Introduction to Newton's law of gravitation

WEBVTT
Kind: captions
Language: en

00:00:00.660 --> 00:00:02.590
We're now going to learn a
little bit about gravity.

00:00:02.590 --> 00:00:05.590
And just so you know, gravity is
something that, especially

00:00:05.590 --> 00:00:08.620
in introductory physics or
even reasonably advanced

00:00:08.620 --> 00:00:11.600
physics, we can learn how to
calculate it, we can learn how

00:00:11.600 --> 00:00:14.650
to realize what are the
important variables in it, but

00:00:14.650 --> 00:00:17.050
it's something that's really
not well understood.

00:00:17.050 --> 00:00:20.940
Even once you learn general
relativity, if you do get

00:00:20.940 --> 00:00:23.880
there, I have to say, you can
kind of say, oh, well, it's

00:00:23.880 --> 00:00:26.120
the warping of space time and
all of this, but it's hard to

00:00:26.120 --> 00:00:31.190
get an intuition of why two
objects, just because they

00:00:31.190 --> 00:00:33.690
have this thing called
mass, they are

00:00:33.690 --> 00:00:34.640
attracted to each other.

00:00:34.640 --> 00:00:38.170
It's really, at least to me,
a little bit mystical.

00:00:38.170 --> 00:00:42.240
But with that said, let's learn
to deal with gravity.

00:00:42.240 --> 00:00:45.140
And we'll do that learning
Newton's Law of Gravity, and

00:00:45.140 --> 00:00:48.620
this works for most purposes.

00:00:48.620 --> 00:00:50.640
So Newton's Law of Gravity says
that the force between

00:00:50.640 --> 00:00:55.140
two masses, and that's the
gravitational force, is equal

00:00:55.140 --> 00:00:58.990
to the gravitational constant G
times the mass of the first

00:00:58.990 --> 00:01:03.500
object times the mass of the
second object divided by the

00:01:03.500 --> 00:01:06.890
distance between the two
objects squared.

00:01:06.890 --> 00:01:07.860
So that's simple enough.

00:01:07.860 --> 00:01:10.320
So let's play around with this,
and see if we can get

00:01:10.320 --> 00:01:13.490
some results that look
reasonably familiar to us.

00:01:13.490 --> 00:01:17.120
So let's use this formula
to figure out what the

00:01:17.120 --> 00:01:21.070
acceleration, the gravitational
acceleration, is

00:01:21.070 --> 00:01:23.120
at the surface of the Earth.

00:01:23.120 --> 00:01:26.130
So let's draw the Earth,
just so we know what

00:01:26.130 --> 00:01:27.380
we're talking about.

00:01:29.830 --> 00:01:31.630
So that's my Earth.

00:01:31.630 --> 00:01:34.060
And let's say we want to figure
out the gravitational

00:01:34.060 --> 00:01:36.260
acceleration on Sal.

00:01:36.260 --> 00:01:37.510
That's me.

00:01:41.000 --> 00:01:45.770
And so how do we apply this
equation to figure out how

00:01:45.770 --> 00:01:49.490
much I'm accelerating down
towards the center of Earth or

00:01:49.490 --> 00:01:51.950
the Earth's center of mass?

00:01:51.950 --> 00:01:55.890
The force is equal to-- so
what's this big G thing?

00:01:55.890 --> 00:02:00.010
The G is the universal
gravitational constant.

00:02:00.010 --> 00:02:03.330
Although, as far as I know, and
I'm not an expert on this,

00:02:03.330 --> 00:02:08.620
I actually think its measurement
can change.

00:02:08.620 --> 00:02:11.390
It's not truly, truly a
constant, or I guess when on

00:02:11.390 --> 00:02:12.840
different scales, it can be
a little bit different.

00:02:12.840 --> 00:02:16.700
But for our purposes, it is a
constant, and the constant in

00:02:16.700 --> 00:02:21.520
most physics classes, is this:
6.67 times 10 to the negative

00:02:21.520 --> 00:02:24.546
11th meters cubed per kilogram
seconds squared.

00:02:24.546 --> 00:02:27.470
I know these units are crazy,
but all you have to realize is

00:02:27.470 --> 00:02:30.005
these are just the units needed,
that when you multiply

00:02:30.005 --> 00:02:33.040
it times a mass and a mass
divided by a distance squared,

00:02:33.040 --> 00:02:35.930
you get Newtons, or kilogram
meters per second squared.

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So we won't worry so much about
the units right now.

00:02:38.360 --> 00:02:40.610
Just realize that you're going
to have to work with meters in

00:02:40.610 --> 00:02:42.420
kilograms seconds.

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So let's just write
that number down.

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I'll change colors to
keep it interesting.

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6.67 times 10 to the negative
11th, and we want to know the

00:02:55.080 --> 00:03:00.190
acceleration on Sal, so
m1 is the mass of Sal.

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And I don't feel like revealing
my mass in this

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video, so I'll just leave
it as a variable.

00:03:05.240 --> 00:03:06.430
And then what's the mass 2?

00:03:06.430 --> 00:03:07.950
It's the mass of Earth.

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And I wrote that here.

00:03:08.820 --> 00:03:10.250
I looked it up on Wikipedia.

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This is the mass of Earth.

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So I multiply it times the
mass of Earth, times 5.97

00:03:18.890 --> 00:03:21.670
times 10 to the 24th kilograms--
weighs a little

00:03:21.670 --> 00:03:25.700
bit, not weighs, is a little
bit more massive than Sal--

00:03:25.700 --> 00:03:27.900
divided by the distance
squared.

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Now, you might say, well, what's
the distance between

00:03:29.420 --> 00:03:30.990
someone standing on the
Earth and the Earth?

00:03:30.990 --> 00:03:33.080
Well, it's zero because they're
touching the Earth.

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But it's important to realize
that the distance between the

00:03:35.130 --> 00:03:38.570
two objects, especially when
we're talking about the

00:03:38.570 --> 00:03:40.910
universal law of gravitation, is
the distance between their

00:03:40.910 --> 00:03:42.310
center of masses.

00:03:42.310 --> 00:03:44.290
For all general purposes, my
center of mass, maybe it's

00:03:44.290 --> 00:03:46.610
like three feet above
the ground, because

00:03:46.610 --> 00:03:48.500
I'm not that tall.

00:03:48.500 --> 00:03:50.510
It's probably a little bit lower
than that, actually.

00:03:50.510 --> 00:03:52.760
Anyway, my center of mass might
be three feet above the

00:03:52.760 --> 00:03:54.380
ground, and where's Earth's
center of mass?

00:03:54.380 --> 00:03:56.330
Well, it's at the center of
Earth, so we have to know the

00:03:56.330 --> 00:03:59.800
radius of Earth, right?

00:03:59.800 --> 00:04:03.120
So the radius of Earth is--
I also looked it up on

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Wikipedia-- 6,371 kilometers.

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How many meters is that?

00:04:08.490 --> 00:04:11.100
It's 6 million meters, right?

00:04:11.100 --> 00:04:13.470
And then, you know, the extra
meter to get to my center of

00:04:13.470 --> 00:04:16.079
mass, we can ignore for now,
because it would be .001, so

00:04:16.079 --> 00:04:16.970
we'll ignore that for now.

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So it's 6-- and soon.

00:04:18.120 --> 00:04:20.290
I'll write it in scientific
notation since everything else

00:04:20.290 --> 00:04:25.230
is in scientific notation--
6.371 times 10 to the sixth

00:04:25.230 --> 00:04:25.860
meters, right?

00:04:25.860 --> 00:04:29.610
6,000 kilometers is
6 million meters.

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So let's write that down.

00:04:30.660 --> 00:04:37.580
So the distance is going
to be 6.37 times 10

00:04:37.580 --> 00:04:40.470
to the sixth meters.

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We have to square that.

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Remember, it's distance
squared.

00:04:44.560 --> 00:04:47.830
So let's see if we can simplify
this a little bit.

00:04:47.830 --> 00:04:52.670
Let's just multiply those top
numbers first. Force is equal

00:04:52.670 --> 00:04:53.910
to-- let's bring the
variable out.

00:04:53.910 --> 00:04:58.550
Mass of Sal times-- let's
do this top part.

00:04:58.550 --> 00:05:17.470
So we have 6.67 times 5.97
is equal to 39.82.

00:05:17.470 --> 00:05:19.670
And I just multiplied this times
this, so now I have to

00:05:19.670 --> 00:05:21.160
multiply the 10's.

00:05:21.160 --> 00:05:23.590
So 10 to the negative 11th times
10 to the negative 24th.

00:05:23.590 --> 00:05:25.065
We can just add the exponents.

00:05:25.065 --> 00:05:26.850
They have the same base.

00:05:26.850 --> 00:05:28.360
So what's 24 minus 11?

00:05:28.360 --> 00:05:31.030
It's 10 to the 13th, right?

00:05:31.030 --> 00:05:32.610
And then what does the
denominator look like?

00:05:32.610 --> 00:05:35.510
It's going to be the 6.37
squared times 10

00:05:35.510 --> 00:05:37.270
to the sixth squared.

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So it's going to be-- whatever
this is is going to be like 37

00:05:40.010 --> 00:05:44.900
or something-- times-- what's
10 to the sixth squared?

00:05:44.900 --> 00:05:47.650
It's 10 to the 12th, right?

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10 to the 12th.

00:05:49.120 --> 00:05:52.586
So let's figure out what
6.37 squared is.

00:05:52.586 --> 00:05:56.434
This little calculator I have
doesn't have squared, so I

00:05:56.434 --> 00:06:09.650
have to-- so it's 40.58.

00:06:09.650 --> 00:06:12.740
And so simplifying it, the force
is equal to the mass of

00:06:12.740 --> 00:06:31.990
Sal times-- let's divide, 39.82
divided by 40.58 is

00:06:31.990 --> 00:06:38.290
equal to 9.81.

00:06:38.290 --> 00:06:40.130
That's just this divided
by this.

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And then 10 to the 13th divided
by 10 to the 12th.

00:06:44.770 --> 00:06:46.200
Actually no, this isn't 9.81.

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Sorry, it's 0.981.

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0.981, and then 10 to the 13th
divided by 10 to the 12th is

00:06:51.700 --> 00:06:52.560
just 10, right?

00:06:52.560 --> 00:06:57.160
10 to the first, times 10,
so what's 0.981 times 10?

00:06:57.160 --> 00:07:03.710
Well, the force is equal to 9.81
times the mass of Sal.

00:07:03.710 --> 00:07:04.550
And where does this get us?

00:07:04.550 --> 00:07:06.780
How can we figure out the
acceleration right now?

00:07:06.780 --> 00:07:11.220
Well, force is just mass times
acceleration, right?

00:07:11.220 --> 00:07:14.800
So that's also going to just be
equal to the acceleration

00:07:14.800 --> 00:07:19.790
of gravity-- that's supposed to
be a small g there-- times

00:07:19.790 --> 00:07:22.310
the mass of Sal, right?

00:07:22.310 --> 00:07:25.265
So we know the gravitational
force is 9.81 times the mass

00:07:25.265 --> 00:07:27.570
of Sal, and we also know that
that's the same thing as the

00:07:27.570 --> 00:07:29.760
acceleration of gravity
times the mass of Sal.

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We can divide both sides by the
mass of Sal, and we have

00:07:31.840 --> 00:07:33.390
the acceleration of gravity.

00:07:33.390 --> 00:07:35.820
And if we had used the units the
whole way, you would have

00:07:35.820 --> 00:07:38.470
seen that it is kilograms meters
per second squared.

00:07:38.470 --> 00:07:41.880
And we have just shown that, at
least based on the numbers

00:07:41.880 --> 00:07:46.090
that they've given in Wikipedia,
the acceleration of

00:07:46.090 --> 00:07:49.170
gravity on the surface of the
Earth is almost exactly what

00:07:49.170 --> 00:07:51.450
we've been using in all the
projectile motion problems.

00:07:51.450 --> 00:07:55.290
It's 9.8 meters per
second squared.

00:07:55.290 --> 00:07:57.570
That's exciting.

00:07:57.570 --> 00:07:59.830
So let's do another quick
problem with gravity, because

00:07:59.830 --> 00:08:02.570
I've got two minutes.

00:08:02.570 --> 00:08:06.390
Let's say there's another
planet called the

00:08:06.390 --> 00:08:08.380
planet Small Earth.

00:08:08.380 --> 00:08:15.760
And let's say the radius of
Small Earth is equal to 1/2

00:08:15.760 --> 00:08:19.380
the radius of Earth and the mass
of Small Earth is equal

00:08:19.380 --> 00:08:22.280
to 1/2 the mass of Earth.

00:08:22.280 --> 00:08:27.750
So what's the pull of gravity
on any object, say same

00:08:27.750 --> 00:08:29.170
object, on this?

00:08:29.170 --> 00:08:31.880
How much smaller would
it be on this planet?

00:08:31.880 --> 00:08:33.952
Well, actually let me save
that to the next video,

00:08:33.952 --> 00:08:34.789
because I hate being rushed.

00:08:34.789 --> 00:08:36.500
So I'll see you


Gravitation (part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.900 --> 00:00:01.550
Welcome back.

00:00:01.550 --> 00:00:05.620
So I was trying to rush and
finish a problem in the last

00:00:05.620 --> 00:00:08.189
two minutes of the video, and
I realize that's just bad

00:00:08.189 --> 00:00:09.980
teaching, because I
end up rushing.

00:00:09.980 --> 00:00:11.810
So this is the problem we were
going to work on, and you'll

00:00:11.810 --> 00:00:12.780
see a lot of these.

00:00:12.780 --> 00:00:17.890
They just want you to become
familiar with the variables in

00:00:17.890 --> 00:00:19.350
Newton's law of gravitation.

00:00:19.350 --> 00:00:23.010
So I said that there's two
planets, one is Earth.

00:00:23.010 --> 00:00:27.100
Now I have time to draw things,
so that's Earth.

00:00:27.100 --> 00:00:28.830
And then there's Small Earth.

00:00:28.830 --> 00:00:32.470
And Small Earth-- well, maybe
I'll just call it the small

00:00:32.470 --> 00:00:34.370
planet, so we don't
get confused.

00:00:34.370 --> 00:00:37.600
It's green, showing that
there's probably

00:00:37.600 --> 00:00:40.300
life on that planet.

00:00:40.300 --> 00:00:42.550
Let's say it has 1/2 the radius,
and 1/2 the mass.

00:00:45.300 --> 00:00:46.625
So if you think about
it, it's probably a

00:00:46.625 --> 00:00:48.000
lot denser than Earth.

00:00:48.000 --> 00:00:49.550
That's a good problem
to think about.

00:00:49.550 --> 00:00:51.150
How much denser is it, right?

00:00:51.150 --> 00:00:55.520
Because if you have 1/2 the
radius, your volume is much

00:00:55.520 --> 00:00:56.560
less than 1/2.

00:00:56.560 --> 00:00:58.173
I don't want to go into that
now, but that's something for

00:00:58.173 --> 00:00:58.950
you to think about.

00:00:58.950 --> 00:01:01.900
But my question is what
fraction, if I'm standing on

00:01:01.900 --> 00:01:06.260
the surface of this-- so the
same person, so Sal, if I'm on

00:01:06.260 --> 00:01:13.900
Earth, what fraction is the pull
when I'm on this small

00:01:13.900 --> 00:01:15.390
green planet?

00:01:15.390 --> 00:01:18.170
So what is the pull
on me on Earth?

00:01:18.170 --> 00:01:23.620
Well, it's just going to be-- my
weight on Earth, the force

00:01:23.620 --> 00:01:28.510
on Earth, is going to be equal
to the gravitational constant

00:01:28.510 --> 00:01:35.160
times my mass, mass of me.

00:01:35.160 --> 00:01:41.930
So m sub m times the mass of
Earth divided by what?

00:01:41.930 --> 00:01:44.010
We learned in the last video,
divided by the distance

00:01:44.010 --> 00:01:45.730
between me and the center
of the mass of Earth.

00:01:45.730 --> 00:01:49.740
Really, my center of mass and
the center of mass of Earth.

00:01:49.740 --> 00:01:52.820
But this is between the surface
of the Earth, and I'd

00:01:52.820 --> 00:01:56.030
like to think that I'm not
short, but it's negligible

00:01:56.030 --> 00:01:58.380
between my center of mass and
the surface, so we'll just

00:01:58.380 --> 00:02:00.690
consider the radius
of the Earth.

00:02:00.690 --> 00:02:05.290
So we divide it by the radius
of the Earth squared.

00:02:08.770 --> 00:02:11.820
Using these same variables,
what's going to be the force

00:02:11.820 --> 00:02:13.900
on this other planet?

00:02:13.900 --> 00:02:16.490
So the force on the other
planet, this green planet--

00:02:16.490 --> 00:02:20.930
I'll do it in green-- and we're
calling it the small

00:02:20.930 --> 00:02:23.710
planet, it equals what?

00:02:23.710 --> 00:02:26.810
It equals the gravitational
constant again.

00:02:26.810 --> 00:02:29.240
And my mass doesn't change when
I go from one planet to

00:02:29.240 --> 00:02:31.850
another, right?

00:02:31.850 --> 00:02:33.620
Its mass now is what?

00:02:33.620 --> 00:02:37.710
We would write it m
sub s here, right?

00:02:37.710 --> 00:02:39.300
This is the small planet.

00:02:39.300 --> 00:02:41.850
And we wrote right here that
it's 1/2 the mass of Earth, so

00:02:41.850 --> 00:02:42.930
I'll just write that.

00:02:42.930 --> 00:02:44.700
So it's 1/2 the mass of Earth.

00:02:48.190 --> 00:02:49.900
And what's its radius?

00:02:49.900 --> 00:02:51.750
What's the radius now?

00:02:51.750 --> 00:02:54.100
I could just write the radius
of the small planet squared,

00:02:54.100 --> 00:02:54.990
but I'll say, well, we know.

00:02:54.990 --> 00:02:57.920
It's 1/2 the radius of Earth,
so let's put that in there.

00:02:57.920 --> 00:03:00.290
So 1/2 radius of Earth.

00:03:00.290 --> 00:03:02.420
We have to square it.

00:03:02.420 --> 00:03:04.270
Let's see what this
simplifies to.

00:03:04.270 --> 00:03:12.760
This equals-- so we can take
this 1/2 here-- 1/2G mass of

00:03:12.760 --> 00:03:18.140
me times mass of Earth over--
what's 1/2 squared?

00:03:18.140 --> 00:03:19.260
It's 1/4.

00:03:19.260 --> 00:03:26.580
Over 1/4 radius of
Earth squared.

00:03:26.580 --> 00:03:30.330
And what's 1/2 divided by 1/4?

00:03:30.330 --> 00:03:32.420
1/4 goes into 1/2 two
times, right?

00:03:32.420 --> 00:03:33.880
Or another way you can think
about it is if you have a

00:03:33.880 --> 00:03:35.500
fraction in the denominator,
when you put it in the

00:03:35.500 --> 00:03:38.020
numerator, you flip it
and it becomes 4.

00:03:38.020 --> 00:03:39.060
So 4 times 1/2 is 2.

00:03:39.060 --> 00:03:41.760
Either way, it's just math.

00:03:41.760 --> 00:03:45.640
So the force on the small planet
is going to be equal to

00:03:45.640 --> 00:03:53.030
1/2 divided by 1/4 is 2 times
G, mass of me, times mass of

00:03:53.030 --> 00:03:56.930
Earth, divided by the radius
of Earth squared.

00:03:56.930 --> 00:04:01.650
And if we look up here,
this is the same

00:04:01.650 --> 00:04:05.445
thing as this, right?

00:04:05.445 --> 00:04:07.130
It's identical.

00:04:07.130 --> 00:04:10.820
So we know that the force that
applied to me when I'm on the

00:04:10.820 --> 00:04:15.660
surface of the small planet is
actually two times the force

00:04:15.660 --> 00:04:19.180
applied on Earth, when
I go to Earth.

00:04:19.180 --> 00:04:20.610
And that's something interesting
to think about,

00:04:20.610 --> 00:04:24.830
because you might have said
initially, wow, you know, the

00:04:24.830 --> 00:04:27.040
mass of the object matters
a lot in gravity.

00:04:27.040 --> 00:04:29.270
The more massive the object,
the more it's

00:04:29.270 --> 00:04:31.330
going to pull on me.

00:04:31.330 --> 00:04:33.640
But what we see here is
that actually, no.

00:04:33.640 --> 00:04:36.180
When I'm on the surface of
this smaller planet, it's

00:04:36.180 --> 00:04:37.830
pulling even harder on me.

00:04:37.830 --> 00:04:38.830
And why is that?

00:04:38.830 --> 00:04:41.990
Well, because I'm actually
closer to its center of mass.

00:04:41.990 --> 00:04:45.260
And as we talked about earlier
in this video, this object is

00:04:45.260 --> 00:04:46.650
probably a lot denser.

00:04:46.650 --> 00:04:50.350
You could say it's only 1/2 the
mass, but it's much less

00:04:50.350 --> 00:04:51.780
than 1/2 of the volume, right?

00:04:51.780 --> 00:04:55.090
Because the volume is the cube
of the radius and all of that.

00:04:55.090 --> 00:04:56.920
I don't want to confuse you, but
this is just something to

00:04:56.920 --> 00:04:57.610
think about.

00:04:57.610 --> 00:04:59.480
So not only does the mass
matter, but the

00:04:59.480 --> 00:05:01.970
radius matters a lot.

00:05:01.970 --> 00:05:03.980
And the radius is actually the
square, so it actually

00:05:03.980 --> 00:05:06.350
matters even more.

00:05:06.350 --> 00:05:09.800
So that's something
that's pretty

00:05:09.800 --> 00:05:10.430
interesting to think about.

00:05:10.430 --> 00:05:14.080
And these are actually very
common problems when they just

00:05:14.080 --> 00:05:16.680
want to tell you, oh, you go to
a planet that is two times

00:05:16.680 --> 00:05:21.140
the mass of another planet, et
cetera, et cetera, what is the

00:05:21.140 --> 00:05:23.090
difference in force
between the two?

00:05:23.090 --> 00:05:25.570
And one thing I want you to
realize, actually, before I

00:05:25.570 --> 00:05:29.050
finish this video since I do
have some extra time, when we

00:05:29.050 --> 00:05:30.890
think about gravity, especially
with planets and

00:05:30.890 --> 00:05:33.500
all of that, you always
feel like, oh, it's

00:05:33.500 --> 00:05:36.400
Earth pulling on me.

00:05:36.400 --> 00:05:41.750
Let's say that this is the
Earth, and the Earth is huge,

00:05:41.750 --> 00:05:45.730
and this is a tiny spaceship
right here.

00:05:45.730 --> 00:05:48.310
It's traveling.

00:05:48.310 --> 00:05:50.190
You always think that
Earth is pulling on

00:05:50.190 --> 00:05:51.150
the spaceship, right?

00:05:51.150 --> 00:05:53.350
The gravitational
force of Earth.

00:05:53.350 --> 00:05:57.070
But it actually turns out, when
we looked at the formula,

00:05:57.070 --> 00:05:57.980
the formula is symmetric.

00:05:57.980 --> 00:05:59.800
It's not really saying one
is pulling on the other.

00:05:59.800 --> 00:06:01.570
They're actually saying
this is the force

00:06:01.570 --> 00:06:03.420
between the two objects.

00:06:03.420 --> 00:06:04.960
They're attracted
to each other.

00:06:04.960 --> 00:06:13.580
So if the Earth is pulling on
me with the force of 500

00:06:13.580 --> 00:06:16.820
Newtons, it actually turns out
that I am pulling on the Earth

00:06:16.820 --> 00:06:18.920
with an equal and opposite
force of 5 Newtons.

00:06:18.920 --> 00:06:20.420
We're pulling towards
each other.

00:06:20.420 --> 00:06:23.140
It just feels like the Earth is,
at least from my point of

00:06:23.140 --> 00:06:25.220
view, that the Earth
is pulling to me.

00:06:25.220 --> 00:06:29.410
And we're actually both being
pulled towards the combined

00:06:29.410 --> 00:06:30.180
center of mass.

00:06:30.180 --> 00:06:33.050
So in this situation, let's say
the Earth is pulling on

00:06:33.050 --> 00:06:37.220
the spaceship with the force
of-- I don't know.

00:06:37.220 --> 00:06:40.210
I'm making up numbers
now, but let's say

00:06:40.210 --> 00:06:43.970
it's 1 million Newtons.

00:06:43.970 --> 00:06:46.320
It actually turns out that the
spaceship will be pulling on

00:06:46.320 --> 00:06:51.950
the Earth with the same force
of 1 million Newtons.

00:06:51.950 --> 00:06:55.610
And they're both going to be
moved to the combined system's

00:06:55.610 --> 00:06:56.900
center of mass.

00:06:56.900 --> 00:07:00.320
And the combined system's center
of mass since the Earth

00:07:00.320 --> 00:07:02.910
is so much more massive is
going to be very close to

00:07:02.910 --> 00:07:03.790
Earth's center of mass.

00:07:03.790 --> 00:07:05.770
It's probably going to
be very close to

00:07:05.770 --> 00:07:06.450
Earth's center of mass.

00:07:06.450 --> 00:07:07.900
It's going to be like
right there, right?

00:07:07.900 --> 00:07:12.070
So in this situation, Earth
won't be doing a lot of

00:07:12.070 --> 00:07:15.930
moving, but it will be pulled
in the direction of the

00:07:15.930 --> 00:07:18.340
spaceship, and the spaceship
will try to go to Earth's

00:07:18.340 --> 00:07:20.680
center of mass, but at some
point, probably the

00:07:20.680 --> 00:07:24.750
atmosphere, or the rock that it
runs into, it won't be able

00:07:24.750 --> 00:07:27.640
to go much further and
it might crash

00:07:27.640 --> 00:07:28.610
right around there.

00:07:28.610 --> 00:07:31.420
Anyway, I wanted just to give
you the sense that it's not

00:07:31.420 --> 00:07:33.240
necessarily one object just
pulling on the other.

00:07:33.240 --> 00:07:35.230
They're pulling towards
each other to their

00:07:35.230 --> 00:07:36.850
combined center of masses.

00:07:36.850 --> 00:07:40.650
It would make a lot more sense
if they had just two people

00:07:40.650 --> 00:07:42.210
floating in space, they actually
would have some

00:07:42.210 --> 00:07:43.900
gravity towards each other.

00:07:43.900 --> 00:07:47.110
It's almost a little romantic.

00:07:47.110 --> 00:07:48.970
They would float
to each other.

00:07:48.970 --> 00:07:52.910
And actually, you could
figure it out.

00:07:52.910 --> 00:07:55.030
I don't have the time to do
it, but you could use this

00:07:55.030 --> 00:07:58.010
formula and use the constant,
and you could figure out,

00:07:58.010 --> 00:08:00.350
well, what is the gravitational
attraction

00:08:00.350 --> 00:08:01.560
between two people?

00:08:01.560 --> 00:08:04.080
And what you'll see is that
between two people floating in

00:08:04.080 --> 00:08:06.850
space, there are other forms
of attraction that are

00:08:06.850 --> 00:08:09.150
probably stronger than their

00:08:09.150 --> 00:08:11.380
gravitational attraction, anyway.

00:08:11.380 --> 00:08:13.520
I'll let you ponder that
and I will see

00:08:13.520 --> 00:08:15.410
you in the next video.


Calculus Proof that a=v^2/r

WEBVTT
Kind: captions
Language: en

00:00:00.760 --> 00:00:01.480
Welcome back.

00:00:01.480 --> 00:00:06.050
Well, I'm now going to prove to
you that the magnitude of

00:00:06.050 --> 00:00:09.630
centripetal acceleration is
equal to the magnitude of the

00:00:09.630 --> 00:00:13.150
velocity when you're going
around the circle divided--

00:00:13.150 --> 00:00:15.590
velocity squared divided
by the radius.

00:00:15.590 --> 00:00:17.590
So let's start with the drawing,
just so that we know

00:00:17.590 --> 00:00:22.200
what we're doing, just as much
for me as it is for you.

00:00:22.200 --> 00:00:24.260
So that's the circle, and
you can guess that's

00:00:24.260 --> 00:00:25.840
going to be our path.

00:00:25.840 --> 00:00:30.280
And let's call p our
position vector.

00:00:30.280 --> 00:00:34.820
And this is the center of
the circle right here.

00:00:34.820 --> 00:00:38.390
Let me do the position
vector in magenta.

00:00:38.390 --> 00:00:42.070
So let's say this is
p, my position.

00:00:42.070 --> 00:00:43.320
Let me draw that bolder.

00:00:50.320 --> 00:00:51.780
So that's the vector p.

00:00:56.840 --> 00:01:00.920
And let me define a few other
things over here.

00:01:00.920 --> 00:01:04.580
So let's say that the angle
that it's forming with the

00:01:04.580 --> 00:01:06.430
positive x-axis-- let's
say this is the

00:01:06.430 --> 00:01:12.580
positive x-axis-- is theta.

00:01:12.580 --> 00:01:13.580
That's theta right there.

00:01:13.580 --> 00:01:15.910
Let's say the radius
of the circle is r.

00:01:15.910 --> 00:01:18.580
So we have an object.

00:01:18.580 --> 00:01:19.630
It's right here.

00:01:19.630 --> 00:01:22.920
This is its position defined by
this position vector, and

00:01:22.920 --> 00:01:23.750
it's spinning around.

00:01:23.750 --> 00:01:26.130
So its position vector is
going-- at some point, the

00:01:26.130 --> 00:01:27.820
arrow's going to be pointed
there and then there.

00:01:27.820 --> 00:01:29.740
It's just going to be going
around and around the circle.

00:01:29.740 --> 00:01:33.520
But this is its position vector
at some moment in time.

00:01:33.520 --> 00:01:39.480
So what is that position vector
in our bracket notation

00:01:39.480 --> 00:01:40.410
of vectors?

00:01:40.410 --> 00:01:42.460
We have to figure out its
x and y-components.

00:01:42.460 --> 00:01:50.320
Its x-component is right here,
or you could almost say its

00:01:50.320 --> 00:01:53.600
i-component, if we were doing
engineering notation.

00:01:53.600 --> 00:01:58.140
That's its x-component and
that's its y-component, right?

00:01:58.140 --> 00:02:00.940
So in our bracket notation or
whatever, I always forget what

00:02:00.940 --> 00:02:05.700
I call things, p is this.

00:02:05.700 --> 00:02:11.240
p, which is our position
vector, what's its

00:02:11.240 --> 00:02:12.150
x-component?

00:02:12.150 --> 00:02:15.550
It's the radius times
the cosine of theta.

00:02:15.550 --> 00:02:18.050
This should be second nature
to you at this point.

00:02:18.050 --> 00:02:20.880
Radius times cosine of theta.

00:02:20.880 --> 00:02:21.980
What's its y-component?

00:02:21.980 --> 00:02:22.750
It's this.

00:02:22.750 --> 00:02:23.480
It's just this, right?

00:02:23.480 --> 00:02:25.020
That's its y-component.

00:02:25.020 --> 00:02:27.670
Radius times the
sine of theta.

00:02:31.240 --> 00:02:32.030
Fair enough.

00:02:32.030 --> 00:02:33.330
Hopefully, that makes
sense to you so far.

00:02:33.330 --> 00:02:37.670
I just defined its position
vector and I drew it out here

00:02:37.670 --> 00:02:41.680
visually, and then I also wrote
it analytically in its x

00:02:41.680 --> 00:02:45.230
and y-components, as a sum
of its x y-components.

00:02:45.230 --> 00:02:46.250
Well, that's good
and everything.

00:02:46.250 --> 00:02:47.420
So let's see if we can
figure out what its

00:02:47.420 --> 00:02:49.330
velocity vector is.

00:02:49.330 --> 00:02:50.600
Well, what is velocity?

00:02:50.600 --> 00:02:52.890
Velocity is actually just
a change in position.

00:02:52.890 --> 00:02:54.480
Actually, now we are actually
dealing with

00:02:54.480 --> 00:02:55.980
velocity, not speed.

00:02:55.980 --> 00:02:57.650
We will actually get a vector.

00:02:57.650 --> 00:02:59.640
So what is the velocity
vector?

00:02:59.640 --> 00:03:01.520
So the velocity vector
is going to be

00:03:01.520 --> 00:03:02.415
at any given point.

00:03:02.415 --> 00:03:04.540
I'll do it in a different
color.

00:03:04.540 --> 00:03:06.550
The velocity vector is going to
be tangent to the circle.

00:03:06.550 --> 00:03:08.530
It's going to look something
like that.

00:03:08.530 --> 00:03:10.770
That's going to be the
velocity vector.

00:03:10.770 --> 00:03:15.240
So the velocity vector is equal
to the change in the

00:03:15.240 --> 00:03:17.750
position over time.

00:03:17.750 --> 00:03:22.120
So let's take the derivative
of the position vector with

00:03:22.120 --> 00:03:24.410
respect to time.

00:03:24.410 --> 00:03:25.500
And how do we take
a derivative?

00:03:25.500 --> 00:03:27.220
Well, we could just take the
derivative of the x and

00:03:27.220 --> 00:03:29.880
y-components separately, and
I'll show you how we do it in

00:03:29.880 --> 00:03:30.520
this notation.

00:03:30.520 --> 00:03:31.540
And if you think about
it, it should make a

00:03:31.540 --> 00:03:33.550
little bit of sense.

00:03:33.550 --> 00:03:35.870
So the radius is constant.

00:03:35.870 --> 00:03:38.040
As we go around the circle,
the radius isn't changing.

00:03:38.040 --> 00:03:39.980
Another thing to keep in mind,
we're going around.

00:03:39.980 --> 00:03:43.010
We're spinning around the circle
at a constant rate.

00:03:43.010 --> 00:03:44.410
So my speed isn't changing.

00:03:44.410 --> 00:03:46.680
My velocity is obviously
changing, because the

00:03:46.680 --> 00:03:49.920
direction is changing, but the
actual rate at which I'm

00:03:49.920 --> 00:03:51.560
spinning, or the angular
velocity, is

00:03:51.560 --> 00:03:52.460
going to be a constant.

00:03:52.460 --> 00:03:54.780
That's something to keep
in mind for what we get

00:03:54.780 --> 00:03:55.820
to in the next step.

00:03:55.820 --> 00:03:58.560
But anyway, let's take
the derivative.

00:03:58.560 --> 00:04:04.260
So let's take the derivative of
the x term first. Well, r

00:04:04.260 --> 00:04:06.050
is just a constant so
it doesn't change.

00:04:06.050 --> 00:04:08.290
We can just take the r out.

00:04:08.290 --> 00:04:11.110
And then what's the derivative
of cosine theta

00:04:11.110 --> 00:04:13.100
with respect to time?

00:04:13.100 --> 00:04:14.180
Not with respect to theta.

00:04:14.180 --> 00:04:15.840
Remember, we're taking
the derivative

00:04:15.840 --> 00:04:17.209
with respect to time.

00:04:17.209 --> 00:04:18.930
So we do the chain rule.

00:04:18.930 --> 00:04:27.860
It's the derivative of theta
with respect to time times the

00:04:27.860 --> 00:04:32.110
derivative of this term
with respect to theta.

00:04:32.110 --> 00:04:33.840
And what's the derivative
of this?

00:04:33.840 --> 00:04:37.910
What's minus sine of theta?

00:04:37.910 --> 00:04:39.560
Let me put the minus out here.

00:04:39.560 --> 00:04:41.670
Minus sine of theta.

00:04:41.670 --> 00:04:43.360
I just didn't want to put the
minus right in front of the

00:04:43.360 --> 00:04:45.320
sine, because you would think
it's minus sine theta.

00:04:45.320 --> 00:04:48.560
So it's minus r times the rate
of change of the angle with

00:04:48.560 --> 00:04:52.620
respect to time times sine of
theta, the rate at which this

00:04:52.620 --> 00:04:54.730
term is changing with
respect to theta.

00:04:54.730 --> 00:04:57.600
Let's do the same thing
on the y side.

00:04:57.600 --> 00:04:59.100
r is a constant.

00:04:59.100 --> 00:05:02.330
Chain rule: the rate at which
theta is changing

00:05:02.330 --> 00:05:03.580
with respect to time.

00:05:06.265 --> 00:05:06.460
you.

00:05:06.460 --> 00:05:07.970
And then what's the derivative
of sine of theta

00:05:07.970 --> 00:05:08.720
with respect to theta?

00:05:08.720 --> 00:05:11.250
Well, that's cosine of theta.

00:05:11.250 --> 00:05:14.570
This was just the chain
rule that I did.

00:05:14.570 --> 00:05:17.230
Let's see if we can simplify
that a little bit.

00:05:17.230 --> 00:05:20.690
So d theta, dt in both of these,
that's the same thing

00:05:20.690 --> 00:05:22.000
as angular velocity.

00:05:22.000 --> 00:05:24.500
And watch the video on angular
velocity if that doesn't make

00:05:24.500 --> 00:05:25.580
sense to you.

00:05:25.580 --> 00:05:30.170
But we can simplify this as just
w, angular velocity, and

00:05:30.170 --> 00:05:32.060
that's going to be a constant.

00:05:32.060 --> 00:05:33.770
And we have an r there.

00:05:33.770 --> 00:05:37.120
Let's take a wr out
of both sides.

00:05:37.120 --> 00:05:45.450
So we have the velocity vector
is equal to-- the velocity

00:05:45.450 --> 00:05:51.410
vector is equal to wr--
actually, let's take minus wr

00:05:51.410 --> 00:05:57.480
out, so minus wr, And then this
term is sine of theta.

00:06:00.345 --> 00:06:03.350
And we're taking a minus wr, so
the wr goes away, and then

00:06:03.350 --> 00:06:05.050
we introduce a minus
sign here, so it's

00:06:05.050 --> 00:06:06.300
minus cosine of theta.

00:06:11.110 --> 00:06:11.470
Good enough.

00:06:11.470 --> 00:06:13.290
And the reason why we were able
to take-- and this w,

00:06:13.290 --> 00:06:14.730
remember, is going
to be a constant.

00:06:14.730 --> 00:06:17.070
It's not changing with
respect to time.

00:06:17.070 --> 00:06:20.630
The angle is changing with
respect to time, but not the

00:06:20.630 --> 00:06:21.660
rate of change of the angle.

00:06:21.660 --> 00:06:23.630
It's spinning at a
constant rate.

00:06:23.630 --> 00:06:25.160
So what's the acceleration
vector going to be?

00:06:25.160 --> 00:06:27.430
The acceleration vector, and
I'll switch colors again to

00:06:27.430 --> 00:06:28.920
keep it interesting.

00:06:28.920 --> 00:06:32.200
The acceleration vector is just
the derivative of the

00:06:32.200 --> 00:06:33.845
velocity vector with
respect to time.

00:06:39.430 --> 00:06:44.066
And that equals-- this is this
constant term, so let's just

00:06:44.066 --> 00:06:44.940
leave it on the outside.

00:06:44.940 --> 00:06:47.770
It's minus wr.

00:06:47.770 --> 00:06:50.040
And chain rule again.

00:06:50.040 --> 00:06:52.228
If we're taking the derivative
with respect to time, first we

00:06:52.228 --> 00:06:53.950
have to take the derivative of
theta with respect to time,

00:06:53.950 --> 00:06:54.950
and we don't know
what that is.

00:06:54.950 --> 00:06:56.240
That's just going to be w.

00:06:56.240 --> 00:07:03.620
d theta dt and then times this
expression, the derivative of

00:07:03.620 --> 00:07:05.835
this expression with
respect to theta.

00:07:05.835 --> 00:07:10.160
The sine of theta derivative
is just cosine of theta.

00:07:10.160 --> 00:07:15.220
And then on the y-side, what's
the derivative of theta with

00:07:15.220 --> 00:07:16.810
respect to time?

00:07:16.810 --> 00:07:20.560
It's just going to be omega.

00:07:20.560 --> 00:07:22.680
And what's the derivative of
minus cosine theta with

00:07:22.680 --> 00:07:23.520
respect to theta?

00:07:23.520 --> 00:07:24.770
Well, that's sine theta.

00:07:27.610 --> 00:07:32.270
And once again, this
is w and this is w.

00:07:32.270 --> 00:07:34.320
We could take the w out
of the equation.

00:07:34.320 --> 00:07:38.090
We get the acceleration vector
is equal to-- take the w out

00:07:38.090 --> 00:07:44.480
of the x and y-components-- is
equal to minus w squared r

00:07:44.480 --> 00:07:51.470
times cosine theta sine theta.

00:07:51.470 --> 00:07:54.680
Or we could take this r and
multiply it times both of the

00:07:54.680 --> 00:07:57.760
x and y-components, and we have
the acceleration vector

00:07:57.760 --> 00:08:05.920
is equal to minus w squared r
cosine theta r sine theta.

00:08:05.920 --> 00:08:08.690
Now does this thing here
look familiar?

00:08:08.690 --> 00:08:09.210
Well, sure.

00:08:09.210 --> 00:08:12.330
That was our original
position vector.

00:08:12.330 --> 00:08:15.420
So we could say that the
acceleration vector is equal

00:08:15.420 --> 00:08:20.450
to minus our angular velocity
squared times

00:08:20.450 --> 00:08:23.390
the position vector.

00:08:23.390 --> 00:08:26.390
And that makes actually a lot of
sense, because w squared is

00:08:26.390 --> 00:08:27.630
going to be a positive term.

00:08:27.630 --> 00:08:32.510
And what it's saying is that the
direction of this vector

00:08:32.510 --> 00:08:34.083
is going to be the negative
of the direction of

00:08:34.083 --> 00:08:34.580
the position vector.

00:08:34.580 --> 00:08:37.054
So if our position vector
is going outward, our

00:08:37.054 --> 00:08:40.559
acceleration vector, which I'll
draw in green, is going

00:08:40.559 --> 00:08:41.880
to be going inward.

00:08:41.880 --> 00:08:43.789
The acceleration vector
is going to be inward.

00:08:43.789 --> 00:08:46.460
So it is what we
wanted to see.

00:08:46.460 --> 00:08:49.160
Let's see if we can express
this as a function of the

00:08:49.160 --> 00:08:50.420
magnitudes.

00:08:50.420 --> 00:08:53.830
So we'll also say that the
magnitude of the acceleration

00:08:53.830 --> 00:08:57.545
vector, and that's just a
without an arrow on top; I

00:08:57.545 --> 00:09:02.120
could put brackets around it,
is equal to the negative

00:09:02.120 --> 00:09:05.050
angular velocity squared
times the magnitude of

00:09:05.050 --> 00:09:07.860
the position vector.

00:09:07.860 --> 00:09:09.890
Well, what's the magnitude
of this position vector?

00:09:09.890 --> 00:09:11.320
How long is it?

00:09:11.320 --> 00:09:14.470
Well, its magnitude is r
by definition on the

00:09:14.470 --> 00:09:15.640
beginning of the thing.

00:09:15.640 --> 00:09:17.450
So this is just r.

00:09:17.450 --> 00:09:21.390
So acceleration is equal to the
negative angular velocity

00:09:21.390 --> 00:09:23.210
squared times r.

00:09:23.210 --> 00:09:25.460
And what's angular velocity?

00:09:25.460 --> 00:09:28.360
Well, we learned in that video
that angular velocity-- I'll

00:09:28.360 --> 00:09:30.970
do it right here--
is equal to v/r.

00:09:30.970 --> 00:09:33.080
If we just talk about the
magnitudes, not the vectors.

00:09:33.080 --> 00:09:34.580
Remember, if we're not drawing
an arrow on top these

00:09:34.580 --> 00:09:36.950
variables, they're just
scalar quantities.

00:09:36.950 --> 00:09:43.550
So the acceleration is equal
to-- actually, we can get rid

00:09:43.550 --> 00:09:45.770
of the negative sign because
we're not worried about

00:09:45.770 --> 00:09:46.940
direction right now.

00:09:46.940 --> 00:09:48.190
Well, we can keep it there.

00:09:48.190 --> 00:09:49.170
It doesn't matter.

00:09:49.170 --> 00:09:56.470
So we get v squared, because
the magnitude of this,

00:09:56.470 --> 00:09:58.850
absolute value is essentially
magnitude.

00:09:58.850 --> 00:10:02.780
So you get v squared over
r squared times r.

00:10:02.780 --> 00:10:05.840
So r times r squared, you get
the acceleration is v

00:10:05.840 --> 00:10:07.780
squared over r.

00:10:07.780 --> 00:10:09.400
And that's what we
set out to prove.

00:10:09.400 --> 00:10:10.470
And I'm out of time now.

00:10:10.470 --> 00:10:11.610
So I'll see you in
the next video.

00:10:11.610 --> 00:10:13.180
Hopefully, I didn't confuse


Conservation of angular momemtum

WEBVTT
Kind: captions
Language: en

00:00:00.770 --> 00:00:07.140
We learned in the videos on
torque that torque equals--

00:00:07.140 --> 00:00:10.940
let me actually draw a picture,
so you remember what

00:00:10.940 --> 00:00:12.170
we're talking about
with torque.

00:00:12.170 --> 00:00:18.580
So let's say-- so that's the
arm, let's say right here is

00:00:18.580 --> 00:00:21.120
what its pivot is.

00:00:21.120 --> 00:00:22.510
That's its pivot.

00:00:22.510 --> 00:00:30.420
And let's say that I'm putting
a force right there, and it's

00:00:30.420 --> 00:00:31.216
perpendicular to this arm.

00:00:31.216 --> 00:00:35.540
And let's say the length
of this arm is r.

00:00:35.540 --> 00:00:45.220
And this force is f.

00:00:45.220 --> 00:00:48.000
So we know that the torque
is this force times

00:00:48.000 --> 00:00:49.730
the distance, right?

00:00:55.510 --> 00:00:59.030
And then we also know that--
what is force?

00:00:59.030 --> 00:01:01.390
Well, that is equal
to mass times

00:01:01.390 --> 00:01:03.980
acceleration times distance.

00:01:03.980 --> 00:01:05.730
So torque is equal
to mass times

00:01:05.730 --> 00:01:08.090
acceleration times distance.

00:01:08.090 --> 00:01:10.900
And then so what is
acceleration?

00:01:10.900 --> 00:01:15.540
Well that's equal to mass times
change in velocity over

00:01:15.540 --> 00:01:22.240
change in time, times
distance, right?

00:01:22.240 --> 00:01:24.020
So we learned all that from our
torque chapter, and you

00:01:24.020 --> 00:01:26.420
might want to review it, just
to get an intuition of what

00:01:26.420 --> 00:01:27.180
torque is good for.

00:01:27.180 --> 00:01:29.850
But in general, if something
isn't spinning, you apply

00:01:29.850 --> 00:01:32.080
torque, and you'll
get it spinning.

00:01:32.080 --> 00:01:34.560
Or if something is spinning
already, if you apply torque

00:01:34.560 --> 00:01:36.560
in the direction that it's
spinning, it'll spin faster.

00:01:36.560 --> 00:01:39.280
Or if you go in the opposite
direction, it'll spin slower.

00:01:39.280 --> 00:01:42.170
And what I'm showing you here,
is that what happens if you

00:01:42.170 --> 00:01:43.690
apply no torque?

00:01:43.690 --> 00:01:47.310
Well if you apply no torque,
then we know that this

00:01:47.310 --> 00:01:51.070
quantity is 0.

00:01:51.070 --> 00:01:54.270
Or another way to think about
it-- actually why did I write

00:01:54.270 --> 00:01:57.530
d, it could be d, but I
shouldn't have called this r,

00:01:57.530 --> 00:01:59.590
it should be d.

00:01:59.590 --> 00:02:03.950
So if this is 0-- if we are
applying no torque, right--

00:02:03.950 --> 00:02:04.920
what do we know?

00:02:04.920 --> 00:02:06.850
We know that the change in
velocity over change in time,

00:02:06.850 --> 00:02:09.660
times this distance,
won't change-- that

00:02:09.660 --> 00:02:11.760
this quantity is 0.

00:02:11.760 --> 00:02:19.020
So we know that the velocity
times the distance is going to

00:02:19.020 --> 00:02:20.270
be a constant.

00:02:24.230 --> 00:02:27.660
And that comes from what
I just talked about.

00:02:27.660 --> 00:02:31.070
It falls out of Newton's laws,
but it applies to spinning.

00:02:31.070 --> 00:02:33.690
An object that's not spinning
will tend to not stay

00:02:33.690 --> 00:02:35.480
spinning, and an object
that is spinning will

00:02:35.480 --> 00:02:37.190
tend to stay spinning.

00:02:37.190 --> 00:02:39.470
So in this case, if this object
at this point right

00:02:39.470 --> 00:02:42.090
here-- so we're in a case where
there's no torque, so

00:02:42.090 --> 00:02:44.770
this force is 0-- there's
no force applying here.

00:02:44.770 --> 00:02:48.190
And whatever this object's
velocity was-- its tangential

00:02:48.190 --> 00:02:52.490
velocity-- it's going to stay
at that velocity, right?

00:02:52.490 --> 00:02:54.300
It's just going to keep spinning
at that velocity.

00:02:54.300 --> 00:02:56.940
If I apply more torque it'll
go even faster, if I apply

00:02:56.940 --> 00:03:00.600
less torque, it'll slow
down a little bit.

00:03:00.600 --> 00:03:02.510
But we know that this
velocity times

00:03:02.510 --> 00:03:03.440
the distance is constant.

00:03:03.440 --> 00:03:05.850
And actually, I don't know why
I took this m out-- we know

00:03:05.850 --> 00:03:08.060
that the mass times the velocity
times the distance,

00:03:08.060 --> 00:03:09.320
is constant.

00:03:09.320 --> 00:03:10.520
Right?

00:03:10.520 --> 00:03:11.350
So what does that tell us?

00:03:11.350 --> 00:03:18.400
Well, we learned in the angular
velocity video-- my

00:03:18.400 --> 00:03:20.530
mind's a little slow
today-- that what?

00:03:20.530 --> 00:03:27.220
Angular velocity is equal to
velocity divided by the

00:03:27.220 --> 00:03:30.330
radius, and this case, the
radius is this distance, so we

00:03:30.330 --> 00:03:32.730
could also write it as velocity
over distance.

00:03:32.730 --> 00:03:34.510
And when I talk about radius,
it's just the radius of the

00:03:34.510 --> 00:03:37.760
circle that you're spinning
around in, right?

00:03:37.760 --> 00:03:39.620
This could be the
circle, up here.

00:03:39.620 --> 00:03:41.920
So this d, that's the same
thing as the radius.

00:03:41.920 --> 00:03:44.280
I'm just switching letters
to confuse you.

00:03:44.280 --> 00:03:48.210
But let's see if we can write
something, if we can change

00:03:48.210 --> 00:03:50.670
this expression to include
angular velocity.

00:03:50.670 --> 00:03:52.800
You'll see where I'm
going in a second.

00:03:52.800 --> 00:03:55.070
So let's solve for v.

00:03:55.070 --> 00:03:58.560
So let's multiply both sides
of this, times d.

00:03:58.560 --> 00:04:01.450
So you get dw is equal to v.

00:04:01.450 --> 00:04:04.890
Right, I just took this d,
put it on this side.

00:04:04.890 --> 00:04:11.740
So let's write that here-- m
times dw-- right, I just

00:04:11.740 --> 00:04:15.560
replaced the v-- times d,
is equal to a constant.

00:04:15.560 --> 00:04:20.660
Assuming that we have no net
torque on the system.

00:04:20.660 --> 00:04:22.130
And so what does that get us?

00:04:22.130 --> 00:04:26.310
Well that gets us m times the
angular velocity times d

00:04:26.310 --> 00:04:28.490
squared is equal
to a constant.

00:04:31.340 --> 00:04:32.170
So what does this tell us?

00:04:32.170 --> 00:04:35.400
This tells us the mass of an
object spinning-- let me

00:04:35.400 --> 00:04:42.050
rewrite this, because I think--
so what we know is

00:04:42.050 --> 00:04:47.300
that the mass of an object
spinning, times how fast it is

00:04:47.300 --> 00:04:52.020
spinning, times the distance
to the center of its

00:04:52.020 --> 00:04:54.110
rotation-- and actually I'm
going to change that d to an

00:04:54.110 --> 00:04:57.370
r, I don't know why I even used
d to begin with-- times

00:04:57.370 --> 00:04:57.720
that squared.

00:04:57.720 --> 00:05:03.950
That's going to be equal to a
constant, assuming no net

00:05:03.950 --> 00:05:05.720
force-- no net torque.

00:05:05.720 --> 00:05:08.970
Another way of looking at that,
if we just wanted to go

00:05:08.970 --> 00:05:11.420
to angular velocity from the
get-go, we could have said

00:05:11.420 --> 00:05:17.870
torque is equal to mass times
change in velocity, times

00:05:17.870 --> 00:05:21.840
change in time, times the radius
to the center of where

00:05:21.840 --> 00:05:23.400
you're rotating around.

00:05:23.400 --> 00:05:29.810
And change in velocity is just
the same thing as mass times

00:05:29.810 --> 00:05:34.920
change in angular velocity,
times r.

00:05:34.920 --> 00:05:36.460
Change in time, and then
there's another

00:05:36.460 --> 00:05:38.480
r, that's this r.

00:05:38.480 --> 00:05:40.996
Because velocity is angular
velocity times r, and we're

00:05:40.996 --> 00:05:43.560
assuming r doesn't change, so
any change in velocity is just

00:05:43.560 --> 00:05:46.260
going to be a change in
the angular velocity.

00:05:46.260 --> 00:05:48.880
And then we would get the same
thing that we just got here.

00:05:48.880 --> 00:05:51.040
If we have no net torque, we're
going to have no change

00:05:51.040 --> 00:05:52.840
in angular velocity, so angular

00:05:52.840 --> 00:05:54.100
velocity will be a constant.

00:05:54.100 --> 00:05:57.690
So you'll get mass times angular
velocity, times-- and

00:05:57.690 --> 00:06:00.780
then you have this r, and this
r-- times r squared, is going

00:06:00.780 --> 00:06:02.790
to be equal to a constant.

00:06:02.790 --> 00:06:04.430
Where am I going with
all of this?

00:06:04.430 --> 00:06:06.930
Well let's think about
something.

00:06:06.930 --> 00:06:12.870
Let's say that I have some
object traveling in a circle.

00:06:12.870 --> 00:06:15.080
I do everything for a reason,
and you'll see my

00:06:15.080 --> 00:06:18.290
reason right now.

00:06:18.290 --> 00:06:34.120
Let's say that-- so let's
say this is some type of

00:06:34.120 --> 00:06:38.740
retractable pole, and
I'm an ice skater.

00:06:38.740 --> 00:06:42.230
And I'm not using the ice
skater's body right now,

00:06:42.230 --> 00:06:43.690
because it'll become
complicated.

00:06:43.690 --> 00:06:47.070
Let's say this is some type of
robot arm, and that's its

00:06:47.070 --> 00:06:48.050
joint right there.

00:06:48.050 --> 00:06:49.440
And it's holding a
mass out here.

00:06:53.030 --> 00:06:55.730
This is neat because after
this concept, you'll

00:06:55.730 --> 00:06:58.660
understand what goes on in
figure skating, and then the

00:06:58.660 --> 00:06:59.930
Olympics-- oh no, the
Winter Olympics are

00:06:59.930 --> 00:07:00.980
over, aren't they?

00:07:00.980 --> 00:07:04.830
Anyway, let's say that this
object is spinning

00:07:04.830 --> 00:07:06.080
around at some rate.

00:07:08.750 --> 00:07:17.320
It's spinning around at
10 radians a second.

00:07:17.320 --> 00:07:19.330
That's its angular velocity.

00:07:19.330 --> 00:07:22.570
And let's say, right
now its rotational

00:07:22.570 --> 00:07:29.560
distance is 10 feet.

00:07:29.560 --> 00:07:38.665
So this is spinning on an ice
skating rink, because you

00:07:38.665 --> 00:07:40.280
don't want friction
and all of that.

00:07:40.280 --> 00:07:43.920
So what's its current
angular momentum?

00:07:43.920 --> 00:07:47.690
So that's what this term right
here is, angular momentum.

00:07:50.400 --> 00:07:52.790
So what's its current
angular momentum?

00:07:52.790 --> 00:08:00.880
Well its mass times 10, times--
10 is its-- actually,

00:08:00.880 --> 00:08:02.920
let's make this radius a
different number, let's call

00:08:02.920 --> 00:08:05.810
it 8 feet, just you know
what I'm doing.

00:08:05.810 --> 00:08:10.310
So its angular velocity is
10, and then its radius

00:08:10.310 --> 00:08:12.560
is 8, so times 64.

00:08:12.560 --> 00:08:15.250
So it equals 640 times mass.

00:08:15.250 --> 00:08:16.535
This is its angular momentum.

00:08:19.790 --> 00:08:22.130
Now what happens if this arm,
for whatever reason,

00:08:22.130 --> 00:08:26.900
shortens-- and maybe it does
something like this, the arm

00:08:26.900 --> 00:08:27.950
kind of bends.

00:08:27.950 --> 00:08:29.760
And then the mass comes
here, it comes in

00:08:29.760 --> 00:08:32.340
closer to the center.

00:08:32.340 --> 00:08:33.900
I'll write that in a
different color.

00:08:33.900 --> 00:08:36.280
The mass comes closer to
the center, so now

00:08:36.280 --> 00:08:37.659
the radius is 4.

00:08:37.659 --> 00:08:40.870
But I've had no net torque on
the system, all I've done is

00:08:40.870 --> 00:08:45.450
change how far it is from
the center of rotation.

00:08:45.450 --> 00:08:47.660
How much faster is it
going to spin now?

00:08:47.660 --> 00:08:48.760
Let's think about it.

00:08:48.760 --> 00:08:51.310
Its angular momentum won't
change, this is a constant,

00:08:51.310 --> 00:08:53.680
which is its angular momentum.

00:08:53.680 --> 00:08:54.940
That won't change.

00:08:54.940 --> 00:09:01.810
So we now know that the mass
times the angular-- the new

00:09:01.810 --> 00:09:05.470
angular momentum, we'll write
that angular momentum 1--

00:09:05.470 --> 00:09:08.890
times the new distance squared--
times 16-- is also

00:09:08.890 --> 00:09:11.780
going to be equal to 640m.

00:09:11.780 --> 00:09:13.860
The angular momentum
doesn't change.

00:09:13.860 --> 00:09:17.000
Let's cross out m from both
sides, and then divide both

00:09:17.000 --> 00:09:18.330
sides by 16.

00:09:18.330 --> 00:09:23.790
We now have that w1 is equal
to 16 goes into 640.

00:09:23.790 --> 00:09:24.390
So what happened?

00:09:24.390 --> 00:09:27.540
Originally, I was going around
at 10 radians per second, when

00:09:27.540 --> 00:09:33.400
I halved the radius-- when I
got half as close to the

00:09:33.400 --> 00:09:36.200
center of my rotation, I'm
actually spinning around 4

00:09:36.200 --> 00:09:37.540
times as much.

00:09:37.540 --> 00:09:40.620
And that's because this term
is a quadratic term.

00:09:40.620 --> 00:09:44.860
And you probably have observed
this behavior before, when you

00:09:44.860 --> 00:09:48.140
see the ice skater skating
around, and they're spinning

00:09:48.140 --> 00:09:50.950
with their arms wide open, and
then they pull their arms in,

00:09:50.950 --> 00:09:53.130
and they go a lot, lot,
lot, lot faster.

00:09:53.130 --> 00:09:56.700
And that's because their angular
velocity, or the rate

00:09:56.700 --> 00:10:01.980
at which they're spinning, is
proportional to the square of

00:10:01.980 --> 00:10:05.680
the radius around their
axis of rotation.

00:10:05.680 --> 00:10:07.770
Anyway, I hope I didn't confuse
you, and I'll do some

00:10:07.770 --> 00:10:09.620
more problems with this in
the future, but I've

00:10:09.620 --> 00:10:10.390
just run out of time.

00:10:10.390 --> 00:10:11.840
See you soon.


Introduction to angular velocity

WEBVTT
Kind: captions
Language: en

00:00:00.740 --> 00:00:04.300
Well, we've done a lot of work
with how fast something moves,

00:00:04.300 --> 00:00:07.250
let's see if we can work with
how fast something spins.

00:00:07.250 --> 00:00:09.820
Let's see what we can do.

00:00:09.820 --> 00:00:11.970
Since we're going to be working
with things spinning,

00:00:11.970 --> 00:00:14.080
let me draw a circle.

00:00:14.080 --> 00:00:17.370
Since things that spin
go in circles.

00:00:21.300 --> 00:00:23.810
And let me just draw the
positive x-axis because it'll

00:00:23.810 --> 00:00:26.340
come in handy in a second.

00:00:26.340 --> 00:00:29.120
That's the positive x-axis.

00:00:29.120 --> 00:00:32.610
And let's say that I have an
object, and the circle is the

00:00:32.610 --> 00:00:33.860
object's path.

00:00:33.860 --> 00:00:36.705
So let's say this
is the object.

00:00:36.705 --> 00:00:38.780
And it's going around
in a circle in a

00:00:38.780 --> 00:00:39.755
counterclockwise direction.

00:00:39.755 --> 00:00:41.030
Not squiggly counterclockwise,
it's just

00:00:41.030 --> 00:00:43.620
going around this way.

00:00:43.620 --> 00:00:46.740
Let's say I wanted to figure
out, or I wanted to quantify

00:00:46.740 --> 00:00:50.300
how much, or how fast this
thing is spinning.

00:00:50.300 --> 00:00:53.300
So one thing that you're
probably familiar with is

00:00:53.300 --> 00:00:56.230
revolutions per second, or
rotations per second.

00:00:56.230 --> 00:00:59.130
So let's write that down, let's
just say for the sake of

00:00:59.130 --> 00:01:03.110
argument that this was moving
at, I don't know, 1 revolution

00:01:03.110 --> 00:01:03.630
per second.

00:01:03.630 --> 00:01:06.380
So after 1 second it goes back,
then another second.

00:01:06.380 --> 00:01:07.810
So that's how fast
it's spinning, 1

00:01:07.810 --> 00:01:09.570
revolution per second.

00:01:09.570 --> 00:01:12.800
1 revolution-- I'll just
put rev-- per second.

00:01:15.420 --> 00:01:17.210
So let's see if we can quantify
that in angles, and

00:01:17.210 --> 00:01:19.130
we'll do it in radians, but you
could always convert it

00:01:19.130 --> 00:01:21.440
back to degrees, if you want.

00:01:24.630 --> 00:01:27.660
I don't know if you
can see that line.

00:01:27.660 --> 00:01:31.350
Let's just say that theta is the
angle between the radius

00:01:31.350 --> 00:01:33.180
from the center to
that object, and

00:01:33.180 --> 00:01:35.840
the positive x-axis.

00:01:35.840 --> 00:01:38.980
So if this object is travelling
at 1 revolution per

00:01:38.980 --> 00:01:43.090
second, how many radians per
second is it traveling?

00:01:43.090 --> 00:01:46.120
Well, how many radians are
there in a revolution?

00:01:46.120 --> 00:01:49.740
Well there's 2 pi radians
in a revolution, right?

00:01:49.740 --> 00:01:52.980
1 go-around in a circle
is 2 pi radians.

00:01:52.980 --> 00:02:01.950
So we could say, so this equals
1 rev per second, times

00:02:01.950 --> 00:02:07.680
2 pi radians per rev, right?

00:02:07.680 --> 00:02:10.060
And then the revolutions
will cancel out.

00:02:10.060 --> 00:02:12.230
And you have 1 times
2 pi, so you have 2

00:02:12.230 --> 00:02:14.620
pi radians per second.

00:02:14.620 --> 00:02:25.354
So this equals 2 pi radians
per second.

00:02:25.354 --> 00:02:30.250
So that's interesting, we now
know exactly after 5 seconds

00:02:30.250 --> 00:02:31.810
how many radians it has gone.

00:02:31.810 --> 00:02:34.050
Or after half a second, how
many radians has it gone.

00:02:34.050 --> 00:02:35.650
But that might be
vaguely useful.

00:02:35.650 --> 00:02:38.770
Let's see if we can somehow
convert from this notion of

00:02:38.770 --> 00:02:44.530
how fast something is spinning
to its actual speed.

00:02:44.530 --> 00:02:46.950
I was tempted to say velocity,
but its velocity is always

00:02:46.950 --> 00:02:49.190
changing, because the direction
is always changing.

00:02:49.190 --> 00:02:51.600
But the magnitude of the
velocity is staying the same,

00:02:51.600 --> 00:02:52.740
so its speed is staying
the same.

00:02:52.740 --> 00:02:55.680
But we'll say v for speed,
because that's what they tend

00:02:55.680 --> 00:02:59.200
to do in most formulas
that you'll see.

00:02:59.200 --> 00:03:02.710
So let's think about it this
way, in 1 revolution-- so

00:03:02.710 --> 00:03:06.760
there's a couple ways you can
think about this, but as we go

00:03:06.760 --> 00:03:11.010
1 revolution, how far has
this object traveled?

00:03:11.010 --> 00:03:15.250
Well, it's traveled the
circumference of this circle.

00:03:15.250 --> 00:03:16.770
And in order to know the
circumference, we have to know

00:03:16.770 --> 00:03:17.680
the radius of the circle.

00:03:17.680 --> 00:03:21.540
So let's say that the radius
is r-- let's say it's in

00:03:21.540 --> 00:03:22.870
meters, r meters.

00:03:25.700 --> 00:03:34.510
So how many meters will I travel
in 1 second, then?

00:03:34.510 --> 00:03:36.870
Well, you could do the
same thing up here.

00:03:36.870 --> 00:03:46.360
1 revolution per second, times
2 pi r, where r is the

00:03:46.360 --> 00:03:50.810
radius-- whoops, 2 pi r, you can
ignore that line-- meters

00:03:50.810 --> 00:03:52.950
per revolution, that's just
the circumference of the

00:03:52.950 --> 00:03:54.820
thing, of the circle.

00:03:54.820 --> 00:03:59.450
And that equals-- the
revolutions cancel out-- 2 pi

00:03:59.450 --> 00:04:04.610
r meters per second.

00:04:04.610 --> 00:04:08.290
So it's interesting, given
the radius and how many

00:04:08.290 --> 00:04:11.690
revolutions per second, we can
now figure out its velocity.

00:04:11.690 --> 00:04:17.490
So this right here is how fast
it's spinning, and this is the

00:04:17.490 --> 00:04:20.290
object's actual speed, right?

00:04:20.290 --> 00:04:23.030
And this term of how fast
something's spinning, that's

00:04:23.030 --> 00:04:24.620
called angular velocity.

00:04:24.620 --> 00:04:26.620
And of course you know that
the term for how fast

00:04:26.620 --> 00:04:30.240
something is actually
moving is velocity.

00:04:30.240 --> 00:04:36.240
And just so you know, the term
for angular velocity is this

00:04:36.240 --> 00:04:40.310
curvy w, I think that's
lower case omega,

00:04:40.310 --> 00:04:43.050
that's angular velocity.

00:04:43.050 --> 00:04:49.070
So in this case, angular
velocity is equal to 2 pi

00:04:49.070 --> 00:04:52.200
radians per second.

00:04:52.200 --> 00:04:54.060
And what's the velocity equal
to, or at least the magnitude

00:04:54.060 --> 00:04:56.590
of the velocity-- I know the
direction's always changing.

00:04:56.590 --> 00:05:02.270
Well, we know that the velocity
is equal to 2 pi r

00:05:02.270 --> 00:05:03.920
meters per second.

00:05:03.920 --> 00:05:05.860
So if we just ignore the units
for a second, where do you see

00:05:05.860 --> 00:05:07.590
the difference between
the angular

00:05:07.590 --> 00:05:09.620
velocity and the velocity?

00:05:09.620 --> 00:05:12.850
The angular velocity in this
case is 2 pi, and the

00:05:12.850 --> 00:05:14.680
velocity is 2 pi r.

00:05:14.680 --> 00:05:17.630
So in general, if you just
multiply the angular velocity

00:05:17.630 --> 00:05:19.530
times r, you get the velocity.

00:05:19.530 --> 00:05:23.650
So angular velocity times the
radius is equal to velocity.

00:05:23.650 --> 00:05:26.275
Or you can divide both sides of
that by r, and you get the

00:05:26.275 --> 00:05:28.920
angular velocity is equal
to the velocity

00:05:28.920 --> 00:05:30.490
divided by the radius.

00:05:30.490 --> 00:05:34.250
And this is a formula that you
should know by heart, although

00:05:34.250 --> 00:05:36.820
it's good to know where
it came from.

00:05:36.820 --> 00:05:39.670
I guess I did it this way to
maybe give you an intuition,

00:05:39.670 --> 00:05:42.580
because I always have to
work with numbers.

00:05:42.580 --> 00:05:44.650
Especially when I'm new to a
concept-- so that's why I said

00:05:44.650 --> 00:05:46.390
1 revolution per second,
instead of just putting

00:05:46.390 --> 00:05:50.170
everything as a variable-- but
another way to think about it

00:05:50.170 --> 00:05:52.840
is, what is the definition
of a radian?

00:05:52.840 --> 00:06:03.370
By definition, a radian-- if
this angle is x radians, it's

00:06:03.370 --> 00:06:09.860
an angle, and it also tells us
that the arc that is kind of

00:06:09.860 --> 00:06:14.590
projected by this angle,
is equal to x radiuses.

00:06:17.100 --> 00:06:21.700
So if each radius is 2 meters,
it would be x times 2 meters.

00:06:21.700 --> 00:06:23.700
So if this is x radians,
then this is going to

00:06:23.700 --> 00:06:26.130
be x times r meters.

00:06:26.130 --> 00:06:34.860
And that actually comes from the
definition of the radian.

00:06:34.860 --> 00:06:37.410
And that might be more intuitive
to you, than the

00:06:37.410 --> 00:06:39.230
original explanation, or
less, so hopefully one

00:06:39.230 --> 00:06:40.250
of those two works.

00:06:40.250 --> 00:06:43.730
But as you can see, if this
angle is x, and this distance

00:06:43.730 --> 00:06:52.050
is x times r, and if omega is
change in that angle, over

00:06:52.050 --> 00:06:53.300
change in time.

00:06:57.450 --> 00:07:02.890
Then we know this is true too,
that velocity is just change

00:07:02.890 --> 00:07:04.160
in this, over change
in time, right.

00:07:04.160 --> 00:07:09.680
Velocity is change in-- the
radius doesn't change-- change

00:07:09.680 --> 00:07:13.280
in x times r, divided
by change in time.

00:07:13.280 --> 00:07:15.760
And we know once again
that this is omega.

00:07:15.760 --> 00:07:19.680
So another way we just showed
again, that omega times the

00:07:19.680 --> 00:07:21.015
radius is equal to
the velocity.

00:07:21.015 --> 00:07:23.860
Or the angular velocity
times the radius is

00:07:23.860 --> 00:07:25.240
equal to the velocity.

00:07:25.240 --> 00:07:27.520
And this is a useful thing to
learn, we'll see it in a

00:07:27.520 --> 00:07:31.020
couple of things, when I do
the proof for centripetal

00:07:31.020 --> 00:07:33.600
acceleration in calculus, I'm
going to use this fact.

00:07:33.600 --> 00:07:36.510
And when I-- and actually I'm
probably going to record that

00:07:36.510 --> 00:07:39.020
video now-- I'm actually going
to show you the law of

00:07:39.020 --> 00:07:41.370
conservation of angular
momentum, which is very

00:07:41.370 --> 00:07:44.260
similar to the law of
conservation of momentum, but

00:07:44.260 --> 00:07:45.250
it deals with things spinning.

00:07:45.250 --> 00:07:47.820
And this notion of angular
velocity is

00:07:47.820 --> 00:07:49.090
going to come in useful.

00:07:49.090 --> 00:07:51.440
So this is the important
takeaway, that w

00:07:51.440 --> 00:07:52.670
equals v over r.

00:07:52.670 --> 00:07:55.070
And hopefully my video has not
confused you, and has shown

00:07:55.070 --> 00:07:59.350
you that w, the rate at which
the angle is changing, is

00:07:59.350 --> 00:08:01.700
equal to the velocity of the
object, or the magnitude of

00:08:01.700 --> 00:08:05.330
the velocity, divided
by the radius of the

00:08:05.330 --> 00:08:07.280
circle that it's spinning.

00:08:07.280 --> 00:08:09.420
I'll see you in the
next video.


Visual Proof: a= v^2/r

WEBVTT
Kind: captions
Language: en

00:00:00.710 --> 00:00:01.550
Welcome back.

00:00:01.550 --> 00:00:03.670
In the last couple of videos,
actually the very first video

00:00:03.670 --> 00:00:06.080
on centripetal acceleration, I
told you that the necessary

00:00:06.080 --> 00:00:08.580
centripetal acceleration, and
we're just talking about the

00:00:08.580 --> 00:00:12.630
magnitude, and actually the c
tells you the direction, it's

00:00:12.630 --> 00:00:16.430
centripetal so its inward
acceleration, it equals the

00:00:16.430 --> 00:00:18.610
velocity squared over
the radius.

00:00:18.610 --> 00:00:21.170
I told you that the real
rigorous proof has to be done

00:00:21.170 --> 00:00:21.620
with calculus.

00:00:21.620 --> 00:00:24.130
But I looked on Wikipedia and
there's actually a pretty neat

00:00:24.130 --> 00:00:28.230
proof, although when you read
Wikipedia, it's not so obvious

00:00:28.230 --> 00:00:29.150
on what they're saying.

00:00:29.150 --> 00:00:30.810
So I thought I would do a video
on it, because this is

00:00:30.810 --> 00:00:33.990
cool and you don't need calculus
to understand it.

00:00:33.990 --> 00:00:35.430
So let's just do something.

00:00:35.430 --> 00:00:40.285
Let's just plot the distance
vector, the velocity vectors

00:00:40.285 --> 00:00:41.890
and the acceleration vectors
as something

00:00:41.890 --> 00:00:42.920
goes around a circle.

00:00:42.920 --> 00:00:45.770
Let me draw two circles.

00:00:45.770 --> 00:00:50.340
So let's say this is
my first circle.

00:00:50.340 --> 00:00:55.190
Let me draw another circle,
another color just for fun.

00:00:55.190 --> 00:00:57.170
OK.

00:00:57.170 --> 00:00:58.420
This is my other circle.

00:01:02.210 --> 00:01:05.770
This is the center
of the circle.

00:01:05.770 --> 00:01:08.410
And this is the center
of the circle.

00:01:08.410 --> 00:01:12.340
And so what's the position
vector in any point in time?

00:01:12.340 --> 00:01:15.450
Well, the position vector,
you could just

00:01:15.450 --> 00:01:17.190
draw it as a radius.

00:01:19.950 --> 00:01:22.940
So the position vector at
any given time looks

00:01:22.940 --> 00:01:23.710
something like this.

00:01:23.710 --> 00:01:25.600
So let's say initially this
is the position vector.

00:01:29.420 --> 00:01:31.200
That's the initial
position vector.

00:01:31.200 --> 00:01:35.680
Its magnitude is the radius
of the circle.

00:01:35.680 --> 00:01:39.360
And this direction is right
here in the positive

00:01:39.360 --> 00:01:40.480
x-direction.

00:01:40.480 --> 00:01:43.380
And at that point, what is
the velocity vector?

00:01:43.380 --> 00:01:44.870
Well, the velocity-- let's
assume we're going

00:01:44.870 --> 00:01:45.880
counterclockwise.

00:01:45.880 --> 00:01:47.740
I don't know why
I assumed that.

00:01:47.740 --> 00:01:49.720
It could go the other way.

00:01:49.720 --> 00:01:52.346
Let's say that this is the
velocity vector at that point.

00:01:52.346 --> 00:01:56.830
The velocity vector is going to
look something like that.

00:01:59.810 --> 00:02:02.580
That is the velocity vector.

00:02:02.580 --> 00:02:04.440
That's v.

00:02:04.440 --> 00:02:07.510
It's going tangent
to the circle.

00:02:07.510 --> 00:02:09.940
Let's plot the velocity vector
as a function of

00:02:09.940 --> 00:02:10.919
time on this circle.

00:02:10.919 --> 00:02:13.870
So if that's the velocity vector
of that time, I can

00:02:13.870 --> 00:02:15.100
draw the velocity vector here.

00:02:15.100 --> 00:02:15.880
This the the same vector.

00:02:15.880 --> 00:02:18.570
Remember, I'm just saying, at
a particular time, what does

00:02:18.570 --> 00:02:20.710
the distance vector look like
and what does the velocity

00:02:20.710 --> 00:02:21.400
vector look like?

00:02:21.400 --> 00:02:25.660
So at that time, the velocity
vector looks like this.

00:02:25.660 --> 00:02:27.570
I'm trying to make it the
same size to show you

00:02:27.570 --> 00:02:28.190
it's the same vector.

00:02:28.190 --> 00:02:29.650
And I'm doing it in the same
color to show you it's the

00:02:29.650 --> 00:02:32.560
same vector.

00:02:32.560 --> 00:02:35.300
This is the exact same vector.

00:02:35.300 --> 00:02:36.360
This is the actual circle.

00:02:36.360 --> 00:02:39.500
Like if you were to draw it,
this is the path of the dot or

00:02:39.500 --> 00:02:40.990
whatever is moving around
the circle.

00:02:40.990 --> 00:02:44.510
And this, you can just view it
as I'm plotting the velocity

00:02:44.510 --> 00:02:47.790
over time, the velocity
vector over time.

00:02:47.790 --> 00:02:52.470
So let's say a few seconds
later, or a few moments later,

00:02:52.470 --> 00:02:55.460
what does the radius
vector look like?

00:02:55.460 --> 00:02:57.030
Well, then the radius vector
looks like this.

00:03:04.216 --> 00:03:07.620
I'm trying to write as
neatly as possible.

00:03:07.620 --> 00:03:10.150
And what does the velocity
vector look like?

00:03:10.150 --> 00:03:12.700
Go back to the purple.

00:03:12.700 --> 00:03:15.850
The velocity vector once again
is tangent to the circle.

00:03:15.850 --> 00:03:19.160
And it'll look something
like that.

00:03:19.160 --> 00:03:20.160
It's going to have the
same magnitude,

00:03:20.160 --> 00:03:21.410
just different direction.

00:03:23.700 --> 00:03:25.870
Actually, let me do it in a
different color, just to show

00:03:25.870 --> 00:03:26.590
you what I'm doing.

00:03:26.590 --> 00:03:31.250
So let me do it in brown.

00:03:31.250 --> 00:03:34.020
So the velocity vector
is going to look

00:03:34.020 --> 00:03:36.190
something like this.

00:03:36.190 --> 00:03:40.040
It's brown, supposed
to be a brown.

00:03:40.040 --> 00:03:40.770
That's the velocity vector.

00:03:40.770 --> 00:03:44.260
So after a few seconds, or a
few moments, where is the

00:03:44.260 --> 00:03:46.270
velocity vector here?

00:03:46.270 --> 00:03:47.300
Well, it's going to
look like this.

00:03:47.300 --> 00:03:48.850
Remember, this vector
I'm just plotting

00:03:48.850 --> 00:03:51.840
here after a few seconds.

00:03:51.840 --> 00:03:54.320
Let me use that line tool.

00:03:54.320 --> 00:03:58.630
So it's the same magnitude and
now the velocity is just at a

00:03:58.630 --> 00:03:59.290
different angle.

00:03:59.290 --> 00:04:01.615
It should be the exact same
angle as what I just drew in

00:04:01.615 --> 00:04:02.865
the other circle.

00:04:07.790 --> 00:04:11.660
So that's this velocity
vector.

00:04:11.660 --> 00:04:13.460
So when we start here, the
velocity is going to change at

00:04:13.460 --> 00:04:14.370
the circle.

00:04:14.370 --> 00:04:17.019
After a few moments, the
object's rotating around the

00:04:17.019 --> 00:04:19.390
circle, so now the velocity
is the same magnitude.

00:04:19.390 --> 00:04:21.380
It's just switched directions.

00:04:21.380 --> 00:04:22.250
And so what's happening here?

00:04:22.250 --> 00:04:25.370
When we plot the velocity vector
over time, it has the

00:04:25.370 --> 00:04:29.750
same magnitude, so it'll draw
out a circle, but its

00:04:29.750 --> 00:04:32.680
direction changes.

00:04:32.680 --> 00:04:36.110
Just as in this case, what takes
us from this point to

00:04:36.110 --> 00:04:36.910
this point?

00:04:36.910 --> 00:04:38.960
This was the velocity vector
and the velocity vector's

00:04:38.960 --> 00:04:39.690
always changing.

00:04:39.690 --> 00:04:42.300
But, in general, this is
the change in position.

00:04:42.300 --> 00:04:43.680
And what causes the change
in position?

00:04:43.680 --> 00:04:45.940
Well, the velocity, or at least
the speed, because the

00:04:45.940 --> 00:04:47.720
vector's always changing.

00:04:47.720 --> 00:04:50.390
So in this case, what's
changing the velocity?

00:04:50.390 --> 00:04:54.270
Well, just like velocity changes
radius, acceleration

00:04:54.270 --> 00:04:56.670
changes velocity.

00:04:56.670 --> 00:05:00.910
Let me draw an acceleration
vector, and acceleration is

00:05:00.910 --> 00:05:04.690
going to be in this direction.

00:05:04.690 --> 00:05:06.930
Because if the velocity is
changing from here to here,

00:05:06.930 --> 00:05:10.410
the acceleration is going along
the direction of the

00:05:10.410 --> 00:05:12.950
change in velocity.

00:05:12.950 --> 00:05:14.380
So the acceleration
vector might look

00:05:14.380 --> 00:05:15.010
something like that.

00:05:15.010 --> 00:05:17.200
It's going to be tangent
to this velocity path.

00:05:21.700 --> 00:05:23.940
And that's interesting, too,
because if this is the

00:05:23.940 --> 00:05:27.000
acceleration vector when the
velocity is here, so when the

00:05:27.000 --> 00:05:29.100
velocity is here, the
acceleration vector's going

00:05:29.100 --> 00:05:32.010
directly to the left, so that
means that acceleration

00:05:32.010 --> 00:05:34.890
vector's going directly to left,
and so this is also the

00:05:34.890 --> 00:05:38.120
acceleration vector, which
coincides with what we learned

00:05:38.120 --> 00:05:39.610
about centripetal
acceleration.

00:05:39.610 --> 00:05:41.610
The acceleration has to
be going inwards.

00:05:41.610 --> 00:05:43.300
And we see that when
we actually plot

00:05:43.300 --> 00:05:44.720
the velocity vector.

00:05:44.720 --> 00:05:49.430
And if we plot acceleration
vector here, once again it's

00:05:49.430 --> 00:05:51.700
going to be going tangent
to this plot of

00:05:51.700 --> 00:05:52.920
the velocity vector.

00:05:52.920 --> 00:05:53.880
Let me do it in a
different color.

00:05:53.880 --> 00:05:54.602
I've already used that color.

00:05:54.602 --> 00:05:56.280
I'll do it in yellow.

00:05:56.280 --> 00:05:57.870
So then the acceleration
vector is going to look

00:05:57.870 --> 00:05:58.460
something like this.

00:05:58.460 --> 00:05:59.530
Remember, acceleration
is nothing

00:05:59.530 --> 00:06:00.930
but change in velocity.

00:06:00.930 --> 00:06:03.915
So at this point, when this is
the velocity vector, the

00:06:03.915 --> 00:06:06.130
acceleration vector
will just be this.

00:06:06.130 --> 00:06:10.540
Once again, it's just the
opposite direction of the

00:06:10.540 --> 00:06:11.500
position vector.

00:06:11.500 --> 00:06:12.720
So why am I doing all this?

00:06:12.720 --> 00:06:14.240
Well, I'm doing all of
this to set up the

00:06:14.240 --> 00:06:16.700
analogy to show you.

00:06:16.700 --> 00:06:22.210
As this object completes one
entire path, what's happening

00:06:22.210 --> 00:06:22.960
on this circle?

00:06:22.960 --> 00:06:25.890
Well, this circle, the velocity
is completing one

00:06:25.890 --> 00:06:27.480
entire path, right?

00:06:27.480 --> 00:06:31.220
However long it takes to go
around this circle is the same

00:06:31.220 --> 00:06:35.840
amount of time it takes to go
from this velocity back to

00:06:35.840 --> 00:06:36.630
this velocity.

00:06:36.630 --> 00:06:38.950
The magnitude is the same the
whole time, but the direction

00:06:38.950 --> 00:06:39.990
is changing.

00:06:39.990 --> 00:06:43.010
So if this takes 10 seconds to
go around the circle in real

00:06:43.010 --> 00:06:46.910
position, then it takes 10
seconds for the acceleration

00:06:46.910 --> 00:06:50.210
to change the direction of this
velocity enough that it

00:06:50.210 --> 00:06:53.910
goes back to the original
velocity direction.

00:06:53.910 --> 00:06:55.950
So why am I doing all of that?

00:06:55.950 --> 00:07:01.110
Well, how long does it take
for the object to do one

00:07:01.110 --> 00:07:03.660
rotation around this path?

00:07:03.660 --> 00:07:06.650
Well, it's just the distance
divided by the velocity.

00:07:06.650 --> 00:07:10.740
So the time to do one revolution
around this path is

00:07:10.740 --> 00:07:12.190
the distance-- well,
that's just the

00:07:12.190 --> 00:07:13.420
circumference of the circle.

00:07:13.420 --> 00:07:21.920
Well, that's 2 pi r divided
by your speed, which is v.

00:07:21.920 --> 00:07:26.190
And how much time does it take
to go around this path?

00:07:26.190 --> 00:07:31.080
Well, we know it's going be the
same time, and now we're

00:07:31.080 --> 00:07:33.400
going to say the time's
going to be the change

00:07:33.400 --> 00:07:34.770
in velocity, right?

00:07:34.770 --> 00:07:37.435
In this circle, this is the
change in distance, but it's

00:07:37.435 --> 00:07:40.170
the change in velocity.

00:07:40.170 --> 00:07:42.530
So here we had this much
change in velocity.

00:07:42.530 --> 00:07:44.450
I know this might be a
little non-intuitive.

00:07:44.450 --> 00:07:46.330
Then we have a little bit more
change in velocity, a little

00:07:46.330 --> 00:07:48.260
bit more change in velocity,
a little bit

00:07:48.260 --> 00:07:49.810
more change in velocity.

00:07:49.810 --> 00:07:53.790
So the total change in velocity
is just going to be

00:07:53.790 --> 00:07:56.740
the circumference
of this circle.

00:07:56.740 --> 00:07:58.720
And what's the circumference
of this circle?

00:07:58.720 --> 00:08:01.870
The radius of the circle is the
magnitude of the velocity.

00:08:01.870 --> 00:08:06.140
So it's 2 pi times the velocity
of the circle.

00:08:06.140 --> 00:08:10.570
And so, what is the amount
of time it takes a do one

00:08:10.570 --> 00:08:11.170
revolution?

00:08:11.170 --> 00:08:13.280
Well, it's the total
change in velocity.

00:08:13.280 --> 00:08:16.010
And that's 2 pi times the
magnitude of the velocity

00:08:16.010 --> 00:08:19.110
divided by acceleration.

00:08:19.110 --> 00:08:21.490
And if this doesn't make
complete sense, just remember,

00:08:21.490 --> 00:08:24.980
acceleration is change in
velocity over change in time.

00:08:29.290 --> 00:08:32.520
I'm glossing over it a little
bit, because you might say,

00:08:32.520 --> 00:08:34.299
oh, well, the net change
in velocity is zero.

00:08:34.299 --> 00:08:38.059
But we're actually more
concerned about what's the

00:08:38.059 --> 00:08:40.100
total change in the velocity?

00:08:40.100 --> 00:08:43.559
So it changes a lot, then it
goes back to its original, so

00:08:43.559 --> 00:08:45.420
that the net is still zero.

00:08:45.420 --> 00:08:48.200
But this should hopefully give
you a little intuition.

00:08:48.200 --> 00:08:49.250
So think about what I said.

00:08:49.250 --> 00:08:52.760
But the bottom line is we know
that this expression and this

00:08:52.760 --> 00:08:56.080
expression have to be equal,
because in the same amount of

00:08:56.080 --> 00:09:00.050
time it takes the object to do
one complete revolution on

00:09:00.050 --> 00:09:03.730
this circle, its velocity,
direction, has also done one

00:09:03.730 --> 00:09:04.560
complete revolution.

00:09:04.560 --> 00:09:07.070
So we can set these two
equal to each other.

00:09:07.070 --> 00:09:16.440
So we could say 2 pi r over v
is equal to 2 pi v over a.

00:09:16.440 --> 00:09:17.750
Let me switch colors.

00:09:17.750 --> 00:09:21.490
We can cross the 2 pi
out on both sides.

00:09:21.490 --> 00:09:25.530
And then we could multiply
both sides times v.

00:09:25.530 --> 00:09:28.970
You get r is equal to
v squared over a.

00:09:28.970 --> 00:09:32.440
Multiply both sides times a,
I'm doing this little bit

00:09:32.440 --> 00:09:36.120
circuitously, and you get ar
is equal to v squared.

00:09:36.120 --> 00:09:38.940
Divide both sides by r, and
you get a is equal to v

00:09:38.940 --> 00:09:40.850
squared over r.

00:09:40.850 --> 00:09:43.010
So hopefully, that gives you a
little bit of intuition about

00:09:43.010 --> 00:09:44.650
why this works.

00:09:44.650 --> 00:09:46.240
And I got this from Wikipedia,
so I want to

00:09:46.240 --> 00:09:47.210
give them proper credit.

00:09:47.210 --> 00:09:50.550
And hopefully, my explanation
helps clarify what the people

00:09:50.550 --> 00:09:52.940
on Wikipedia are talking
about a little bit.

00:09:52.940 --> 00:09:55.880
Anyway, I'll also do the
calculus proof, because that's

00:09:55.880 --> 00:09:57.720
more just straight up math.

00:09:57.720 --> 00:10:00.070
And I'll see you in
the next video.


Centripetal Acceleration (part 3)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.460
Welcome back.

00:00:01.460 --> 00:00:04.250
I thought I'd do a couple of
centripetal acceleration

00:00:04.250 --> 00:00:06.250
problems. Some of the more
common ones that you

00:00:06.250 --> 00:00:07.300
maybe see in school.

00:00:07.300 --> 00:00:10.015
So let's say I have a, well it
could be a hot wheel, or it

00:00:10.015 --> 00:00:11.640
could be a car, a real car.

00:00:11.640 --> 00:00:13.195
And say it's on a track.

00:00:13.195 --> 00:00:15.520
So we draw the track.

00:00:15.520 --> 00:00:16.550
What's a good color
for a track.

00:00:16.550 --> 00:00:17.390
I don't know, yellow.

00:00:17.390 --> 00:00:18.480
It's a yellow track.

00:00:18.480 --> 00:00:20.770
So this is going to
be a side view.

00:00:20.770 --> 00:00:23.160
And we all remember this from
our Hot Wheels days.

00:00:23.160 --> 00:00:26.840
And the track does a
loop-d-loop, like that.

00:00:26.840 --> 00:00:28.840
And then I have a car, and
let's say it's going at a

00:00:28.840 --> 00:00:29.700
constant velocity.

00:00:29.700 --> 00:00:31.980
We're not going to worry about
the car's, or at least a

00:00:31.980 --> 00:00:34.270
constant speed, right, cause
the direction of velocity

00:00:34.270 --> 00:00:35.080
might change.

00:00:35.080 --> 00:00:36.410
Just not the magnitude.

00:00:36.410 --> 00:00:38.460
So this is the car.

00:00:38.460 --> 00:00:41.120
And my question to you is, how
fast does it have to be going?

00:00:41.120 --> 00:00:44.580
What's its speed have to be
for it to not fall when it

00:00:44.580 --> 00:00:46.800
gets to this point in
the loop-d-loop?

00:00:46.800 --> 00:00:47.950
So it's going to go like this.

00:00:47.950 --> 00:00:50.960
This is going to be the
path of the car.

00:00:50.960 --> 00:00:53.800
How fast does it need to go?

00:00:53.800 --> 00:00:56.180
So let's just think about it.

00:00:56.180 --> 00:00:58.340
Of course, I have to give a
little bit of information.

00:00:58.340 --> 00:01:07.590
Let's say the radius of this
loop-d-loop, I don't know,

00:01:07.590 --> 00:01:08.790
let's say it's 20 feet.

00:01:08.790 --> 00:01:09.930
I'm making this up on the fly.

00:01:09.930 --> 00:01:11.760
I hope the numbers
work out well.

00:01:11.760 --> 00:01:16.020
So what we have to figure out
is essentially, what is the

00:01:16.020 --> 00:01:18.260
centripetal acceleration going
to be on this car?

00:01:18.260 --> 00:01:19.980
And if we know the centripetal
acceleration and we know the

00:01:19.980 --> 00:01:22.370
radius, we can figure
out the velocity.

00:01:22.370 --> 00:01:24.710
So let's just think about what
happens to the car as it goes

00:01:24.710 --> 00:01:26.400
up the loop-d-loop.

00:01:26.400 --> 00:01:28.950
At this point, let's say when
the car's right here-- so this

00:01:28.950 --> 00:01:33.490
is the car-- what is making
the car's velocity change?

00:01:33.490 --> 00:01:36.170
Cause at this point the car's
velocity is like this.

00:01:36.170 --> 00:01:38.820
And then at this point, the
car's velocity is like this.

00:01:38.820 --> 00:01:40.750
At this point, the car's
velocity is like this.

00:01:40.750 --> 00:01:44.720
It's always going to be tangent
to the loop-d-loop.

00:01:44.720 --> 00:01:46.950
Well, this point, down here,
is actually going to be the

00:01:46.950 --> 00:01:49.580
normal force of the
loop-d-loop right?

00:01:49.580 --> 00:01:53.650
As the car is kind of driving
into the slope, the slope is

00:01:53.650 --> 00:01:56.590
putting upward pressure on the
car that's making it go in the

00:01:56.590 --> 00:01:58.280
circular path.

00:01:58.280 --> 00:02:01.460
But as we go beyond this
point, we see something

00:02:01.460 --> 00:02:02.820
interesting happening.

00:02:02.820 --> 00:02:08.639
There's going to be the normal
force of the loop-d-loop

00:02:08.639 --> 00:02:09.949
itself, of the surface.

00:02:09.949 --> 00:02:12.030
And then we're also going to
start having gravity pulling

00:02:12.030 --> 00:02:13.050
down on the car.

00:02:13.050 --> 00:02:15.590
And some portion of the gravity
will kind of pull to

00:02:15.590 --> 00:02:16.350
the center.

00:02:16.350 --> 00:02:18.440
And it's a little complicated
at this point.

00:02:18.440 --> 00:02:23.400
But at this point-- let me draw
the car, let me draw the

00:02:23.400 --> 00:02:29.540
upside down car-- what
are the forces?

00:02:29.540 --> 00:02:35.630
Well, if we want the car to just
not fall, right, the only

00:02:35.630 --> 00:02:37.480
force is going to be the force
of gravity, or the

00:02:37.480 --> 00:02:39.380
acceleration of gravity.

00:02:39.380 --> 00:02:42.110
If the car is going fast
enough to offset the

00:02:42.110 --> 00:02:46.060
acceleration of gravity, it'll
stay on the loop-d-loop.

00:02:46.060 --> 00:02:49.310
So let's figure out how fast it
has to be going to offset

00:02:49.310 --> 00:02:51.060
the acceleration of gravity.

00:02:51.060 --> 00:02:53.620
Well, we know from the previous
video, that the

00:02:53.620 --> 00:02:58.020
centripetal acceleration is
equal to the magnitude-- I'm

00:02:58.020 --> 00:02:59.800
going to put an absolute value
around that so you know it's

00:02:59.800 --> 00:03:01.180
not V for velocity
vector, it's the

00:03:01.180 --> 00:03:02.850
magnitude of the velocity.

00:03:02.850 --> 00:03:04.870
I know I keep changing
conventions, but that's good

00:03:04.870 --> 00:03:07.720
for you cause you'll see
different things-- divided by

00:03:07.720 --> 00:03:08.200
the radius.

00:03:08.200 --> 00:03:09.260
Velocity squared over radius.

00:03:09.260 --> 00:03:10.250
Remember, this isn't
the vector.

00:03:10.250 --> 00:03:12.140
This is just the magnitude.

00:03:12.140 --> 00:03:14.710
And once again this is also
just the magnitude of the

00:03:14.710 --> 00:03:15.680
acceleration vector.

00:03:15.680 --> 00:03:16.780
So we know what the
acceleration

00:03:16.780 --> 00:03:18.030
vector is at this point.

00:03:20.730 --> 00:03:23.340
Well, we're just worried
about the magnitude.

00:03:23.340 --> 00:03:25.670
We'll say it's 32 feet
per second squared.

00:03:25.670 --> 00:03:28.890
That's the acceleration of
gravity on the surface of the

00:03:28.890 --> 00:03:29.910
earth, at sea level.

00:03:29.910 --> 00:03:32.300
30 feet per second squared.

00:03:32.300 --> 00:03:35.860
And that equals the velocity
squared-- I'll get rid of the

00:03:35.860 --> 00:03:40.410
absolute value sign, or the
magnitude sign-- divided by

00:03:40.410 --> 00:03:43.270
the radius, divided by 20.

00:03:43.270 --> 00:03:45.380
And so multiply both
sides by 20.

00:03:45.380 --> 00:03:50.645
I get 640 is equal to
velocity squared.

00:03:53.240 --> 00:04:04.170
And so 640 square
root is 25.3.

00:04:04.170 --> 00:04:11.366
So the car has to be going
25.3 feet per second.

00:04:11.366 --> 00:04:12.780
So now I'll ask you
a question.

00:04:12.780 --> 00:04:15.500
What happens if it's going
at 20 feet per second?

00:04:15.500 --> 00:04:17.640
Well, if it's going 20 feet per
second, some place around

00:04:17.640 --> 00:04:21.470
here, let me draw the path of
the slower car going 20 feet

00:04:21.470 --> 00:04:22.240
per second.

00:04:22.240 --> 00:04:24.830
If the car's going 20 feet per
second, it'll probably make it

00:04:24.830 --> 00:04:26.870
pretty far, and then some place
around here, it's just

00:04:26.870 --> 00:04:29.870
going to start falling and
then it'll fall down.

00:04:29.870 --> 00:04:31.490
That's a car going 20
feet per second.

00:04:34.210 --> 00:04:35.390
I'll ask you another question.

00:04:35.390 --> 00:04:38.590
What's going to happen if
the car goes faster?

00:04:38.590 --> 00:04:40.950
Let's say what happens if the
velocity is-- and I'll do it

00:04:40.950 --> 00:04:46.990
in a different color-- I don't
know, 50 feet per second?

00:04:46.990 --> 00:04:48.750
So this is a super fast car.

00:04:52.660 --> 00:04:54.800
So if this was just
a simple orbit

00:04:54.800 --> 00:04:56.080
problem, what would happen?

00:04:56.080 --> 00:04:59.550
We have this centripetal
acceleration from gravity, but

00:04:59.550 --> 00:05:02.600
that alone isn't enough to
offset its velocity.

00:05:02.600 --> 00:05:05.120
So if we have no other forces
other than gravity, the car

00:05:05.120 --> 00:05:09.520
would kind of exit its orbit,
you could kind of say right?

00:05:09.520 --> 00:05:11.620
It would actually go
out of its path.

00:05:11.620 --> 00:05:13.190
It might do something
like this.

00:05:13.190 --> 00:05:15.160
Whoops, its path might look
something like this.

00:05:15.160 --> 00:05:17.390
I know I draw these very messy
things, but it might actually

00:05:17.390 --> 00:05:19.170
fly out, right?

00:05:19.170 --> 00:05:21.700
So what's keeping it
from flying out?

00:05:21.700 --> 00:05:25.430
Well, the actual surface of
the loop-d-loop right?

00:05:25.430 --> 00:05:29.150
So we have something very
interesting going on here.

00:05:29.150 --> 00:05:30.300
Well, let's figure it out.

00:05:30.300 --> 00:05:32.580
What is the centripetal
acceleration have to be?

00:05:32.580 --> 00:05:34.630
Well, if the car's going 50 feet
per second, in order to

00:05:34.630 --> 00:05:38.840
keep it going in a circle, the
acceleration has to be 50

00:05:38.840 --> 00:05:43.860
squared over the radius,
which is 20.

00:05:43.860 --> 00:05:50.630
50 times 50, so that's
25, two 0's over 20.

00:05:50.630 --> 00:05:52.020
The 0's cancel.

00:05:52.020 --> 00:05:53.960
So the acceleration is
going have to be

00:05:53.960 --> 00:05:59.490
125 feet per second.

00:05:59.490 --> 00:06:02.090
The inward acceleration, the
centripetal acceleration has

00:06:02.090 --> 00:06:04.980
to be 125 feet per second.

00:06:04.980 --> 00:06:06.340
So what's going on?

00:06:06.340 --> 00:06:07.930
Gravity is only going
to provide 32 feet

00:06:07.930 --> 00:06:09.220
per second of that.

00:06:09.220 --> 00:06:12.330
So the rest of it is actually
going to be the normal force

00:06:12.330 --> 00:06:15.740
of the surface of the
loop-d-loop or of the surface

00:06:15.740 --> 00:06:17.690
of the road, cause the
car is going to push.

00:06:17.690 --> 00:06:22.020
The car is going to want to
kind of exit its orbit.

00:06:22.020 --> 00:06:23.570
It's going to do something
like that.

00:06:23.570 --> 00:06:26.610
And what's keeping it from
doing that is the road.

00:06:26.610 --> 00:06:28.630
The road is keeping
the car contained.

00:06:28.630 --> 00:06:34.530
And so it's essentially putting
enough extra normal

00:06:34.530 --> 00:06:39.050
force onto the car and so that
force will kind of be applied

00:06:39.050 --> 00:06:41.850
to the tires to offset that.

00:06:41.850 --> 00:06:44.580
And there you have it.

00:06:44.580 --> 00:06:47.680
So think of it this way.

00:06:47.680 --> 00:06:54.380
If you were tied to the
loop-d-loop on the inside,

00:06:54.380 --> 00:06:55.700
right here.

00:06:55.700 --> 00:06:57.930
If this is some kind of western
movie where the

00:06:57.930 --> 00:07:01.130
heroine is about to die, and
there's a car coming that's

00:07:01.130 --> 00:07:02.780
going to roll her over.

00:07:02.780 --> 00:07:08.950
If that car is going at exactly
25.3 feet per second,

00:07:08.950 --> 00:07:10.130
it'll probably just
kind of bump

00:07:10.130 --> 00:07:11.140
over the heroine actually.

00:07:11.140 --> 00:07:12.780
The heroine probably will cause
the car to fall and she

00:07:12.780 --> 00:07:14.500
probably won't die.

00:07:14.500 --> 00:07:18.120
But if that car's going at 50
feet per second, then there's

00:07:18.120 --> 00:07:23.750
enough normal force so the road
is pushing on the car,

00:07:23.750 --> 00:07:25.720
and of course is an equal and
opposite force, so the car is

00:07:25.720 --> 00:07:26.600
pushing down.

00:07:26.600 --> 00:07:28.410
And that amount of the
car pushing down

00:07:28.410 --> 00:07:31.030
would squish the heroine.

00:07:31.030 --> 00:07:34.240
So I don't know if that was
disturbing or useful or

00:07:34.240 --> 00:07:36.060
confusing, but this is
just another way

00:07:36.060 --> 00:07:37.000
to think about things.

00:07:37.000 --> 00:07:38.550
Anyway, I'll see you
in the next video.