop.

00:05:13.860 --> 00:05:17.610
Right The bowling ball's
going to stop.

00:05:17.610 --> 00:05:21.080
If I were to push something as
hard as I could, maybe it

00:05:21.080 --> 00:05:23.900
travels for a couple of feet,
but then it's going to stop.

00:05:23.900 --> 00:05:25.730
And he'll say, oh, well you
know, there's these forces

00:05:25.730 --> 00:05:27.230
that you're not realizing
there's a force.

00:05:27.230 --> 00:05:31.100
There's the wind resistance in
the bowling ball example.

00:05:31.100 --> 00:05:34.250
There's the force of friction
in the example where I just

00:05:34.250 --> 00:05:34.540
pushed something.

00:05:34.540 --> 00:05:36.370
And I would've said, well
Newton, you're just making up

00:05:36.370 --> 00:05:37.340
these forces again.

00:05:37.340 --> 00:05:38.690
And this is why this is
so not intuitive.

00:05:38.690 --> 00:05:42.400
Because he had to essentially
realize that there were all of

00:05:42.400 --> 00:05:45.280
these forces acting on something
when to someone at

00:05:45.280 --> 00:05:47.110
that time, you wouldn't have
realized that and you wouldn't

00:05:47.110 --> 00:05:49.420
have been able to even conceive
that there's a place

00:05:49.420 --> 00:05:52.830
called space, for example,
where these

00:05:52.830 --> 00:05:54.100
things wouldn't happen.

00:05:54.100 --> 00:05:57.820
If I push something in space,
it will keep going.

00:05:57.820 --> 00:06:00.900
It would be an object in motion
and it will keep that

00:06:00.900 --> 00:06:03.800
velocity until some other
force acts on it.

00:06:03.800 --> 00:06:05.860
So it wasn't that intuitive.

00:06:05.860 --> 00:06:13.050
And so a more modern way to
write this is to say that

00:06:13.050 --> 00:06:16.440
there is a frame of reference,
there exists a frame of

00:06:16.440 --> 00:06:18.460
reference-- and I'll explain
what a frame of reference is.

00:06:18.460 --> 00:06:22.810
But there exists a frame of
reference where this is true.

00:06:22.810 --> 00:06:25.530
That could be the new way
of saying Newton's

00:06:25.530 --> 00:06:27.160
first law of motion.

00:06:27.160 --> 00:06:29.850
So what's a frame
of reference?

00:06:29.850 --> 00:06:32.430
So everything in physics--
if I'm moving,

00:06:32.430 --> 00:06:33.710
moving relative to what?

00:06:33.710 --> 00:06:35.290
Moving relative to
the observer?

00:06:35.290 --> 00:06:36.480
Moving relative to the earth?

00:06:36.480 --> 00:06:37.490
You don't know.

00:06:37.490 --> 00:06:40.700
So a frame of reference is what
is the observer doing?

00:06:40.700 --> 00:06:45.870
So example: when I'm in space
and I let go of the apple, me

00:06:45.870 --> 00:06:48.810
and the apple are kind of in
this-- I am observing the

00:06:48.810 --> 00:06:52.420
apple from what I call an
inertial frame of reference.

00:06:52.420 --> 00:06:53.850
So this is a frame of reference
actually where

00:06:53.850 --> 00:06:55.380
Newton's laws hold.

00:06:55.380 --> 00:07:00.210
If I take the apple on earth and
I let go and it drops, the

00:07:00.210 --> 00:07:02.450
reason why this first law didn't
hold is because I'm not

00:07:02.450 --> 00:07:04.450
really in an inertial
frame of reference.

00:07:04.450 --> 00:07:08.190
Because me and the apple are
both constantly being pulled

00:07:08.190 --> 00:07:10.590
on by this force
called gravity.

00:07:10.590 --> 00:07:13.610
So although it looks like
nothing's going on, me and the

00:07:13.610 --> 00:07:17.260
apple are in the same--
nothing's really acting on us.

00:07:17.260 --> 00:07:17.830
There is.

00:07:17.830 --> 00:07:19.380
There's this force
of acceleration.

00:07:19.380 --> 00:07:21.930
Similarly, if I'm in a
car and that car is

00:07:21.930 --> 00:07:22.880
accelerating, right?

00:07:22.880 --> 00:07:28.085
So let's say the car-- looks
more like a pickup truck, so

00:07:28.085 --> 00:07:29.335
I'll go with the pickup truck.

00:07:33.430 --> 00:07:36.425
Let's say you have a pair of
dice hanging from your rear

00:07:36.425 --> 00:07:37.510
view mirror.

00:07:37.510 --> 00:07:38.950
This is the dice right here.

00:07:38.950 --> 00:07:41.890
What happens when the
car accelerates?

00:07:41.890 --> 00:07:44.150
Well the dice move
back, right?

00:07:44.150 --> 00:07:48.810
And so when you're sitting in
the truck itself, it looks

00:07:48.810 --> 00:07:50.670
like the dice are just
moving back.

00:07:50.670 --> 00:07:53.110
No one's really doing
anything to it.

00:07:53.110 --> 00:07:55.950
Let's say the car had no windows
and you would just all

00:07:55.950 --> 00:07:58.573
of a sudden mysteriously feel--
well, you'd feel a

00:07:58.573 --> 00:08:00.600
little squeezing on your chest
too, but you would also just

00:08:00.600 --> 00:08:02.020
see these dice move back.

00:08:02.020 --> 00:08:03.300
And you'd say, hey.

00:08:03.300 --> 00:08:05.040
Newton's first law
doesn't hold.

00:08:05.040 --> 00:08:07.100
And what I would say is well
that's because you're in a

00:08:07.100 --> 00:08:08.980
non-inertial frame
of reference.

00:08:08.980 --> 00:08:11.170
To someone outside of the truck,
they would see, oh

00:08:11.170 --> 00:08:13.660
well, the truck is moving, the
truck is actually accelerating

00:08:13.660 --> 00:08:16.150
and that's why the
dice move back.

00:08:16.150 --> 00:08:20.210
So in I guess you could say the
horizontal dimension, and

00:08:20.210 --> 00:08:21.980
I'm probably just confusing you,
but I want to give you a

00:08:21.980 --> 00:08:25.660
really intuitive feel about why
this isn't so intuitive.

00:08:25.660 --> 00:08:28.940
In the horizontal direction,
because there are no forces of

00:08:28.940 --> 00:08:32.760
gravity or whatever acting in
this direction, and if I'm

00:08:32.760 --> 00:08:35.880
outside of the truck, I could
then-- I would be in an

00:08:35.880 --> 00:08:37.900
inertial frame of reference
in at least

00:08:37.900 --> 00:08:38.860
the horizontal dimension.

00:08:38.860 --> 00:08:41.159
I mean we always have gravity
pulling down on us.

00:08:41.159 --> 00:08:43.870
But from the outside of the
truck, I could observe that

00:08:43.870 --> 00:08:48.080
oh, you know, Newton's law holds
because the whole frame

00:08:48.080 --> 00:08:51.340
of reference, this truck is
actually being accelerated.

00:08:51.340 --> 00:08:54.440
So me being outside of that, I
would be in an inertial frame

00:08:54.440 --> 00:08:55.870
of reference.

00:08:55.870 --> 00:08:58.570
Hopefully I haven't confused
you too much.

00:08:58.570 --> 00:09:00.710
The way to think about it is
that an inertial frame of

00:09:00.710 --> 00:09:03.860
reference is just a frame of
reference where there's no

00:09:03.860 --> 00:09:07.070
outside forces acting on the
whole frame of reference.

00:09:07.070 --> 00:09:10.870
And a frame of reference
is just what is

00:09:10.870 --> 00:09:11.680
the observer doing?

00:09:11.680 --> 00:09:12.450
What am I doing?

00:09:12.450 --> 00:09:13.900
Am I moving with the object?

00:09:13.900 --> 00:09:15.760
Am I being accelerated
with the object?

00:09:15.760 --> 00:09:19.840
Or, are neither me nor the
object being acted upon?

00:09:19.840 --> 00:09:22.245
And that's the way to
think about it.

00:09:22.245 --> 00:09:23.270
Oh, I already ran out of time.

00:09:23.270 --> 00:09:24.540
I only got one law done.

00:09:24.540 --> 00:09:26.570
So I'll see you in
the next video.


Ferris Wheel Trig Problem (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:03.410
In the last part of the problem
we figured out that the

00:00:03.410 --> 00:00:06.420
function of the height of the
ferris wheel, the people at the

00:00:06.420 --> 00:00:10.140
ferris wheel at any time is a
function of t. h of t is equal

00:00:10.140 --> 00:00:18.590
to 9 minus 8 cosine of 18t
where t is in seconds.

00:00:18.590 --> 00:00:21.730
Now the second part of this
problem they want us to graph h

00:00:21.730 --> 00:00:28.530
as a function of t between 0 is
less than or equal to t, is

00:00:28.530 --> 00:00:30.800
less than or equal to 30.

00:00:30.800 --> 00:00:34.500
So let me draw axes.

00:00:34.500 --> 00:00:41.690
So let's say that
that's my h-axis.

00:00:41.690 --> 00:00:43.660
Let's say that this
is my t-axis.

00:00:53.280 --> 00:00:56.845
So this is t equals 0, and this
is t is equal to 30 seconds.

00:01:00.740 --> 00:01:03.660
I get confused when I see
this 18 here or whatever.

00:01:03.660 --> 00:01:06.040
So what I'm going to do first
of all is I'm going to graph a

00:01:06.040 --> 00:01:08.230
different function, slightly
different function, then I'll

00:01:08.230 --> 00:01:09.870
translate it to this function.

00:01:09.870 --> 00:01:15.820
I'm going to graph h of
theta is equal to 9

00:01:15.820 --> 00:01:23.430
minus 8 cosine of theta.

00:01:23.430 --> 00:01:25.010
I think you'll see where
I'm going with this

00:01:25.010 --> 00:01:26.320
when I'm all done.

00:01:26.320 --> 00:01:28.600
So let's try to graph h
of theta is equal to 9

00:01:28.600 --> 00:01:31.260
minus 8 cosine of theta.

00:01:31.260 --> 00:01:33.810
So when t is equal
to 30 seconds, what

00:01:33.810 --> 00:01:35.090
is theta equal to?

00:01:35.090 --> 00:01:38.800
So 30 times 18, that's 540.

00:01:38.800 --> 00:01:40.780
So this is 540 degrees.

00:01:40.780 --> 00:01:41.370
Same thing.

00:01:41.370 --> 00:01:44.340
I'll write the thetas in
red above the t-axis.

00:01:44.340 --> 00:01:47.310
This is 540 degrees, so
that's like two times

00:01:47.310 --> 00:01:47.940
around the circle.

00:01:47.940 --> 00:01:51.470
So that's 540 degrees,
then this is going to

00:01:51.470 --> 00:01:53.890
be roughly 270 degrees.

00:01:58.400 --> 00:02:01.450
So, 90 degrees will be
about 1/3 of this.

00:02:01.450 --> 00:02:04.830
That would be 90 degrees, that
would be 180, so that would be

00:02:04.830 --> 00:02:11.680
90 degrees, that would be 180
degrees, this would be 360

00:02:11.680 --> 00:02:19.910
degrees, and this would be 360
plus 90 so this will

00:02:19.910 --> 00:02:22.140
be 450 degrees.

00:02:22.140 --> 00:02:24.390
If you wanted to figure out
the corresponding time,

00:02:24.390 --> 00:02:27.680
you just take this
degree and divide by 18.

00:02:27.680 --> 00:02:31.300
So it's 90 divided
by 18 is what?

00:02:31.300 --> 00:02:34.175
It's five, right?

00:02:34.175 --> 00:02:39.360
So if I were to write here,
this is at 5 seconds, this is

00:02:39.360 --> 00:02:45.400
at 10 seconds, this is 15
seconds, this is 20 seconds,

00:02:45.400 --> 00:02:51.910
this is -- sorry, this is 25
seconds, this is 30 seconds.

00:02:51.910 --> 00:02:56.140
Actually, a simple thing we
could do is let's just figure

00:02:56.140 --> 00:02:59.640
out what the value of the
function is at these points.

00:02:59.640 --> 00:03:02.690
Because these are pretty
easy degrees to figure out

00:03:02.690 --> 00:03:04.810
what the cosine value is.

00:03:04.810 --> 00:03:09.160
So let's figure out --
let me draw a table.

00:03:09.160 --> 00:03:11.740
Tables are always good and
I'll do it in yellow.

00:03:11.740 --> 00:03:17.520
So I'll draw a t theta and h.

00:03:20.720 --> 00:03:23.160
This might be kind of an
unconventional way of doing

00:03:23.160 --> 00:03:26.170
things, but I have a simple
mind so this is actually

00:03:26.170 --> 00:03:27.680
how I like to do it.

00:03:27.680 --> 00:03:34.460
So I like to think of
theta as 0, 90, 180,

00:03:34.460 --> 00:03:42.810
270, 360, 450 and 540.

00:03:42.810 --> 00:03:47.160
And t, the corresponding
time of those, as 0,

00:03:47.160 --> 00:03:54.170
5, 10, 15, 20, 25, 30.

00:03:54.170 --> 00:03:55.850
It's not rocket science here.

00:03:55.850 --> 00:04:00.470
When t equals 15 seconds,
15 times 18, we're trying

00:04:00.470 --> 00:04:04.120
to find the cosine of
270 degrees, right?

00:04:04.120 --> 00:04:06.080
15 times 18 is 270 degrees.

00:04:06.080 --> 00:04:07.740
I'm just doing this because I
don't have a calculator and

00:04:07.740 --> 00:04:10.480
this will help me
pick good points.

00:04:10.480 --> 00:04:13.850
So when t is equal to
0, what is height?

00:04:13.850 --> 00:04:17.750
Or t is equal to 0, theta is
equal to 0, so cosine of theta

00:04:17.750 --> 00:04:19.800
is -- cosine of 0 is 1.

00:04:19.800 --> 00:04:23.090
So 9 minus 8 is 1.

00:04:23.090 --> 00:04:24.610
I'm going to do h in
a different color.

00:04:24.610 --> 00:04:26.540
So this is 1.

00:04:26.540 --> 00:04:28.430
Cosine of 90 degrees?

00:04:28.430 --> 00:04:30.180
Cosine of 90 degrees is 0.

00:04:30.180 --> 00:04:34.170
So 9 minus 0 is 9.

00:04:34.170 --> 00:04:35.885
Cosine of 180 degrees?

00:04:35.885 --> 00:04:37.960
So we're going all the way
around the unit circle.

00:04:37.960 --> 00:04:40.310
Cosine of 180
degrees is minus 1.

00:04:40.310 --> 00:04:47.770
So minus 1 times minus 8 is
plus 8, so 9 plus 8, that's 15.

00:04:47.770 --> 00:04:50.500
Cosine of 270 degrees are
pointing straight down, so the

00:04:50.500 --> 00:04:51.670
x-coord is going to be 0.

00:04:51.670 --> 00:04:55.540
So once again,
we're at 9 again.

00:04:55.540 --> 00:04:56.930
9 minus 0.

00:04:56.930 --> 00:04:57.900
360 degrees.

00:04:57.900 --> 00:04:59.900
Cosine of 360 degrees
is the same thing as

00:04:59.900 --> 00:05:02.390
cosine of 0, right?

00:05:02.390 --> 00:05:07.080
So once again, I mean we've
gone around the circle once.

00:05:07.080 --> 00:05:10.480
So it's going to be the same
as 0, so it's going to be 1.

00:05:10.480 --> 00:05:12.420
And 450 is going to be
the same thing as 90.

00:05:12.420 --> 00:05:16.250
So it's going to be 9
and then 15 degrees.

00:05:16.250 --> 00:05:19.690
So let's plot these points.

00:05:19.690 --> 00:05:22.980
Actually, let me just
draw 15 up here.

00:05:22.980 --> 00:05:24.820
So what are the points
that keep showing up?

00:05:24.820 --> 00:05:33.310
So this is 1, that's 1, that's
1, and then we have 9.

00:05:33.310 --> 00:05:44.685
1, there's 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12, 13, 14, 15.

00:05:48.320 --> 00:05:50.970
Fair enough.

00:05:50.970 --> 00:05:54.380
So let me draw some
guidelines just to help us.

00:05:57.760 --> 00:06:02.270
Actually, let me do them kind
of hard to see, because I

00:06:02.270 --> 00:06:07.400
don't want to draw too much
attention to the guidelines.

00:06:07.400 --> 00:06:10.100
I could do one guideline there.

00:06:12.970 --> 00:06:16.075
Then I'll do a bottom
guideline right there.

00:06:20.040 --> 00:06:21.690
Then the 9 keeps showing up.

00:06:25.020 --> 00:06:26.260
Oh, you know what, I can't add.

00:06:26.260 --> 00:06:28.060
What's 9 plus 8?

00:06:28.060 --> 00:06:30.330
It's not 15, it's 17.

00:06:30.330 --> 00:06:37.580
Sorry, clearly, I need to
practice my addition.

00:06:37.580 --> 00:06:39.970
So this is 9 plus
8, this is 17.

00:06:39.970 --> 00:06:41.890
And I realized that because I
was like, well 9 should be in

00:06:41.890 --> 00:06:43.820
the middle, so this
is actually 17.

00:06:43.820 --> 00:06:45.820
Ignore my little marks here.

00:06:45.820 --> 00:06:48.520
That's 17, this is 1.

00:06:48.520 --> 00:06:49.470
Ignore the marks.

00:06:49.470 --> 00:06:54.130
9 would be right in the
middle between 1 and 17.

00:06:54.130 --> 00:06:58.290
So let me draw kind of
mediant point right there.

00:06:58.290 --> 00:06:59.820
So this is 9.

00:06:59.820 --> 00:07:03.150
Sorry I can't add properly.

00:07:03.150 --> 00:07:06.230
Then let's draw the graph
or at least plot the

00:07:06.230 --> 00:07:07.820
points on the graph.

00:07:07.820 --> 00:07:15.240
So, t equals 0 where h equals
1, so that's this point.

00:07:15.240 --> 00:07:18.020
That's right here.

00:07:18.020 --> 00:07:23.540
When t equals 5, h is
equal to 9, right here.

00:07:23.540 --> 00:07:26.350
When t is equal to 10,
h is equal to 17.

00:07:30.380 --> 00:07:34.280
When t is equal to
15, h is 9 again.

00:07:34.280 --> 00:07:37.510
So it's right here.

00:07:37.510 --> 00:07:40.040
At 20 we're back at 1.

00:07:40.040 --> 00:07:41.590
I think you see the pattern.

00:07:41.590 --> 00:07:44.570
At 25 we're back at 9.

00:07:44.570 --> 00:07:49.500
And then at 30 we're back at
17, not 15, because now I

00:07:49.500 --> 00:07:51.330
have corrected my mistake.

00:07:51.330 --> 00:07:53.270
And this is going to be sined
graph, it's going to look

00:07:53.270 --> 00:07:56.360
something like this.

00:07:56.360 --> 00:07:59.080
Let me do it in a vibrant color
so I can overwrite everything

00:07:59.080 --> 00:08:01.130
and it's going to look
something like this.

00:08:01.130 --> 00:08:05.466
Go oops, and then up
and them down here.

00:08:11.180 --> 00:08:19.100
Curve up, come back down, curve
up and then come back down.

00:08:19.100 --> 00:08:20.060
Like that.

00:08:20.060 --> 00:08:21.490
So that's our graph.

00:08:21.490 --> 00:08:23.850
I think in the problem they
tell us to approximate.

00:08:23.850 --> 00:08:30.910
Actually, let me open up my
cousin's problem -- my other

00:08:30.910 --> 00:08:35.120
account has timed-out on
me while I recorded this.

00:08:35.120 --> 00:08:40.080
They wanted to approximate when
t equals 4 what the height is.

00:08:40.080 --> 00:08:43.030
So when t equals 4 the
height is like right

00:08:43.030 --> 00:08:44.440
around there, right?

00:08:44.440 --> 00:08:46.650
So the height is a
little bit less than 9.

00:08:46.650 --> 00:08:50.500
And I don't know, 7 or
8 meters in the air.

00:08:50.500 --> 00:08:53.770
And when time is equal to 10 --
well, time equal 10, we figured

00:08:53.770 --> 00:08:57.930
out exactly, we know that there
are 17 meters in the air.

00:08:57.930 --> 00:09:00.510
So I know this was kind of a
little messy and graphing

00:09:00.510 --> 00:09:03.770
trick functions tend to be,
but hopefully you found

00:09:03.770 --> 00:09:06.160
this vaguely useful.

00:09:06.160 --> 00:09:07.620
Have fun.


Ferris Wheel Trig Problem

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:01.890
All right, I have
a problem here.

00:00:01.890 --> 00:00:04.840
Jacob and Emily ride a
ferris wheel at a carnival

00:00:04.840 --> 00:00:06.290
in Billings, Montana.

00:00:06.290 --> 00:00:09.280
The wheel has a
16-meter diameter.

00:00:09.280 --> 00:00:10.440
So let me draw the wheel.

00:00:10.440 --> 00:00:13.110
It has a 16-meter diameter.

00:00:13.110 --> 00:00:15.225
So let me draw it big.

00:00:15.225 --> 00:00:16.886
Give me a lot of space.

00:00:16.886 --> 00:00:20.910
So it has a 16-meter
diameter, so what's its

00:00:20.910 --> 00:00:21.710
radius going to be?

00:00:21.710 --> 00:00:24.570
Its radius is going to
be half of that, right?

00:00:24.570 --> 00:00:33.280
So if I were to draw its
radius, just draw it like that.

00:00:36.370 --> 00:00:39.780
It's a 16-meter diameter, so
it's radius is going to be 8

00:00:39.780 --> 00:00:44.390
meters with its lowest point
above the ground-- oh,

00:00:44.390 --> 00:00:46.540
with its lowest point 1
meter above the ground.

00:00:46.540 --> 00:00:49.690
So its lowest point
is right here.

00:00:49.690 --> 00:00:53.580
This is its lowest point,
and that is 1 meter

00:00:53.580 --> 00:00:54.270
above the ground.

00:00:54.270 --> 00:00:57.920
So this distance right
here is 1 meter.

00:00:57.920 --> 00:00:58.570
Fair enough.

00:01:01.180 --> 00:01:05.335
Assume that Jacob and Emily's
height h above the ground is a

00:01:05.335 --> 00:01:10.100
sinusoidal function of time
where t equals 0 represents the

00:01:10.100 --> 00:01:11.700
lowest point on the wheel.

00:01:11.700 --> 00:01:15.720
So this is at point t equals 0
right here. t equals 0 is the

00:01:15.720 --> 00:01:16.860
lowest point of the wheel.

00:01:16.860 --> 00:01:18.440
Write an equation for h.

00:01:18.440 --> 00:01:21.060
Oh, I think I forgot,
so let me reread it.

00:01:21.060 --> 00:01:23.490
Jacob and Emily ride a
ferris wheel at a carnival

00:01:23.490 --> 00:01:24.170
in Billings, Montana.

00:01:24.170 --> 00:01:26.030
The wheel has a 16-meter
diameter-- we did

00:01:26.030 --> 00:01:29.450
that-- and turns at 3
revolutions per minute.

00:01:29.450 --> 00:01:34.910
So it turns at 3 revolutions
per minute with its lowest

00:01:34.910 --> 00:01:36.570
point 1 meter above the ground.

00:01:36.570 --> 00:01:39.790
Assume that Jacob and Emily's
height h above the ground is

00:01:39.790 --> 00:01:43.250
a sinusoidal function of
time, where t equals 0.

00:01:43.250 --> 00:01:46.160
So we need to write a function
of h, their distance above the

00:01:46.160 --> 00:01:49.200
ground, as a function of time,
and they're saying that

00:01:49.200 --> 00:01:51.280
time is given in seconds.

00:01:51.280 --> 00:01:53.860
So, first of all, they're
telling us 3 revolutions

00:01:53.860 --> 00:01:55.760
every minute, right?

00:01:55.760 --> 00:02:03.950
So that's 3 revolutions per
60 seconds, and that's the

00:02:03.950 --> 00:02:08.500
same thing as 1 revolution
per 20 seconds, right?

00:02:08.500 --> 00:02:13.630
I just divide both sides of
the per by 3 for 20 seconds.

00:02:13.630 --> 00:02:15.710
And one revolution is
how many degrees?

00:02:15.710 --> 00:02:17.840
One revolution is 360 degrees.

00:02:17.840 --> 00:02:27.420
So it's 360 degrees
per 20 seconds.

00:02:27.420 --> 00:02:30.280
And if you're going 360 degrees
per 20 seconds, let's divide--

00:02:30.280 --> 00:02:32.980
you know, the per you can just
kind of use as an equal

00:02:32.980 --> 00:02:35.270
sign of the equation.

00:02:35.270 --> 00:02:37.180
360 degrees for 20 seconds.

00:02:37.180 --> 00:02:38.270
That means you're
going to go what?

00:02:38.270 --> 00:02:40.740
18 degrees, Just divide
both sides by 20.

00:02:40.740 --> 00:02:46.895
18 degrees per second.

00:02:49.430 --> 00:02:51.450
And we could have done it with
a numerator and a denominator.

00:02:51.450 --> 00:02:54.260
3 revs per-- you know,
you could have said 3

00:02:54.260 --> 00:02:57.940
revs over 60 seconds.

00:02:57.940 --> 00:02:59.140
That's actually how I
should have done it.

00:02:59.140 --> 00:03:05.430
3 revs over 60 seconds is equal
to 1 rev over 20 seconds, which

00:03:05.430 --> 00:03:11.600
is equal to 360 degrees over 20
seconds, which is equal to 18

00:03:11.600 --> 00:03:15.790
degrees per second, right?

00:03:15.790 --> 00:03:18.350
So we're going to travel
18 degrees per second.

00:03:18.350 --> 00:03:23.510
So the total number of degrees
we've traveled in t seconds is

00:03:23.510 --> 00:03:29.890
going to be-- so see, if we say
the angle, that's the angle

00:03:29.890 --> 00:03:31.180
from our starting point.

00:03:31.180 --> 00:03:37.270
So let's say we've traveled t
seconds, and we're right there.

00:03:37.270 --> 00:03:42.460
What is-- let's drop a
little altitude right here.

00:03:42.460 --> 00:03:46.270
What is this angle going to be,
where this angle is right here?

00:03:46.270 --> 00:03:48.070
What is this angle going to be?

00:03:48.070 --> 00:03:51.000
How many degrees
have we traveled?

00:03:51.000 --> 00:03:54.180
Well, we say we traveled 18
degrees per second, so if we

00:03:54.180 --> 00:04:03.090
travel t seconds, this is going
to be 18t degrees, right?

00:04:03.090 --> 00:04:05.920
All right, so let's see if we
can figure out how their height

00:04:05.920 --> 00:04:09.600
as a function of this-- well,
as a function of t or as

00:04:09.600 --> 00:04:12.170
a function of this
angle right here.

00:04:12.170 --> 00:04:16.270
So what is this
height right here?

00:04:16.270 --> 00:04:17.220
Up here?

00:04:17.220 --> 00:04:20.420
It's 1 meter plus the radius
because this distance

00:04:20.420 --> 00:04:21.710
right here is 8.

00:04:21.710 --> 00:04:25.000
So we could say that this is--
this point right here is h is

00:04:25.000 --> 00:04:27.420
equal to 9 at this
point, right?

00:04:27.420 --> 00:04:29.650
We could almost view
that as the h axis.

00:04:29.650 --> 00:04:30.990
So that's h is equal to 9.

00:04:30.990 --> 00:04:34.020
So at this point,
how high are they?

00:04:34.020 --> 00:04:37.430
If this is h-- so right
now, let me draw a little

00:04:37.430 --> 00:04:41.810
drop and go flat here.

00:04:41.810 --> 00:04:47.510
So their height above the
ground is this distance h,

00:04:47.510 --> 00:04:53.280
which is the same thing
as this distance h.

00:04:53.280 --> 00:04:55.050
So what is that distance?

00:04:55.050 --> 00:04:58.100
Well, it's going to be-- well,
if this distance is h, what is

00:04:58.100 --> 00:05:00.590
this distance going to be?

00:05:00.590 --> 00:05:03.540
This distance is going
to be 9 minus h.

00:05:03.540 --> 00:05:04.060
How do I know this?

00:05:04.060 --> 00:05:06.170
This whole distance is 9.

00:05:06.170 --> 00:05:09.770
This distance is h, so-- let me
do it in a better color-- so

00:05:09.770 --> 00:05:15.690
that this distance right
here is 9 minus h.

00:05:15.690 --> 00:05:17.410
So let's see what we can do.

00:05:17.410 --> 00:05:17.860
What do we know?

00:05:17.860 --> 00:05:19.550
We know this distance.

00:05:19.550 --> 00:05:23.300
We know this angle
is 18t degrees.

00:05:23.300 --> 00:05:24.720
And do we know this side?

00:05:24.720 --> 00:05:25.010
Sure.

00:05:25.010 --> 00:05:25.730
That's the radius.

00:05:25.730 --> 00:05:26.490
That's 8.

00:05:26.490 --> 00:05:28.610
8 meters.

00:05:28.610 --> 00:05:32.780
9 minus h meters, 8
meters, and 18 degrees.

00:05:32.780 --> 00:05:36.260
And what are these sides
relative to this angle?

00:05:36.260 --> 00:05:42.800
Well, if we were to draw a
triangle here relative to this

00:05:42.800 --> 00:05:47.400
angle right here, the 9 minus h
is adjacent, and the 8 meters

00:05:47.400 --> 00:05:51.430
is, of course, the
hypotenuse, right?

00:05:51.430 --> 00:05:55.780
So what trig function deals
with adjacent and hypotenuse.

00:05:55.780 --> 00:05:57.500
SOHCAHTOA.

00:05:57.500 --> 00:06:01.120
CAH, cosine is adjacent
over hypotenuse.

00:06:01.120 --> 00:06:05.900
So we could say the cosine at
18 degrees, the cosine of 18t

00:06:05.900 --> 00:06:11.445
degrees, is equal to
its adjacent side.

00:06:11.445 --> 00:06:14.462
The adjacent side is 9 minus h.

00:06:14.462 --> 00:06:21.670
It's equal to 9 minus h over
the hypotenuse, over 8.

00:06:21.670 --> 00:06:24.720
And now we can solve for
h, and we'll have h

00:06:24.720 --> 00:06:25.910
as a function of t.

00:06:25.910 --> 00:06:27.810
So we multiply both sides by 8.

00:06:27.810 --> 00:06:37.090
You get 8 cosine of 18t
is equal to 9 minus h.

00:06:37.090 --> 00:06:39.870
Maybe we could subtract
9 from both sides.

00:06:39.870 --> 00:06:48.190
So we get minus 9 plus 8 cosine
of 18t is equal to minus h, and

00:06:48.190 --> 00:06:52.340
then multiply both sides by
negative 1, and then you get

00:06:52.340 --> 00:07:00.130
9-- positive 9, right-- minus 8
cosine of 18t is equals

00:07:00.130 --> 00:07:10.500
to h, or h is equal to 9
minus 8 cosine of 18t.

00:07:10.500 --> 00:07:11.330
So there we have it.

00:07:11.330 --> 00:07:16.230
We have expressed h
as a function of t.

00:07:16.230 --> 00:07:18.050
And in the next video,
I'm actually going to

00:07:18.050 --> 00:07:18.835
graph this function.

00:07:18.835 --> 00:07:20.420
See you soon.