Ratios on coordinate plane

WEBVTT
Kind: captions
Language: en

00:00:01.007 --> 00:00:02.604
- [Instructor] We are told that a baker

00:00:02.604 --> 00:00:06.392
uses eight cups of flour to
make one batch of muffins

00:00:06.392 --> 00:00:08.301
for his bakery.

00:00:08.301 --> 00:00:11.787
Complete the table for the given ratio.

00:00:11.787 --> 00:00:14.376
So they're saying that for every batch,

00:00:14.376 --> 00:00:16.098
he needs eight cups of flour

00:00:16.098 --> 00:00:19.102
or he needs eight cups
of flour for every batch.

00:00:19.102 --> 00:00:20.982
So if you have two batches

00:00:20.982 --> 00:00:22.608
how many cups of flour would that be?

00:00:22.608 --> 00:00:24.781
Pause the video and try to figure it out.

00:00:24.781 --> 00:00:26.964
Well if he has twice as many
batches, he's gonna have

00:00:26.964 --> 00:00:29.516
twice the number of cups of flour.

00:00:29.516 --> 00:00:32.685
So instead of eight, it
would be 16 cups of flour.

00:00:32.685 --> 00:00:35.285
And if he had three times
the number of batches,

00:00:35.285 --> 00:00:37.934
it would be three times the
number of cups of flour.

00:00:37.934 --> 00:00:42.263
So instead of eight, it would
be eight times three, or 24.

00:00:42.263 --> 00:00:46.034
Now down here, they say
plot the ordered pairs

00:00:46.034 --> 00:00:49.731
x comma y from the table
on the following graph.

00:00:49.731 --> 00:00:53.337
So we wanna graph one batch, eight cups,

00:00:53.337 --> 00:00:57.461
two batches, 16 cups,
three batches, 24 cups.

00:00:57.461 --> 00:00:59.017
So let's do that.

00:00:59.017 --> 00:01:00.365
Let's see if we can, okay.

00:01:00.365 --> 00:01:04.308
So right here, I'm assuming
on the horizontal axis

00:01:04.308 --> 00:01:06.959
that is our batches, and
then our vertical axis

00:01:06.959 --> 00:01:08.269
is cups of flour.

00:01:08.269 --> 00:01:11.886
So for every batch, we
need eight cups of flour.

00:01:11.886 --> 00:01:14.365
So one batch, this is
eight right over here,

00:01:14.365 --> 00:01:16.620
five, six, seven, eight,

00:01:16.620 --> 00:01:20.298
and then for two batches,
we're going to need

00:01:20.298 --> 00:01:22.817
16 cups of flour so that puts us

00:01:22.817 --> 00:01:25.084
right over there, that's 16.

00:01:25.084 --> 00:01:28.648
And then for three batches,
we are going to need

00:01:28.648 --> 00:01:31.876
24 cups of flour, and that
actually goes slightly off

00:01:31.876 --> 00:01:34.855
of our screen here, let
me scroll up a little bit.

00:01:34.855 --> 00:01:38.412
So for three batches, we're
going to have to bring that

00:01:38.412 --> 00:01:42.403
to 24 which is right here and I can see

00:01:42.403 --> 00:01:44.383
the 25 right above that.

00:01:44.383 --> 00:01:47.787
And what you'll see,
because the ratio between

00:01:47.787 --> 00:01:50.761
our batches and our cups
of flour are constant,

00:01:50.761 --> 00:01:53.618
that all of these points,
you could connect them all

00:01:53.618 --> 00:01:57.722
with one straight line
because we have a fixed ratio.

00:01:57.722 --> 00:02:00.831
Every time we move one to the right,

00:02:00.831 --> 00:02:02.529
we're gonna move eight up.

00:02:02.529 --> 00:02:04.509
Every time we add another
batch, we're gonna have

00:02:04.509 --> 00:02:07.751
eight more cups of flour,
every time we add a batch,

00:02:07.751 --> 00:02:10.060
eight more cups of flour.

00:02:10.060 --> 00:02:11.616
Lets do another example.

00:02:11.616 --> 00:02:14.786
Here we're told Drew
earns money washing cars

00:02:14.786 --> 00:02:16.992
for his neighbors on the weekends.

00:02:16.992 --> 00:02:20.435
Drew charges a set rate
for each car he washes.

00:02:20.435 --> 00:02:22.936
The points on the
following coordinate plane

00:02:22.936 --> 00:02:27.000
show how much Drew charges
for two, five, and eight cars.

00:02:27.000 --> 00:02:28.706
So let's see what's going on over here.

00:02:28.706 --> 00:02:33.082
So when he washes two cars,
he looks like he charges $15.

00:02:33.082 --> 00:02:37.097
When he washes five cars,
it looks like he's charging,

00:02:37.097 --> 00:02:40.519
well it looks like some
place between 35 and $40,

00:02:40.519 --> 00:02:42.470
and he charges eight cars,

00:02:42.470 --> 00:02:45.639
it looks like he's charging $60.

00:02:45.639 --> 00:02:48.597
So one way to think about it is the ratio

00:02:48.597 --> 00:02:50.671
between the number of cars he's washing

00:02:50.671 --> 00:02:53.921
and the dollars, it stays at two to 15,

00:02:54.877 --> 00:02:57.457
notice two cars, for every $15.

00:02:57.457 --> 00:03:00.829
I guess I could say
$15 for every two cars.

00:03:00.829 --> 00:03:02.882
And so, when you go to eight cars,

00:03:02.882 --> 00:03:05.695
you're multiplying by
four the number of cars

00:03:05.695 --> 00:03:09.035
and you're also multiplying
by four the number of dollars.

00:03:09.035 --> 00:03:11.810
And so once again, since
we have a fixed ratio here,

00:03:11.810 --> 00:03:14.411
all three of these points
sit on the same line.

00:03:14.411 --> 00:03:16.971
But then they ask us down here,

00:03:16.971 --> 00:03:21.502
they say how much does
Drew charge for four cars?

00:03:21.502 --> 00:03:25.382
Well if it's $15 for two cars, well then

00:03:25.382 --> 00:03:27.759
four cars would be twice as much.

00:03:27.759 --> 00:03:30.848
So it would be $30 for four cars.

00:03:30.848 --> 00:03:33.094
We have the same ratio.

00:03:33.094 --> 00:03:35.600
Let's do one more example.

00:03:35.600 --> 00:03:38.609
Here we're told McKenna
earns money each time

00:03:38.609 --> 00:03:41.401
she shovels snow for her
neighbors, as she should.

00:03:41.401 --> 00:03:44.074
McKenna plots points on
the coordinate plane below

00:03:44.074 --> 00:03:46.057
to show how much she earns
for different numbers

00:03:46.057 --> 00:03:48.704
of times she shovels snow.

00:03:48.704 --> 00:03:50.161
Alright.

00:03:50.161 --> 00:03:53.351
So let's see, when she
shovels snow three times,

00:03:53.351 --> 00:03:56.488
looks like she gets,
halfway between 16 and 20,

00:03:56.488 --> 00:03:58.680
looks like she gets $18,

00:03:58.680 --> 00:04:00.421
four times, looks like it's $24,

00:04:00.421 --> 00:04:04.588
so looks like the ratio is
staying constant at three to 18.

00:04:06.286 --> 00:04:10.119
The ratio of three to
18 is the same thing as,

00:04:11.025 --> 00:04:13.998
one way to think about it
is $18 for every three times

00:04:13.998 --> 00:04:17.025
she shovels snow, that would
be equivalent of six dollars

00:04:17.025 --> 00:04:19.341
for every time she shovels snow.

00:04:19.341 --> 00:04:22.908
So let's go down here to
see what they're asking us.

00:04:22.908 --> 00:04:25.021
They say which of the
following ordered pairs

00:04:25.021 --> 00:04:27.683
could McKenna add to the graph?

00:04:27.683 --> 00:04:31.145
So this would be one
times she shoveling snow,

00:04:31.145 --> 00:04:33.920
would she get $10 for it.

00:04:33.920 --> 00:04:36.704
So does she get $10 for
every time she shovels snow?

00:04:36.704 --> 00:04:39.354
No, that wouldn't be
consistent with the data here.

00:04:39.354 --> 00:04:43.194
She got $18 for shoveling
snow three times.

00:04:43.194 --> 00:04:45.350
So that looks like she's
getting six dollars

00:04:45.350 --> 00:04:48.103
for every time she shovels snow.

00:04:48.103 --> 00:04:51.953
So I would rule out choice A, choice B,

00:04:51.953 --> 00:04:54.799
shoveling snow twice, she gets $12.

00:04:54.799 --> 00:04:56.496
Well that makes sense,
if she gets six dollars

00:04:56.496 --> 00:04:58.182
every time she shovels snow.

00:04:58.182 --> 00:05:01.323
If the ratio of shoveling of the times

00:05:01.323 --> 00:05:04.990
to the dollars is three
to 18 or one to six,

00:05:06.350 --> 00:05:09.420
that would be equivalent to two to 12

00:05:09.420 --> 00:05:11.399
and we would pick that choice.


Ratios with tape diagrams

WEBVTT
Kind: captions
Language: en

00:00:00.984 --> 00:00:03.040
- [Instructor] We're
told Kenzie makes quilts

00:00:03.040 --> 00:00:06.079
with some blue squares
and some green squares.

00:00:06.079 --> 00:00:09.713
The ratio of blue squares
to green squares is shown

00:00:09.713 --> 00:00:11.299
in the diagram.

00:00:11.299 --> 00:00:13.684
The table shows the number of
blue squares and the number

00:00:13.684 --> 00:00:17.830
of green squares that Kenzie
will make on two of her quilts.

00:00:17.830 --> 00:00:19.912
All right, this is the
table they're talking about.

00:00:19.912 --> 00:00:22.534
Based on the ratio,
complete the missing values

00:00:22.534 --> 00:00:24.056
in the table.

00:00:24.056 --> 00:00:25.468
So why don't you pause this video and see

00:00:25.468 --> 00:00:27.517
if you can figure it out.

00:00:27.517 --> 00:00:28.575
Well, first, let's think

00:00:28.575 --> 00:00:31.082
about the ratio of blue to green squares.

00:00:31.082 --> 00:00:33.691
So for every three blue squares

00:00:33.691 --> 00:00:37.435
or that seem a similar color,
for every three blue squares,

00:00:37.435 --> 00:00:40.367
we are going to have
one, two, three, four,

00:00:40.367 --> 00:00:42.353
five green squares.

00:00:42.353 --> 00:00:46.750
So the ratio of blue to
green is three to five,

00:00:46.750 --> 00:00:50.333
and so in quilt A, she
has 21 blue squares.

00:00:52.100 --> 00:00:53.931
So she has 21 blue squares.

00:00:53.931 --> 00:00:55.825
How many green squares would she have?

00:00:55.825 --> 00:00:58.028
Well, to go from three to 21,

00:00:58.028 --> 00:01:00.359
you have to multiply by seven,

00:01:00.359 --> 00:01:04.526
and so you would take five and
then multiply that by seven.

00:01:05.368 --> 00:01:09.843
So you'd multiply five
times seven to get to 35.

00:01:09.843 --> 00:01:12.542
As long as you multiply both
of these by the same number

00:01:12.542 --> 00:01:13.914
or divide them by the same number,

00:01:13.914 --> 00:01:16.006
you're going to get an equivalent ratio.

00:01:16.006 --> 00:01:20.391
So 21 to 35 is the same
thing as three to five.

00:01:20.391 --> 00:01:22.148
Now we have a situation in quilt B.

00:01:22.148 --> 00:01:25.330
They've given us the number of
green squares, so that's 20.

00:01:25.330 --> 00:01:27.233
Well, how do we get 20 from five?

00:01:27.233 --> 00:01:30.042
Well, we would multiply by four.

00:01:30.042 --> 00:01:32.685
So if you multiply the number
of green squares by four,

00:01:32.685 --> 00:01:34.035
then you would do the same thing

00:01:34.035 --> 00:01:35.364
for the number of blue squares.

00:01:35.364 --> 00:01:37.611
Three times four...

00:01:37.611 --> 00:01:39.910
Three times four is
going to be equal to 12.

00:01:39.910 --> 00:01:44.319
12 blue squares for every 20
green squares is the same ratio

00:01:44.319 --> 00:01:48.559
as three blue squares for
every five green squares.

00:01:48.559 --> 00:01:50.642
Let's do another example.

00:01:51.624 --> 00:01:55.482
Here, we are told the following
diagram describes the number

00:01:55.482 --> 00:01:59.142
of cups of blue and
red paint in a mixture.

00:01:59.142 --> 00:02:02.781
What is the ratio of blue paint
to red paint in the mixture?

00:02:02.781 --> 00:02:04.435
So try to work it out.

00:02:04.435 --> 00:02:05.674
All right, so let's just see.

00:02:05.674 --> 00:02:08.978
We have one, two, three,
four, five, six, seven,

00:02:08.978 --> 00:02:11.052
eight, nine, 10...

00:02:11.052 --> 00:02:13.158
10 cups of blue paint

00:02:13.158 --> 00:02:17.075
for every one, two, three,
four, five, six cups

00:02:19.012 --> 00:02:20.793
of red paint.

00:02:20.793 --> 00:02:24.138
So this would be a legitimate
ratio, a ratio of 10 cups

00:02:24.138 --> 00:02:27.324
of blue paint for every
six cups of red paint,

00:02:27.324 --> 00:02:30.166
but this isn't in I guess
you could say lowest terms

00:02:30.166 --> 00:02:32.171
or most simplified terms

00:02:32.171 --> 00:02:35.680
because we can actually divide
both of these numbers by two,

00:02:35.680 --> 00:02:39.327
so if you divide 10 by two, you get five.

00:02:39.327 --> 00:02:41.399
I'll do that in blue color,

00:02:41.399 --> 00:02:45.722
and if you divide six
by two, you get three.

00:02:45.722 --> 00:02:49.843
So one way to think about it
is for every five blue squares,

00:02:49.843 --> 00:02:53.029
you have three red
squares in this diagram,

00:02:53.029 --> 00:02:55.845
in this tape diagram
that's sometimes called,

00:02:55.845 --> 00:02:58.878
or you could say for every
five cups of blue paint,

00:02:58.878 --> 00:03:01.840
you have three cups of
red paint in our mixture,

00:03:01.840 --> 00:03:03.331
and you could even see that here.

00:03:03.331 --> 00:03:05.877
So three cups of red paint

00:03:05.877 --> 00:03:08.641
and one, two, three, four, five...

00:03:08.641 --> 00:03:11.601
And five cups of blue paint,

00:03:11.601 --> 00:03:15.574
and you see that again right over here.

00:03:15.574 --> 00:03:18.423
Let's do another example.

00:03:18.423 --> 00:03:22.356
Here, we're told Luna and
Ginny each cast magic spells.

00:03:22.356 --> 00:03:24.559
The ratio of spells Luna casts

00:03:24.559 --> 00:03:28.665
to spells Ginny casts is
represented in this tape diagram.

00:03:28.665 --> 00:03:31.154
All right, based on the
ratio, what is the number

00:03:31.154 --> 00:03:35.431
of spells Ginny casts
when Luna casts 20 spells?

00:03:35.431 --> 00:03:38.531
Pause this video to see
if you can work it out.

00:03:38.531 --> 00:03:39.888
All right, so let's
just see the ratio here.

00:03:39.888 --> 00:03:44.055
For every one, two, three,
four spells that Luna casts,

00:03:46.756 --> 00:03:50.673
Ginny casts one, two,
three, four, five spells.

00:03:51.810 --> 00:03:54.358
So the ratio is four to five,

00:03:54.358 --> 00:03:57.407
but if Luna casts 20 spells...

00:03:57.407 --> 00:04:01.852
So if a Luna casts 20 spells,
well to go from four to 20,

00:04:01.852 --> 00:04:04.614
we had to multiply by five,

00:04:04.614 --> 00:04:06.740
and so we would do the
same thing with the number

00:04:06.740 --> 00:04:08.652
of spells Ginny casts.

00:04:08.652 --> 00:04:12.335
You'd multiply that by five, so it's 25.

00:04:12.335 --> 00:04:13.966
So four Luna spells

00:04:13.966 --> 00:04:16.319
for every five Ginny
spells is the same thing

00:04:16.319 --> 00:04:20.270
as 20 Luna spells for
every 25 Ginny spells,

00:04:20.270 --> 00:04:23.067
and so how many spells does Ginny cast

00:04:23.067 --> 00:04:26.016
when Luna casts 20 spells?

00:04:26.016 --> 00:04:28.433
She casts 25, and we're done.


Ratios and measurement

WEBVTT
Kind: captions
Language: en

00:00:00.831 --> 00:00:02.932
- [Instructor] We're told
to complete the ratio table

00:00:02.932 --> 00:00:06.094
to convert the units of
measure from hours to weeks,

00:00:06.094 --> 00:00:08.180
or weeks to hours.

00:00:08.180 --> 00:00:10.916
So we see here, they've told us already

00:00:10.916 --> 00:00:14.525
that there's 168 hours for every one week.

00:00:14.525 --> 00:00:17.591
One way to think about
it is the ratio of hours

00:00:17.591 --> 00:00:21.708
for every week is 168 to
one, and then they calculate,

00:00:21.708 --> 00:00:24.041
well if we have 1,176 hours,

00:00:25.080 --> 00:00:27.181
how many weeks is that going to be?

00:00:27.181 --> 00:00:30.039
So pause this video and see
if you can figure it out.

00:00:30.039 --> 00:00:32.956
Let's see, to go from 168 to 1,176,

00:00:34.820 --> 00:00:36.921
what do we have to multiply by?

00:00:36.921 --> 00:00:39.977
So, let's see, that looks
like we might be multiplying

00:00:39.977 --> 00:00:40.810
by seven.

00:00:41.870 --> 00:00:42.893
Let me try that out.

00:00:42.893 --> 00:00:45.643
So 168 times seven is equal to...

00:00:47.386 --> 00:00:49.386
Eight times seven is 56,

00:00:50.806 --> 00:00:53.973
six times seven is 42 plus five is 47,

00:00:55.122 --> 00:00:59.361
and then one times seven
plus four is indeed 1,176.

00:00:59.361 --> 00:01:01.949
So we multiplied by seven,
we multiplied the number

00:01:01.949 --> 00:01:03.041
of hours by seven,

00:01:03.041 --> 00:01:06.572
so that means we're gonna have
seven times as many weeks.

00:01:06.572 --> 00:01:10.826
So, one times seven is
just that is seven weeks.

00:01:10.826 --> 00:01:13.146
Now, what about a situation
where we have three weeks,

00:01:13.146 --> 00:01:15.539
how many hours is that going to be?

00:01:15.539 --> 00:01:19.521
Well we are multiplying
our weeks by three,

00:01:19.521 --> 00:01:22.802
so we would wanna multiply our hours.

00:01:22.802 --> 00:01:25.937
We would wanna multiply
our hours times three.

00:01:25.937 --> 00:01:27.604
So, 168 times three,

00:01:29.984 --> 00:01:32.614
eight times three is 24,

00:01:32.614 --> 00:01:35.781
six times three is 18, plus two is 20,

00:01:37.690 --> 00:01:40.997
and then one times three
is three, plus two is five.

00:01:40.997 --> 00:01:43.330
So, that would be 504 hours.

00:01:44.931 --> 00:01:47.014
Let's do another example.

00:01:48.081 --> 00:01:51.501
So, here they tell us the double
number lines show the ratio

00:01:51.501 --> 00:01:53.510
of yards to miles.

00:01:53.510 --> 00:01:56.075
So, the ratio of yards to miles.

00:01:56.075 --> 00:02:00.242
It looks like we have 3,520
yards for every two miles.

00:02:02.887 --> 00:02:03.800
For every two miles,

00:02:03.800 --> 00:02:06.543
and you see that on this double
number line right over here.

00:02:06.543 --> 00:02:09.567
Then they say how may
yards are in five miles?

00:02:09.567 --> 00:02:13.143
So, why don't you pause this
video and try to figure it out.

00:02:13.143 --> 00:02:14.932
Well the way my brain wants to do it is,

00:02:14.932 --> 00:02:18.126
well let's just think about
how many yard are in each mile.

00:02:18.126 --> 00:02:20.376
So, if the ratio's 3,520:2,

00:02:21.677 --> 00:02:25.159
how could I rewrite this
ratio so it is how many yards

00:02:25.159 --> 00:02:26.556
for every one mile.

00:02:26.556 --> 00:02:29.642
So, to go from two to
one, I am dividing by two,

00:02:29.642 --> 00:02:32.566
so I would wanna divide
this by two as well.

00:02:32.566 --> 00:02:35.483
So, two goes into 3,520, let's see,

00:02:39.649 --> 00:02:43.073
two goes into three one
time, one times two is two,

00:02:43.073 --> 00:02:46.561
you subtract, you bring down the five.

00:02:46.561 --> 00:02:50.146
Two goes into 15 seven
times, seven times two is 14,

00:02:50.146 --> 00:02:54.472
subtract, we have one,
bring down that two,

00:02:54.472 --> 00:02:59.066
two goes into 12 six
times, six times two is 12,

00:02:59.066 --> 00:03:00.973
and we subtract, no remainder,

00:03:00.973 --> 00:03:02.626
but then we're gonna
have one more zero here,

00:03:02.626 --> 00:03:04.494
cuz we bring down that zero.

00:03:04.494 --> 00:03:06.224
We say two goes into zero zero times,

00:03:06.224 --> 00:03:08.659
zero times two is zero,
and we have no remainder,

00:03:08.659 --> 00:03:11.187
and so this is 1,760.

00:03:11.187 --> 00:03:13.698
So, we could put that here
on our double number lines.

00:03:13.698 --> 00:03:17.365
So, if we have one mile
that is 1,760 yards.

00:03:18.577 --> 00:03:20.312
Now, they're asking about five miles,

00:03:20.312 --> 00:03:24.153
so three, four, five,
so we have five miles.

00:03:24.153 --> 00:03:25.985
What is the number of yards?

00:03:25.985 --> 00:03:29.392
Well if you multiply by five here,

00:03:29.392 --> 00:03:33.362
you're also going to multiply
by five right over there.

00:03:33.362 --> 00:03:35.947
So, what's 1,760 times five?

00:03:35.947 --> 00:03:36.873
Well just figure it out.

00:03:36.873 --> 00:03:40.373
1,760 times five, five times zero is zero,

00:03:42.618 --> 00:03:44.368
five times six is 30.

00:03:46.623 --> 00:03:49.093
Regroup that three cuz it's really 300s,

00:03:49.093 --> 00:03:51.459
five times seven is 35,

00:03:51.459 --> 00:03:55.478
plus three is 38, five times one is five,

00:03:55.478 --> 00:03:56.747
plus three is eight.

00:03:56.747 --> 00:03:59.247
So, there you go, 8,800 yards.

00:04:01.840 --> 00:04:04.257
Let's do a few more examples.

00:04:05.641 --> 00:04:09.808
Here, we're told there are
914.4 millimeters in a yard.

00:04:10.934 --> 00:04:13.321
There are three feet in a yard.

00:04:13.321 --> 00:04:16.238
How many millimeters are in a foot?

00:04:17.514 --> 00:04:18.834
Okay, so one way to think about it,

00:04:18.834 --> 00:04:22.917
you could say there's
914.4 millimeters per yard,

00:04:26.814 --> 00:04:30.981
or you could say 914.4
millimeters per three feet,

00:04:34.954 --> 00:04:37.828
since three feet and a
yard is the same thing.

00:04:37.828 --> 00:04:39.289
So, if you wanna know per foot,

00:04:39.289 --> 00:04:41.987
you would just divide
both of these by three.

00:04:41.987 --> 00:04:42.820
So, let's do that,

00:04:42.820 --> 00:04:45.125
and I'll just do it in
a different color here.

00:04:45.125 --> 00:04:49.292
Three goes into 914.4, Three
goes into nine three times,

00:04:52.368 --> 00:04:54.462
three times three is nine, subtract,

00:04:54.462 --> 00:04:57.130
we get a zero bring down the one,

00:04:57.130 --> 00:05:01.231
three goes into one zero times,
zero times three is zero,

00:05:01.231 --> 00:05:05.059
subtract, get a one, and
bring down that four,

00:05:05.059 --> 00:05:08.199
three goes into 14 four times,

00:05:08.199 --> 00:05:09.427
gonna have this decimal right over here,

00:05:09.427 --> 00:05:13.842
four times three is 12,
you subtract, and then,

00:05:13.842 --> 00:05:16.532
so you get a two, bring down this four,

00:05:16.532 --> 00:05:18.204
you get a 24, and lucky for us,

00:05:18.204 --> 00:05:21.605
three goes perfectly into 24 eight times,

00:05:21.605 --> 00:05:23.605
eight times three is 24,

00:05:25.341 --> 00:05:26.674
24 you subtract,

00:05:27.702 --> 00:05:28.815
now we have no remainder.

00:05:28.815 --> 00:05:32.565
So, we have 304.8
millimeters for every foot.

00:05:35.197 --> 00:05:37.364
Let's do one last example.

00:05:49.090 --> 00:05:52.264
So pause this video and see
if you can figure it out.

00:05:52.264 --> 00:05:54.193
So, let's just write this out in words.

00:05:54.193 --> 00:05:57.110
So, it's $12 per pound of confetti.

00:05:59.179 --> 00:06:02.762
So, you could do this
as $12 per 16 ounces.

00:06:04.186 --> 00:06:08.353
16 ounces of confetti, and
so if we want it per ounce.

00:06:10.788 --> 00:06:13.458
So, you could do this as a 12 to 16 ratio,

00:06:13.458 --> 00:06:16.071
but we wanna say something to one ratio.

00:06:16.071 --> 00:06:18.488
So, if you say per one ounce,

00:06:21.218 --> 00:06:23.831
well we're dividing by 16 there,

00:06:23.831 --> 00:06:26.976
so we would wanna divide by 16 as well.

00:06:26.976 --> 00:06:30.024
So, this is going to be 12 divided by 16.

00:06:30.024 --> 00:06:31.532
So, 12 divided by...

00:06:31.532 --> 00:06:33.422
Well let me write it over here.

00:06:33.422 --> 00:06:37.589
So 12 divided by 16 is
the same thing as 3/4,

00:06:39.707 --> 00:06:43.874
just divide both of them by
four, and so this is 0.75,

00:06:45.244 --> 00:06:47.911
or 75 cents, 75 cents per ounce,

00:06:50.439 --> 00:06:53.143
or 75 hundredths of a dollar per ounce.

00:06:53.143 --> 00:06:55.226
So, 0.75, and we're done.


Ratios and double number lines

WEBVTT
Kind: captions
Language: en

00:00:00.614 --> 00:00:02.851
- [Narrator] We're told that
the double number line shows

00:00:02.851 --> 00:00:05.934
that five pounds of avocados cost $9.

00:00:07.836 --> 00:00:10.452
And so what is going on here
with this double number line?

00:00:10.452 --> 00:00:14.640
This shows how as we increase
the number of avocados,

00:00:14.640 --> 00:00:16.583
how the cost increases.

00:00:16.583 --> 00:00:19.875
So for example, when we have
zero pounds of avocados,

00:00:19.875 --> 00:00:21.763
it costs us $0.

00:00:21.763 --> 00:00:24.492
When we have five pounds of avocados,

00:00:24.492 --> 00:00:25.742
it costs us $9.

00:00:27.570 --> 00:00:29.537
And so, if you look at
any point over here,

00:00:29.537 --> 00:00:30.855
let's say you look right over here.

00:00:30.855 --> 00:00:32.910
This would be, let's
see, this is one, right?

00:00:32.910 --> 00:00:34.910
This is one, two, three, four, five.

00:00:34.910 --> 00:00:38.512
If you look at one, this
point on the cost number line

00:00:38.512 --> 00:00:43.131
would tell you how much one
pound of avocado would cost.

00:00:43.131 --> 00:00:45.632
Two pounds of avocados,
how much would that cost?

00:00:45.632 --> 00:00:48.590
You would look at this second
number line right over there.

00:00:48.590 --> 00:00:51.923
So they ask us, based on the ratio shown

00:00:54.195 --> 00:00:57.257
in the double number
line, what is the cost

00:00:57.257 --> 00:00:59.424
for one pound of avocados?

00:01:00.641 --> 00:01:02.240
So pause the video and think about it.

00:01:02.240 --> 00:01:05.152
Remember, one pound of avocados
on this top number line,

00:01:05.152 --> 00:01:06.633
we look at this second number line,

00:01:06.633 --> 00:01:08.049
the cost would be right over here.

00:01:08.049 --> 00:01:10.132
What is this going to be?

00:01:11.863 --> 00:01:14.380
Well, we could just set it up as a ratio.

00:01:14.380 --> 00:01:17.146
The ratio of pounds of avocados to cost

00:01:17.146 --> 00:01:20.621
is going to be, let me
do this in some colors.

00:01:20.621 --> 00:01:23.704
So if I have five pounds of avocados,

00:01:24.941 --> 00:01:27.108
it is going to cost me $9.

00:01:28.061 --> 00:01:31.490
So the ratio of pounds to
dollars is five to nine.

00:01:31.490 --> 00:01:35.111
So if I were to have one pound of avocado,

00:01:35.111 --> 00:01:37.358
one pound of avocado,

00:01:37.358 --> 00:01:42.103
I have divided by five to
get one pound of avocado.

00:01:42.103 --> 00:01:44.803
I would have to do the
same thing for the cost.

00:01:44.803 --> 00:01:47.136
So if I divide nine by five,

00:01:48.469 --> 00:01:52.523
this is going to be nine fifths dollars.

00:01:52.523 --> 00:01:55.581
Nine fifths dollars would
be the cost of one pound.

00:01:55.581 --> 00:01:57.567
Well, nine fifths isn't
always the most natural way

00:01:57.567 --> 00:02:01.930
to write money, so you
can view this nine fifths

00:02:01.930 --> 00:02:05.017
is equal to one and four fifths,

00:02:05.017 --> 00:02:07.708
which is equal to one and eight 10ths,

00:02:07.708 --> 00:02:09.934
which is equal to 1.8,

00:02:09.934 --> 00:02:12.517
or you could say this is $1.80.

00:02:13.786 --> 00:02:15.750
So if you were to go onto
this double number line,

00:02:15.750 --> 00:02:18.258
the cost of one pound of avocado,

00:02:18.258 --> 00:02:21.729
this point right over here would be $1.80.

00:02:21.729 --> 00:02:23.199
If you said two pounds of avocados,

00:02:23.199 --> 00:02:24.819
well, now you would double it.

00:02:24.819 --> 00:02:26.736
So this would be $3.60,

00:02:28.203 --> 00:02:30.043
and you would go on and on and on

00:02:30.043 --> 00:02:33.618
all the way until you got to $9 here.

00:02:33.618 --> 00:02:35.701
Let's do another example.

00:02:36.880 --> 00:02:40.604
So here, we are told the
double number line shows

00:02:40.604 --> 00:02:44.646
how many model trains
Irene can build in a week.

00:02:44.646 --> 00:02:47.802
So we can see, in zero weeks
she can't build any trains,

00:02:47.802 --> 00:02:50.978
but in one week she can build nine trains.

00:02:50.978 --> 00:02:53.865
And they asked us, which
table represents the rate

00:02:53.865 --> 00:02:56.448
of Irene building model trains?

00:02:57.606 --> 00:03:00.949
So pause this video and see
if you can figure it out.

00:03:00.949 --> 00:03:04.160
So once again, every week
she can build nine trains.

00:03:04.160 --> 00:03:05.931
So one way to think about it is

00:03:05.931 --> 00:03:08.568
the ratio of week to trains

00:03:08.568 --> 00:03:10.256
would be one

00:03:10.256 --> 00:03:11.089
to nine.

00:03:12.451 --> 00:03:13.455
So let's see.

00:03:13.455 --> 00:03:15.925
If I look at this table, I
just wanna see where the ratio

00:03:15.925 --> 00:03:19.546
between weeks to trains
stays at one to nine.

00:03:19.546 --> 00:03:23.330
So five to 45, that is still one to nine.

00:03:23.330 --> 00:03:27.395
To go from one to five,
I've multiplied by five.

00:03:27.395 --> 00:03:32.218
And then to go to nine to 45,
I've also multiplied by five.

00:03:32.218 --> 00:03:34.423
So this one checks out.

00:03:34.423 --> 00:03:38.961
And another way to think about
it, is 45 is nine times five.

00:03:38.961 --> 00:03:40.782
So nine times.

00:03:40.782 --> 00:03:42.946
That might be an easier
way to think about it.

00:03:42.946 --> 00:03:46.307
Over here, 12 to 108, that's, once again,

00:03:46.307 --> 00:03:48.057
12 times nine is 108.

00:03:49.325 --> 00:03:50.908
And then 26 to 234.

00:03:52.280 --> 00:03:56.332
Let's see, 26 times 10
would be 260, minus 26.

00:03:56.332 --> 00:03:58.126
Yeah, it would be 234.

00:03:58.126 --> 00:03:59.482
So this is nine times.

00:03:59.482 --> 00:04:02.199
So in all of these cases,
the ratio of trains

00:04:02.199 --> 00:04:06.130
or the ratio of weeks to
trains is one to nine.

00:04:06.130 --> 00:04:08.938
So this one is looking good,
so I'll just circle that in.

00:04:08.938 --> 00:04:11.188
But let's just make sure
that this one doesn't work.

00:04:11.188 --> 00:04:15.960
Over here, the ratio of weeks
to trains is nine to one,

00:04:15.960 --> 00:04:17.989
not one to nine.

00:04:17.989 --> 00:04:20.761
The train should be nine times the weeks,

00:04:20.761 --> 00:04:23.106
while here the weeks is
nine times the train.

00:04:23.106 --> 00:04:24.378
So just looking at that first one,

00:04:24.378 --> 00:04:28.002
we know that this is
not going to work out.

00:04:28.002 --> 00:04:30.169
Let's do one last example.

00:04:31.447 --> 00:04:35.217
The double number line shows
how many snowballs Jacob

00:04:35.217 --> 00:04:37.719
and his friends can make in one minute.

00:04:37.719 --> 00:04:40.137
No minutes, they can make zero snowballs.

00:04:40.137 --> 00:04:42.328
In one minute, they can make 12.

00:04:42.328 --> 00:04:44.699
Complete the table to
show the same information

00:04:44.699 --> 00:04:47.108
as the double number line.

00:04:47.108 --> 00:04:48.202
So once again, pause this video

00:04:48.202 --> 00:04:50.711
and see if you can work this out.

00:04:50.711 --> 00:04:53.339
Well, we can think about it as a ratio.

00:04:53.339 --> 00:04:55.776
The ratio of minutes

00:04:55.776 --> 00:04:57.943
to snowballs is one to 12.

00:04:59.583 --> 00:05:02.666
So minutes to snowballs is one to 12.

00:05:03.967 --> 00:05:05.438
Or another way to think about it is,

00:05:05.438 --> 00:05:09.637
the snowballs is going to
be 12 times the minutes.

00:05:09.637 --> 00:05:12.763
And so over here, if I have 12 snowballs,

00:05:12.763 --> 00:05:16.107
we already know that's
going to be one minute.

00:05:16.107 --> 00:05:19.390
If I have 48 snowballs,
let's just think about it.

00:05:19.390 --> 00:05:22.540
To go from 12 to 48 you
have to multiply by four.

00:05:22.540 --> 00:05:23.991
So you have four times as many snowballs,

00:05:23.991 --> 00:05:27.969
it's gonna take you four
times as many minutes.

00:05:27.969 --> 00:05:30.923
And then if you go to five,
to go from one to five

00:05:30.923 --> 00:05:34.479
you multiply by five, so
you're gonna have five times

00:05:34.479 --> 00:05:37.083
as many snowballs as you
would be able to make

00:05:37.083 --> 00:05:37.987
in one minute.

00:05:37.987 --> 00:05:41.369
So five times 12 is 60, and we're done.

00:05:41.369 --> 00:05:45.202
It could be one to 12,
four to 48, five to 60.

00:05:46.271 --> 00:05:50.703
For every five minutes,
you can make 60 snowballs,

00:05:50.703 --> 00:05:52.243
or they make 60 snowballs.


Worked example: separable differential equation (with taking exp of both sides)

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.774
- [Tutor] What we're
going to do in this video

00:00:01.774 --> 00:00:04.426
is see if we can solve
the differential equation,

00:00:04.426 --> 00:00:07.287
the derivative of y with respect to x

00:00:07.287 --> 00:00:09.370
is equal to x times y,

00:00:10.637 --> 00:00:11.824
pause this video

00:00:11.824 --> 00:00:16.484
and see if you can find
a general solution here.

00:00:16.484 --> 00:00:18.580
So the first thing that
my brain likes to do,

00:00:18.580 --> 00:00:20.749
when I see a differential
equation is to say,

00:00:20.749 --> 00:00:23.780
hey, is this separable
and when I say separable,

00:00:23.780 --> 00:00:26.420
can I get all the ys and dys on one side

00:00:26.420 --> 00:00:28.889
and all the xs and dxs on the other side?

00:00:28.889 --> 00:00:31.153
And you can indeed do that,

00:00:31.153 --> 00:00:33.381
if we treat our differentials like,

00:00:33.381 --> 00:00:35.982
if we could treat 'em
like algebraic variables,

00:00:35.982 --> 00:00:37.666
which is fair game, when you're dealing

00:00:37.666 --> 00:00:39.502
with differential equations like this,

00:00:39.502 --> 00:00:42.419
we could multiple both sides by dx,

00:00:43.561 --> 00:00:45.529
so multiple both sides by dx

00:00:45.529 --> 00:00:47.838
and we could divide both sides by y,

00:00:47.838 --> 00:00:50.750
let me move this over a little
bit, so we have some space,

00:00:50.750 --> 00:00:54.764
so we could also multiple
both sides by one over y,

00:00:54.764 --> 00:00:58.151
one over y and so what that does is

00:00:58.151 --> 00:01:02.548
these dxs cancel out and this
y and one over y cancels out

00:01:02.548 --> 00:01:04.995
and we are left with, let
me write all the things

00:01:04.995 --> 00:01:06.892
in terms of y on the
left-hand side in blue,

00:01:06.892 --> 00:01:08.809
so we have one over you

00:01:10.479 --> 00:01:11.729
dy is equal to,

00:01:12.983 --> 00:01:15.234
and then I'll do all this stuff in orange,

00:01:15.234 --> 00:01:16.922
we have is equal to, we're just left

00:01:16.922 --> 00:01:20.247
with an x and a dx, x dx

00:01:20.247 --> 00:01:22.309
and then we'll wanna take

00:01:22.309 --> 00:01:25.670
the indefinite integral of both sides,

00:01:25.670 --> 00:01:29.553
now what's the
anti-derivative of one over y?

00:01:29.553 --> 00:01:32.076
Well, if we want it in
the most general form,

00:01:32.076 --> 00:01:36.660
this would be the natural log
of the absolute value of y

00:01:36.660 --> 00:01:39.117
and then this is going to be equal to

00:01:39.117 --> 00:01:41.117
the anti-derivative of x

00:01:43.158 --> 00:01:44.879
is x squared over two

00:01:44.879 --> 00:01:46.947
and then we wanna put a
constant on either side,

00:01:46.947 --> 00:01:50.343
I'll just put it on the
right-hand side, plus C,

00:01:50.343 --> 00:01:53.224
this ensures that we're dealing
with the general solution,

00:01:53.224 --> 00:01:55.671
now if we wanna solve explicitly for y,

00:01:55.671 --> 00:01:59.009
we could raise e to both sides' power,

00:01:59.009 --> 00:02:01.062
another way to think about it is,

00:02:01.062 --> 00:02:02.838
if this is equal to that,

00:02:02.838 --> 00:02:05.040
then e to this power

00:02:05.040 --> 00:02:08.972
is going to be the same
thing as e to that power,

00:02:08.972 --> 00:02:10.578
now, what does that do for us?

00:02:10.578 --> 00:02:12.294
Well, what is e to the natural log

00:02:12.294 --> 00:02:14.652
of the absolute value of y?

00:02:14.652 --> 00:02:16.283
Well, I'm raising e to the power

00:02:16.283 --> 00:02:18.498
that I would have to raise e too

00:02:18.498 --> 00:02:20.511
to to get to the absolute value of y,

00:02:20.511 --> 00:02:22.504
so the left-hand side here,

00:02:22.504 --> 00:02:25.940
it simplifies to the absolute value of y

00:02:25.940 --> 00:02:28.509
and we get that as being equal to,

00:02:28.509 --> 00:02:30.754
now, we could use our exponent properties,

00:02:30.754 --> 00:02:33.671
this over here is the same thing as

00:02:35.253 --> 00:02:37.753
e to the x squared over two

00:02:40.093 --> 00:02:42.926
times e to the C times e to the C,

00:02:44.039 --> 00:02:46.672
I am just using our
exponent properties here,

00:02:46.672 --> 00:02:48.566
well, e to the C, we could just view that

00:02:48.566 --> 00:02:52.639
as some other type of constant,
this is just sum constant C,

00:02:52.639 --> 00:02:56.049
so we could rewrite this whole thing as

00:02:56.049 --> 00:02:57.966
C e to the x squared

00:03:01.231 --> 00:03:03.515
x squared over two, hopefully
you see what I'm doing there,

00:03:03.515 --> 00:03:05.507
I just use my exponent properties

00:03:05.507 --> 00:03:07.978
e dah dah dah, sum of two things

00:03:07.978 --> 00:03:10.722
is equal to e to the first thing
times e to the second thing

00:03:10.722 --> 00:03:14.510
and I just said, well, e to
the power of sum constant C,

00:03:14.510 --> 00:03:17.577
we could just relabel that
as, let's call that our blue C

00:03:17.577 --> 00:03:21.094
and so this simplifies to blue C times e

00:03:21.094 --> 00:03:24.054
to the x squared over two,

00:03:24.054 --> 00:03:26.842
now we saw this absolute value sign here,

00:03:26.842 --> 00:03:29.509
so this essentially means that y

00:03:31.244 --> 00:03:34.445
could be equal to, we
could write it this way,

00:03:34.445 --> 00:03:37.445
y could be equal to plus or minus C,

00:03:40.676 --> 00:03:43.426
C e to the x squared over two,

00:03:46.224 --> 00:03:48.107
x squared over two,

00:03:48.107 --> 00:03:50.719
but once again, we don't
know what this constant is,

00:03:50.719 --> 00:03:52.164
I didn't say that it was
positive or negative,

00:03:52.164 --> 00:03:54.609
so when you see a plus or
minus of a constant here,

00:03:54.609 --> 00:03:56.734
you could really just subsume all of this,

00:03:56.734 --> 00:03:59.362
I'll just call this red C,

00:03:59.362 --> 00:04:02.115
so we could say that y is equal to,

00:04:02.115 --> 00:04:05.452
I'll just rewrite it over again for fun,

00:04:05.452 --> 00:04:07.119
y is equal to red C,

00:04:08.661 --> 00:04:12.954
not the Red Sea, but a red C (laughs)

00:04:12.954 --> 00:04:16.037
times e to the x squared over two,

00:04:18.371 --> 00:04:21.329
this right over here,
it's the general solution

00:04:21.329 --> 00:04:25.412
to the original, separable
differential equation.


Linear velocity comparison from radius and angular velocity: Worked example

WEBVTT
Kind: captions
Language: en

00:00:00.358 --> 00:00:03.042
- [Instructor] Let's say that
we have two pumpkin catapults,

00:00:03.042 --> 00:00:05.658
so let me just draw the ground here.

00:00:05.658 --> 00:00:08.450
And so the first pumpkin catapult,

00:00:08.450 --> 00:00:11.417
let me just draw it right
over here, that's its space.

00:00:11.417 --> 00:00:13.438
And then this is the part

00:00:13.438 --> 00:00:16.059
that actually catapult the pumpkin.

00:00:16.059 --> 00:00:18.942
So that's what it looks like.

00:00:18.942 --> 00:00:21.942
It holds the pumpkin right over here.

00:00:21.942 --> 00:00:24.758
That's the pumpkin that's
about to be shot away.

00:00:24.758 --> 00:00:27.467
And what it does is once it releases,

00:00:27.467 --> 00:00:30.392
I guess someone presses a
button or pulls a lever,

00:00:30.392 --> 00:00:33.658
the pumpkin catapult is going to release,

00:00:33.658 --> 00:00:37.009
and so this arm is going
to turn a certain amount

00:00:37.009 --> 00:00:40.633
and then just stop
immediately right over there.

00:00:40.633 --> 00:00:44.083
And then that pumpkin, that
pumpkin is going to be released

00:00:44.083 --> 00:00:47.150
with some type of initial velocity.

00:00:47.150 --> 00:00:49.775
So your pumpkin is just
gonna go like that.

00:00:49.775 --> 00:00:52.925
So this is our small pumpkin catapult.

00:00:52.925 --> 00:00:54.242
This is our small one.

00:00:54.242 --> 00:00:57.533
But let's say we also have
a large pumpkin catapult.

00:00:57.533 --> 00:00:58.667
Let me draw that.

00:00:58.667 --> 00:01:02.208
So our large pumpkin
catapult right over here,

00:01:02.208 --> 00:01:03.708
similar mechanism.

00:01:05.242 --> 00:01:08.442
But let's say its arm
is four times as long.

00:01:08.442 --> 00:01:13.417
So, that looks about four
times as long right over there.

00:01:13.417 --> 00:01:15.584
So this is the larger one.

00:01:16.675 --> 00:01:19.458
The large pumpkin catapult.

00:01:19.458 --> 00:01:21.417
Let me make sure to draw the pumpkin

00:01:21.417 --> 00:01:23.850
to remember what we are catapulting.

00:01:23.850 --> 00:01:28.017
And then it will go through
actually the exact same angle,

00:01:29.058 --> 00:01:30.808
you go to the exact same angle,

00:01:30.808 --> 00:01:32.792
and then let go of its pumpkin.

00:01:32.792 --> 00:01:34.025
And this is gonna be a very useful video

00:01:34.025 --> 00:01:35.266
because you will find yourself

00:01:35.266 --> 00:01:39.463
making many pumpkin catapults
in your life. (laughs)

00:01:39.463 --> 00:01:41.417
And then it will let it go,

00:01:41.417 --> 00:01:44.983
and you will have some linear velocity.

00:01:44.983 --> 00:01:48.208
Now we know a few things
about these pumpkin catapults.

00:01:48.208 --> 00:01:50.425
Let's say the small pumpkin catapult,

00:01:50.425 --> 00:01:53.583
the radius between the
center of the pumpkin,

00:01:53.583 --> 00:01:56.283
the radius between the
center of the pumpkin

00:01:56.283 --> 00:01:58.808
and the center of
rotation right over there,

00:01:58.808 --> 00:02:00.500
let's say that this is r.

00:02:00.500 --> 00:02:03.533
While for the large one, this one is,

00:02:03.533 --> 00:02:07.308
this distance right over here, is four r.

00:02:07.308 --> 00:02:10.109
We also know the angular velocity

00:02:10.109 --> 00:02:12.508
while this thing is moving.

00:02:12.508 --> 00:02:14.975
So we know that the angular velocity here,

00:02:14.975 --> 00:02:18.567
let's say the magnitude of
the angular velocity is omega.

00:02:18.567 --> 00:02:19.925
It would actually be negative

00:02:19.925 --> 00:02:21.283
if we were to write it as a vector

00:02:21.283 --> 00:02:23.217
'cause we're going in
the clockwise direction

00:02:23.217 --> 00:02:25.250
because that's the convention.

00:02:25.250 --> 00:02:26.083
But this right over here

00:02:26.083 --> 00:02:28.533
is the magnitude of the angular velocity

00:02:28.533 --> 00:02:31.283
and just to make that tangible for you,

00:02:31.283 --> 00:02:35.366
let's say that this is
two pi radians per second.

00:02:37.250 --> 00:02:41.050
And let's say this thing,
while it's in motion,

00:02:41.050 --> 00:02:43.975
it also has the same magnitude
of its angular velocity,

00:02:43.975 --> 00:02:46.058
so this thing right over here is also.

00:02:46.058 --> 00:02:47.250
The magnitude of angular velocity

00:02:47.250 --> 00:02:50.892
is once again two pi radians per second.

00:02:50.892 --> 00:02:55.583
So my question to you is
how would the velocity,

00:02:55.583 --> 00:02:58.150
the magnitude of the
velocity of the pumpkin

00:02:58.150 --> 00:03:01.342
being released from the small catapult,

00:03:01.342 --> 00:03:02.942
so v sub small.

00:03:02.942 --> 00:03:05.208
If I put a little arrow on top of it,

00:03:05.208 --> 00:03:06.704
we'll be talking about velocity.

00:03:06.704 --> 00:03:07.617
Since I didn't put an arrow,

00:03:07.617 --> 00:03:09.275
we're talking about just
the magnitude of velocity.

00:03:09.275 --> 00:03:11.500
You can think about this as the speed.

00:03:11.500 --> 00:03:14.583
How does this compare to v sub large?

00:03:17.508 --> 00:03:19.833
We have the same angular velocity

00:03:19.833 --> 00:03:22.433
but we have different radii.

00:03:22.433 --> 00:03:26.150
Pause the video and see if
you can figure that out.

00:03:26.150 --> 00:03:28.492
Well, the key thing to realize here,

00:03:28.492 --> 00:03:31.212
and we've seen this in multiple
videos, is the relationship

00:03:31.212 --> 00:03:34.042
between the magnitude of angular velocity

00:03:34.042 --> 00:03:37.742
and the magnitude of
velocity, linear velocity.

00:03:37.742 --> 00:03:41.909
The magnitude of angular
velocity times your radius

00:03:43.100 --> 00:03:47.267
is going to give you the
magnitude of your linear velocity.

00:03:48.142 --> 00:03:50.892
So for the small right over here,

00:03:52.767 --> 00:03:56.017
we could write this as v small

00:03:56.017 --> 00:03:58.450
is going to be equal to omega.

00:03:58.450 --> 00:03:59.467
These are the same omega,

00:03:59.467 --> 00:04:01.167
that omega and that omega is the same.

00:04:01.167 --> 00:04:03.108
In fact, we don't even
have to know what this is.

00:04:03.108 --> 00:04:06.167
We could say v sub small is equal to omega

00:04:06.167 --> 00:04:08.584
times our radius, which is r.

00:04:13.592 --> 00:04:16.033
And what's v sub large going to be?

00:04:16.033 --> 00:04:17.450
Well, v sub large

00:04:18.583 --> 00:04:21.117
is going to be equal to that same omega.

00:04:21.117 --> 00:04:23.525
So I'm talking about this
particular omega right over here.

00:04:23.525 --> 00:04:25.566
So it's that same omega

00:04:25.566 --> 00:04:29.358
but now, our radius isn't r, it is four r,

00:04:29.358 --> 00:04:32.608
so we're talking about, so times four r

00:04:33.725 --> 00:04:35.750
or if we were to rewrite this,

00:04:35.750 --> 00:04:39.750
this would be equal to
four times omega times r.

00:04:42.617 --> 00:04:45.383
Four times omega times r.

00:04:45.383 --> 00:04:48.433
And what is this right over here?

00:04:48.433 --> 00:04:51.841
Omega times r, that is the
magnitude of the velocity

00:04:51.841 --> 00:04:54.367
of our smaller catapult

00:04:54.367 --> 00:04:58.108
or the pumpkin being released
from the smaller catapult.

00:04:58.108 --> 00:04:59.033
So just like that,

00:04:59.033 --> 00:05:02.450
you see by having the
same angular velocity,

00:05:02.450 --> 00:05:06.192
but if you increase your arm
length by a factor of four,

00:05:06.192 --> 00:05:09.400
your velocity is going to
increase by a factor of four.

00:05:09.400 --> 00:05:13.725
And so you have the magnitude
of velocity of the pumpkin

00:05:13.725 --> 00:05:16.367
being released from the large catapult

00:05:16.367 --> 00:05:20.367
is going to be equal to
four times the magnitude

00:05:21.208 --> 00:05:24.958
of the velocity of the
pumpkin being released

00:05:26.083 --> 00:05:28.250
from the smaller catapult.


Radius comparison from velocity and angular velocity: Worked example

WEBVTT
Kind: captions
Language: en

00:00:00.663 --> 00:00:02.399
- [Instructor] We are
told a red disc spins

00:00:02.399 --> 00:00:05.892
with angular velocity omega,
and a point on the edge

00:00:05.892 --> 00:00:08.032
moves at velocity V.

00:00:08.032 --> 00:00:09.805
So they're giving us angular velocity

00:00:09.805 --> 00:00:12.944
and also you could view
this as linear velocity,

00:00:12.944 --> 00:00:15.619
and they are both vectors,
that's why they are bolded.

00:00:15.619 --> 00:00:20.361
A blue disc spins with
angular velocity two omega,

00:00:20.361 --> 00:00:22.619
so that's twice the angular velocity,

00:00:22.619 --> 00:00:25.830
so its angle is changing twice as fast,

00:00:25.830 --> 00:00:29.937
with a point on the edge
moving at velocity 2V,

00:00:29.937 --> 00:00:33.115
so the linear velocity is also twice,

00:00:33.115 --> 00:00:35.994
twice the linear velocity of the red disc.

00:00:35.994 --> 00:00:39.324
And they ask us which
disc has a larger radius.

00:00:39.324 --> 00:00:41.668
So pause this video and
see if you can answer that

00:00:41.668 --> 00:00:44.458
before we work through it together.

00:00:44.458 --> 00:00:48.107
So let's first, let's
just visualize things.

00:00:48.107 --> 00:00:51.699
So if this is our red disc, my
best shot at drawing a disc,

00:00:51.699 --> 00:00:52.847
this should be a perfect circle,

00:00:52.847 --> 00:00:55.286
but I can't hand-draw a perfect circle,

00:00:55.286 --> 00:00:56.758
but you get the idea.

00:00:56.758 --> 00:01:00.506
And let me draw the
radius of our red disc,

00:01:00.506 --> 00:01:04.089
right over there, I'll
call that R sub red,

00:01:05.338 --> 00:01:06.880
and we know a few things.

00:01:06.880 --> 00:01:11.363
We know that a point right
over here has a velocity V,

00:01:11.363 --> 00:01:13.131
so let me draw that.

00:01:13.131 --> 00:01:15.824
So let's say it's going
right in that direction,

00:01:15.824 --> 00:01:19.324
right at that moment, it has a velocity V,

00:01:21.487 --> 00:01:26.051
and we also know that it has
an angular velocity of omega.

00:01:26.051 --> 00:01:29.884
So our angular velocity
is omega in this case,

00:01:31.246 --> 00:01:34.044
in terms of how fast it is rotating,

00:01:34.044 --> 00:01:36.280
and then we have our blue disc.

00:01:36.280 --> 00:01:38.865
So let me draw the blue
disc, and I'm gonna draw it

00:01:38.865 --> 00:01:40.816
at an arbitrary radius,
'cause we haven't figured out

00:01:40.816 --> 00:01:44.934
how the radii relate
to each other just yet.

00:01:44.934 --> 00:01:47.814
This is, should be a
circle, but I'm trying

00:01:47.814 --> 00:01:49.431
to hand-draw it, alright.

00:01:49.431 --> 00:01:53.388
So this is the radius of the blue disc,

00:01:53.388 --> 00:01:56.971
so I will call that R
sub blue, R sub blue,

00:01:59.256 --> 00:02:02.017
and the velocity of an analogous point

00:02:02.017 --> 00:02:04.297
right over here is 2V.

00:02:04.297 --> 00:02:07.333
So I should make that
vector twice as tall.

00:02:07.333 --> 00:02:10.045
So this is 2V right over here,

00:02:10.045 --> 00:02:13.712
and it has an angular
velocity of two omega,

00:02:15.784 --> 00:02:19.467
so that tells us how fast
it is actually rotating.

00:02:19.467 --> 00:02:21.555
So how do we make the statement,

00:02:21.555 --> 00:02:23.391
which disc has a larger radius?

00:02:23.391 --> 00:02:25.180
Or how do we decide that?

00:02:25.180 --> 00:02:29.060
Well the key realization
is the relationship

00:02:29.060 --> 00:02:32.430
between the magnitude of angular velocity

00:02:32.430 --> 00:02:34.286
and the magnitude of velocity.

00:02:34.286 --> 00:02:35.896
A couple ways to think about it is,

00:02:35.896 --> 00:02:37.818
the magnitude of angular velocity,

00:02:37.818 --> 00:02:39.736
notice I didn't put an arrow
on top, so I'm just talking

00:02:39.736 --> 00:02:42.170
about the magnitude of
our angular velocity,

00:02:42.170 --> 00:02:45.404
times our radius is going to be equal to

00:02:45.404 --> 00:02:47.759
the magnitude of our velocity,

00:02:47.759 --> 00:02:49.476
or is going to be equal to our speed.

00:02:49.476 --> 00:02:50.983
Or another way of thinking about it,

00:02:50.983 --> 00:02:53.034
if you divide both sides by R,

00:02:53.034 --> 00:02:55.069
the magnitude of our angular velocity

00:02:55.069 --> 00:02:58.435
is going to be equal to the
magnitude of our velocity,

00:02:58.435 --> 00:03:00.929
or our speed, over R.

00:03:00.929 --> 00:03:04.185
Or we can say that R
is equal to the speed,

00:03:04.185 --> 00:03:06.617
magnitude of velocity, over the magnitude

00:03:06.617 --> 00:03:09.347
of our angular velocity.

00:03:09.347 --> 00:03:11.450
So in this situation right over here,

00:03:11.450 --> 00:03:15.283
for this red disc, we
could say that R sub red

00:03:16.728 --> 00:03:20.061
is equal to V over omega, and over here,

00:03:23.399 --> 00:03:26.816
we could say that R sub blue is equal to,

00:03:28.078 --> 00:03:30.277
well the magnitude of the velocity,

00:03:30.277 --> 00:03:34.649
the speed is going to
be 2V over the magnitude

00:03:34.649 --> 00:03:38.859
of our angular velocity,
is going to be two omega.

00:03:38.859 --> 00:03:41.785
Well notice, the twos cancel out,

00:03:41.785 --> 00:03:44.833
so this is just going to
be V over omega again.

00:03:44.833 --> 00:03:46.881
These two things are identical.

00:03:46.881 --> 00:03:49.336
That is equal to that.

00:03:49.336 --> 00:03:53.612
So they are actually going to
have the exact same radius.

00:03:53.612 --> 00:03:55.945
So they are the same radius.


Change in angular velocity when velocity doubles

WEBVTT
Kind: captions
Language: en

00:00:00.234 --> 00:00:01.475
- [Instructor] We're told that a car

00:00:01.475 --> 00:00:05.642
with wheel radius r moves
at a linear velocity, v,

00:00:06.608 --> 00:00:09.450
and this is a bolded v, to
show that it's a vector.

00:00:09.450 --> 00:00:12.942
Suddenly, the car
accelerates to velocity 2v.

00:00:12.942 --> 00:00:16.867
How does the angular velocity
of the wheels change?

00:00:16.867 --> 00:00:18.050
So pause this video

00:00:18.050 --> 00:00:21.408
and see if you can figure
this out on your own.

00:00:21.408 --> 00:00:23.800
All right, now let's work
through this together.

00:00:23.800 --> 00:00:25.658
So just to visualize what's going on,

00:00:25.658 --> 00:00:29.808
let's just draw at least the
front wheel of a car here,

00:00:29.808 --> 00:00:33.141
so let me draw the wheel first actually.

00:00:34.025 --> 00:00:37.108
So this is the front wheel of the car

00:00:38.467 --> 00:00:41.300
and I'll draw the part of the car.

00:00:42.408 --> 00:00:44.866
So this is our car right over there.

00:00:44.866 --> 00:00:48.050
It might look something like that.

00:00:48.050 --> 00:00:49.950
We got the windshield right over there,

00:00:49.950 --> 00:00:52.450
so you can visualize this car.

00:00:53.933 --> 00:00:55.683
So our car looks something like,

00:00:55.683 --> 00:00:57.725
we got our headlights and all the rest.

00:00:57.725 --> 00:00:59.958
All right, enough of drawing the car.

00:00:59.958 --> 00:01:02.825
So let's imagine this car right over here

00:01:02.825 --> 00:01:05.908
has a velocity v, let's
say it's going forward,

00:01:05.908 --> 00:01:07.226
so let me draw that.

00:01:07.226 --> 00:01:10.208
So its velocity, I'll just
draw it right over here,

00:01:10.208 --> 00:01:11.041
it is v.

00:01:12.942 --> 00:01:15.650
So this is a velocity of v.

00:01:15.650 --> 00:01:18.500
And since it's hard to
hand draw something bolded,

00:01:18.500 --> 00:01:19.975
I put a little arrow on it,

00:01:19.975 --> 00:01:22.192
but this is equivalent
to this bolded vector v

00:01:22.192 --> 00:01:23.925
that you might see
sometimes in your textbook.

00:01:23.925 --> 00:01:24.883
Bold would be a vector

00:01:24.883 --> 00:01:27.541
or you put an arrow on
it to show it's a vector.

00:01:27.541 --> 00:01:29.350
Now I wanna make notation clear.

00:01:29.350 --> 00:01:32.950
If we talk about the
magnitude of this vector

00:01:32.950 --> 00:01:35.700
and the magnitude of
velocity is just speed,

00:01:35.700 --> 00:01:40.186
I'm gonna denote that as
just v without an arrow on it

00:01:40.186 --> 00:01:42.817
and not bolded, so this is
going to be the same thing

00:01:42.817 --> 00:01:45.419
as the magnitude of our velocity vector,

00:01:45.419 --> 00:01:48.917
which is the same thing as our speed.

00:01:48.917 --> 00:01:53.325
Now let's take a point on
the outer end of our wheel,

00:01:53.325 --> 00:01:54.808
any point on the outer end of the wheel.

00:01:54.808 --> 00:01:56.983
Let's say we take this
point right over here

00:01:56.983 --> 00:01:59.950
and the center of our
wheel is right over here.

00:01:59.950 --> 00:02:01.692
So if the car is moving forward

00:02:01.692 --> 00:02:03.867
to the right with velocity v,

00:02:03.867 --> 00:02:07.189
what's the velocity of
this point right over here?

00:02:07.189 --> 00:02:08.850
Well, the velocity of this point,

00:02:08.850 --> 00:02:11.700
it is going to have the same magnitude

00:02:11.700 --> 00:02:13.300
but in the opposite direction,

00:02:13.300 --> 00:02:15.383
so it's gonna look something like that.

00:02:15.383 --> 00:02:16.900
I don't know if those
lines are exactly equal

00:02:16.900 --> 00:02:19.075
but these are supposed to
have the same magnitude.

00:02:19.075 --> 00:02:20.942
And so you can even imagine this would be,

00:02:20.942 --> 00:02:24.550
you could say that this would
be negative v, just like that.

00:02:24.550 --> 00:02:28.391
But for the purposes of this question,

00:02:28.391 --> 00:02:30.342
I wanna think about its speed.

00:02:30.342 --> 00:02:32.208
So the speed here,

00:02:32.208 --> 00:02:35.791
the speed here is just
going to be v again.

00:02:36.717 --> 00:02:37.933
So the speed of this point,

00:02:37.933 --> 00:02:40.083
which is gonna be the same
as the speed of this point,

00:02:40.083 --> 00:02:41.608
the direction here is different,

00:02:41.608 --> 00:02:43.958
but all of these points along the rim

00:02:43.958 --> 00:02:45.983
or along the outside of
the tire, I should say,

00:02:45.983 --> 00:02:50.000
are all gonna have the
same speed, speed v.

00:02:50.000 --> 00:02:52.392
And we already have a
relationship that we've explored

00:02:52.392 --> 00:02:57.375
in previous videos between
speed and angular velocity.

00:02:57.375 --> 00:02:59.700
We have seen that the speed,

00:02:59.700 --> 00:03:02.142
I'll do it in a new color here.

00:03:02.142 --> 00:03:05.342
I'll say v for speed with
no arrow, not bolded,

00:03:05.342 --> 00:03:06.792
is going to be equal

00:03:06.792 --> 00:03:10.942
to the magnitude of our angular velocity,

00:03:10.942 --> 00:03:15.033
so the magnitude of our angular velocity

00:03:15.033 --> 00:03:16.450
times our radius.

00:03:18.908 --> 00:03:20.325
Times our radius.

00:03:21.316 --> 00:03:22.734
Or another way of saying this,

00:03:22.734 --> 00:03:26.067
we could say that our speed is equal to,

00:03:26.067 --> 00:03:27.900
instead of writing it like this,

00:03:27.900 --> 00:03:31.350
let me just write just an
omega without an arrow over it

00:03:31.350 --> 00:03:34.717
to say this is the magnitude
of our angular velocity.

00:03:34.717 --> 00:03:36.800
So that times our radius.

00:03:37.858 --> 00:03:40.041
So here I'm dealing with
all scalar quantities.

00:03:40.041 --> 00:03:41.307
I'm saying speed is equal

00:03:41.307 --> 00:03:44.033
to magnitude of my angular velocity

00:03:44.033 --> 00:03:45.366
times my radius.

00:03:46.808 --> 00:03:51.167
So if I wanted angular velocity
as a function of speed,

00:03:51.167 --> 00:03:53.592
I could just divide both sides by r.

00:03:53.592 --> 00:03:56.167
And so I could say angular velocity

00:03:56.167 --> 00:03:58.834
is going to be equal to my speed

00:04:01.250 --> 00:04:02.083
over r.

00:04:03.267 --> 00:04:05.175
So let's think about what
happens, and let me be clear,

00:04:05.175 --> 00:04:07.983
this right over, this distance
right over here, that is r.

00:04:07.983 --> 00:04:10.467
That is r right over there.

00:04:10.467 --> 00:04:13.358
So if our velocity goes to 2v,

00:04:13.358 --> 00:04:16.342
so if this goes from v to,

00:04:16.342 --> 00:04:17.858
I'm gonna use a different color,

00:04:17.858 --> 00:04:20.025
if that goes from v to 2v,

00:04:22.792 --> 00:04:26.959
then this is gonna go from
negative v to negative 2v.

00:04:28.525 --> 00:04:31.025
And our speed is gonna go from v

00:04:31.025 --> 00:04:33.692
to just 2v, the twice the speed.

00:04:35.583 --> 00:04:37.817
And if you have twice the speed,

00:04:37.817 --> 00:04:40.275
if this goes by two,

00:04:40.275 --> 00:04:43.400
well, then the angular velocity

00:04:43.400 --> 00:04:46.775
is going to increase by a factor of two.

00:04:46.775 --> 00:04:48.633
So this is going to
increase by a factor of two,

00:04:48.633 --> 00:04:50.558
not squaring this or something else.

00:04:50.558 --> 00:04:52.900
You're just multiplying
this by one over r.

00:04:52.900 --> 00:04:54.775
You're just dividing it by r.

00:04:54.775 --> 00:04:58.250
And r is not changing,
just the speed is changing.

00:04:58.250 --> 00:05:00.175
And since omega is equal to v over r,

00:05:00.175 --> 00:05:03.200
if you double v, you're
gonna double omega.

00:05:03.200 --> 00:05:06.342
So how does the angular
velocity of the wheels change?

00:05:06.342 --> 00:05:08.025
Well, it's going to double.

00:05:08.025 --> 00:05:09.275
It will double.

00:05:13.267 --> 00:05:14.600
And we are done.


Angular velocity and speed

WEBVTT
Kind: captions
Language: en

00:00:00.081 --> 00:00:01.592
- [Instructor] What we're
going to do in this video

00:00:01.592 --> 00:00:03.333
is look at the tangible example

00:00:03.333 --> 00:00:05.814
where we calculate angular velocity,

00:00:05.814 --> 00:00:08.316
but then we're going to
see if we can connect that

00:00:08.316 --> 00:00:10.554
to the notion of speed.

00:00:10.554 --> 00:00:13.085
So let's start with this
example where once again

00:00:13.085 --> 00:00:15.606
we have some type of a ball tethered

00:00:15.606 --> 00:00:18.689
to some type of center of
rotation right over here.

00:00:18.689 --> 00:00:21.161
Let's say this is connected with a string.

00:00:21.161 --> 00:00:22.776
And so if you were to
move the ball around,

00:00:22.776 --> 00:00:27.356
it would move along this blue
circle in either direction.

00:00:27.356 --> 00:00:28.958
Let's just say for the sake of argument,

00:00:28.958 --> 00:00:32.154
the length of the string is seven meters.

00:00:32.154 --> 00:00:36.071
We know that at time is
equal to three seconds,

00:00:37.225 --> 00:00:39.058
our angle is equal to,

00:00:41.230 --> 00:00:44.871
theta is equal to pi over two radians,

00:00:44.871 --> 00:00:46.286
which we've seen in previous videos.

00:00:46.286 --> 00:00:51.063
We would measure from the
positive x-axis, just like that.

00:00:51.063 --> 00:00:55.888
And let's say that at T
is equal to six seconds,

00:00:55.888 --> 00:01:00.055
T is equal to six seconds,
theta is equal to pi radians.

00:01:01.630 --> 00:01:05.797
And so after three seconds, the
ball is now right over here.

00:01:08.269 --> 00:01:12.158
And so if we wanted to actually
visualize how that happens,

00:01:12.158 --> 00:01:15.893
let me see if I can rotate
this ball in three seconds.

00:01:15.893 --> 00:01:17.571
So it would look like this.

00:01:17.571 --> 00:01:20.897
One Mississippi, two
Mississippi, three Mississippi.

00:01:20.897 --> 00:01:22.261
Let's do that again.

00:01:22.261 --> 00:01:23.428
It would be...

00:01:24.502 --> 00:01:28.679
One Mississippi, two
Mississippi, three Mississippi.

00:01:28.679 --> 00:01:30.822
So now that we can visualize

00:01:30.822 --> 00:01:33.175
or conceptualize what's going on,

00:01:33.175 --> 00:01:37.018
see if you can pause this
video and calculate two things.

00:01:37.018 --> 00:01:40.007
So the first thing that I
want you to calculate is

00:01:40.007 --> 00:01:43.424
what is the angular velocity of the ball?

00:01:44.315 --> 00:01:45.388
And actually, it would be the ball

00:01:45.388 --> 00:01:47.372
and every point on that string.

00:01:47.372 --> 00:01:51.437
What is that angular velocity
which we denote with omega?

00:01:51.437 --> 00:01:53.202
And then I want you to figure out

00:01:53.202 --> 00:01:56.324
what is the speed of the ball?

00:01:56.324 --> 00:01:58.389
So what is the speed?

00:01:58.389 --> 00:02:00.406
See if you can figure
out both of those things,

00:02:00.406 --> 00:02:03.005
and for extra points,
see if you can figure out

00:02:03.005 --> 00:02:05.555
a relationship between the two.

00:02:05.555 --> 00:02:07.906
Alright, well let's go
angular velocity first.

00:02:07.906 --> 00:02:09.897
I'm assuming you've had a go at it.

00:02:09.897 --> 00:02:12.742
Angular velocity, you might remember

00:02:12.742 --> 00:02:16.968
is just going to be equal
to our angular displacement,

00:02:16.968 --> 00:02:19.417
which we could say is delta theta,

00:02:19.417 --> 00:02:21.894
and it is a vector quantity.

00:02:21.894 --> 00:02:26.061
And we are going to divide
that by our change in time.

00:02:27.939 --> 00:02:29.580
So delta T.

00:02:29.580 --> 00:02:30.869
And so what is this going to be?

00:02:30.869 --> 00:02:34.102
Well this is going to be
our angular displacement.

00:02:34.102 --> 00:02:35.935
Our final angle is pi.

00:02:36.985 --> 00:02:41.824
Pi radians minus our initial
angle, pi over two radians.

00:02:41.824 --> 00:02:45.017
And then all of that's going
to be over our change in time,

00:02:45.017 --> 00:02:47.530
which is six seconds,
which is our final time

00:02:47.530 --> 00:02:49.978
minus our initial time,
minus three seconds.

00:02:49.978 --> 00:02:53.080
And so we are going to
get in the numerator,

00:02:53.080 --> 00:02:56.140
we have been rotated in
the positive direction,

00:02:56.140 --> 00:02:58.067
Pi over two radians.

00:02:58.067 --> 00:03:01.631
Because it's positive, we
know it's counterclockwise.

00:03:01.631 --> 00:03:04.714
And that happened over three seconds.

00:03:05.561 --> 00:03:08.197
And so we could rewrite this as

00:03:08.197 --> 00:03:11.457
this is going to be equal to pi over six.

00:03:11.457 --> 00:03:13.740
And let's remind
ourselves about the units.

00:03:13.740 --> 00:03:17.159
Our change in angle, that's
going to be in radians,

00:03:17.159 --> 00:03:20.997
and then that is going to be per second.

00:03:20.997 --> 00:03:23.596
So we're going pi over
six radians per second,

00:03:23.596 --> 00:03:25.400
and if you do that over three seconds,

00:03:25.400 --> 00:03:29.483
well then you're going to
go pi over two radians.

00:03:31.039 --> 00:03:32.930
Now with that out of the way,

00:03:32.930 --> 00:03:34.696
let's see if we can calculate speed.

00:03:34.696 --> 00:03:36.958
If you haven't done so
already, pause this video

00:03:36.958 --> 00:03:39.031
and see if you can calculate it.

00:03:39.031 --> 00:03:43.003
Well speed is going to
be equal to the distance

00:03:43.003 --> 00:03:45.691
the ball travels, and we've
touched on that in other videos.

00:03:45.691 --> 00:03:48.257
I encourage you to watch
those if you haven't already.

00:03:48.257 --> 00:03:52.823
The distance that we travel
we could denote with S.

00:03:52.823 --> 00:03:55.065
S is sometimes used to denote arc length,

00:03:55.065 --> 00:03:57.345
or the distance traveled right over here.

00:03:57.345 --> 00:03:59.841
So the speed is going to be our arc length

00:03:59.841 --> 00:04:02.839
divided by our change in time.

00:04:02.839 --> 00:04:05.449
Divided by our change in time.

00:04:05.449 --> 00:04:08.148
But what is our arc length going to be?

00:04:08.148 --> 00:04:09.898
Well we saw in a previous video

00:04:09.898 --> 00:04:13.253
where we related angular
displacement to arc length,

00:04:13.253 --> 00:04:16.144
or distance, that our arc
length is nothing more

00:04:16.144 --> 00:04:20.311
than the absolute value of
our angular displacement,

00:04:23.779 --> 00:04:27.576
of our angular displacement,
times the radius.

00:04:27.576 --> 00:04:29.120
Times the radius.

00:04:29.120 --> 00:04:31.807
And in this case, our radius
would be seven meters.

00:04:31.807 --> 00:04:34.756
So if we substitute all of this up here,

00:04:34.756 --> 00:04:36.310
what are we going to get?

00:04:36.310 --> 00:04:39.227
We are going to get that our speed,

00:04:40.205 --> 00:04:42.459
I'm writing it out because
I don't want to overuse,

00:04:42.459 --> 00:04:43.655
well I am overusing S,

00:04:43.655 --> 00:04:45.257
but I don't want people to get confused.

00:04:45.257 --> 00:04:48.153
Our speed is going to be equal
to the distance we travel,

00:04:48.153 --> 00:04:50.788
which we just wrote is the magnitude

00:04:50.788 --> 00:04:52.182
of our angular displacement.

00:04:52.182 --> 00:04:53.546
And this is all fancy notation,

00:04:53.546 --> 00:04:56.432
but when you actually apply it,
it's pretty straightforward.

00:04:56.432 --> 00:04:59.015
Times the radius of the circle.

00:04:59.892 --> 00:05:03.921
I guess you can say we
are traveling along.

00:05:03.921 --> 00:05:06.142
Let me write that in a different color.

00:05:06.142 --> 00:05:08.634
So times the radius.

00:05:08.634 --> 00:05:11.634
All of that over our change in time.

00:05:12.657 --> 00:05:15.066
All of that over our change in time.

00:05:15.066 --> 00:05:17.480
Well we could put in the
numbers right over here.

00:05:17.480 --> 00:05:21.028
We know that this is
going to be pi over two.

00:05:21.028 --> 00:05:22.211
You take the absolute value of that.

00:05:22.211 --> 00:05:23.981
It's still going to be pi over two.

00:05:23.981 --> 00:05:26.413
We know that our radius in this case

00:05:26.413 --> 00:05:27.762
is the length of that string.

00:05:27.762 --> 00:05:29.286
It is seven meters.

00:05:29.286 --> 00:05:31.628
And we know that our change in time here,

00:05:31.628 --> 00:05:34.836
we know that this over here
is going to be three seconds,

00:05:34.836 --> 00:05:36.678
and we can calculate everything.

00:05:36.678 --> 00:05:39.665
But what's even more
interesting is to recognize

00:05:39.665 --> 00:05:42.498
that what is this right over here?

00:05:45.804 --> 00:05:48.503
What is the absolute value
of our angular displacement

00:05:48.503 --> 00:05:50.170
over change in time?

00:05:51.057 --> 00:05:53.066
Well, this is just the absolute value

00:05:53.066 --> 00:05:55.568
of our angular velocity.

00:05:55.568 --> 00:05:57.735
So we could say that speed

00:05:59.915 --> 00:06:04.715
is equal to the absolute
value of our angular velocity.

00:06:04.715 --> 00:06:09.604
Absolute value of our angular
velocity, times our radius.

00:06:09.604 --> 00:06:11.021
Times our radius.

00:06:12.296 --> 00:06:14.428
And now so this is super useful.

00:06:14.428 --> 00:06:19.281
Our speed in this case,
is going to be pi over six

00:06:19.281 --> 00:06:20.864
radians per second.

00:06:22.373 --> 00:06:23.290
So pi over,

00:06:25.966 --> 00:06:26.966
pi over six.

00:06:28.273 --> 00:06:30.146
Times the radius.

00:06:30.146 --> 00:06:32.404
Times seven meters.

00:06:32.404 --> 00:06:34.472
Times seven meters.

00:06:34.472 --> 00:06:36.304
And so what do we get?

00:06:36.304 --> 00:06:40.387
We are going to get seven
pi over six meters per,

00:06:43.761 --> 00:06:47.928
meters per second, which will
be our units for speed here.

00:06:48.828 --> 00:06:50.936
And the reason why we're
doing the absolute value

00:06:50.936 --> 00:06:53.612
is 'cause remember, speed
is a scalar quantity.

00:06:53.612 --> 00:06:55.490
We're not specifying the direction.

00:06:55.490 --> 00:06:56.849
In fact, the whole time we're traveling,

00:06:56.849 --> 00:06:59.225
our direction is constantly changing.

00:06:59.225 --> 00:07:00.414
So there you have it.

00:07:00.414 --> 00:07:03.489
There's multiple ways to approach
these types of questions,

00:07:03.489 --> 00:07:05.476
but the big takeaway here is one,

00:07:05.476 --> 00:07:07.683
how we calculated angular velocity,

00:07:07.683 --> 00:07:12.592
and then how we can relate
angular velocity to speed.

00:07:12.592 --> 00:07:15.849
And what's nice is there's a
nice, simple formula for it.

00:07:15.849 --> 00:07:18.004
And all of this just came out of something

00:07:18.004 --> 00:07:20.082
that relates to what we
learned in seventh grade

00:07:20.082 --> 00:07:21.811
around the circumference of the circle,

00:07:21.811 --> 00:07:23.836
which we touched on in the video

00:07:23.836 --> 00:07:27.675
relating angular
displacement to arc length,

00:07:27.675 --> 00:07:29.425
or distance traveled.


Distance or arc length from angular displacement

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.785
- [Instructor] What we're
going to do in this video

00:00:01.785 --> 00:00:04.867
is try to draw connections
between angular displacement

00:00:04.867 --> 00:00:09.250
and notions of arc length
or distance traveled.

00:00:09.250 --> 00:00:12.104
Right over here, let's
imagine I have some type

00:00:12.104 --> 00:00:13.643
of a tennis ball or something,

00:00:13.643 --> 00:00:16.814
and it is tethered with a
rope to some type of a nail.

00:00:16.814 --> 00:00:19.235
If you were to try to
move this tennis ball,

00:00:19.235 --> 00:00:21.246
it would just rotate around that nail.

00:00:21.246 --> 00:00:24.459
It would go along this blue circular path.

00:00:24.459 --> 00:00:26.298
Let's just say for the sake of argument

00:00:26.298 --> 00:00:28.681
the radius of this blue
circle right over here,

00:00:28.681 --> 00:00:30.891
let's say it is six meters.

00:00:30.891 --> 00:00:33.698
You could view that as
the length of the string.

00:00:33.698 --> 00:00:35.196
We know what's going on here.

00:00:35.196 --> 00:00:39.448
Our initial angle, theta
initial, the convention is

00:00:39.448 --> 00:00:42.087
to measure it relative
to the positive x-axis.

00:00:42.087 --> 00:00:45.357
Our theta initial is pi over two.

00:00:45.357 --> 00:00:49.457
Let's say we were to then
rotate it by two pi radians,

00:00:49.457 --> 00:00:50.991
positive two pi radians.

00:00:50.991 --> 00:00:54.612
We would rotate in the
counterclockwise direction two pi.

00:00:54.612 --> 00:00:58.487
Then the ball would end up
where it started before.

00:00:58.487 --> 00:01:02.731
Theta final would just
be this plus two pi,

00:01:02.731 --> 00:01:05.789
so that would be five pi over two.

00:01:05.789 --> 00:01:07.499
Of course, we just said that we rotated

00:01:07.499 --> 00:01:11.612
in the positive
counterclockwise direction.

00:01:11.612 --> 00:01:13.804
We said we did it by two pi radians,

00:01:13.804 --> 00:01:15.883
so I kind of gave you the answer

00:01:15.883 --> 00:01:17.781
of what the angular displacement is.

00:01:17.781 --> 00:01:21.630
The angular displacement
in this situation,

00:01:21.630 --> 00:01:25.797
delta theta, is going to
be equal to two pi radians.

00:01:26.974 --> 00:01:29.341
The next question I'm going to ask you is

00:01:29.341 --> 00:01:32.291
what is the distance the
ball would have traveled?

00:01:32.291 --> 00:01:35.953
Remember, distance where we
actually care about the path.

00:01:35.953 --> 00:01:39.814
The distance traveled would be essentially

00:01:39.814 --> 00:01:42.830
the circumference of this circle.

00:01:42.830 --> 00:01:43.999
Think about what that is,

00:01:43.999 --> 00:01:46.279
and actually while you pause
the video and think about

00:01:46.279 --> 00:01:48.671
what the distance the ball traveled is,

00:01:48.671 --> 00:01:50.848
also think about what
would be the displacement

00:01:50.848 --> 00:01:52.765
that that ball travels.

00:01:54.564 --> 00:01:56.943
Well the easier answer, I'm
assuming you've had a go at it,

00:01:56.943 --> 00:01:57.776
is the displacement.

00:01:57.776 --> 00:01:59.614
The ball ends up where it
starts, so the displacement

00:01:59.614 --> 00:02:01.643
in this situation is as.

00:02:01.643 --> 00:02:02.989
We're not talking about
angular displacement.

00:02:02.989 --> 00:02:05.226
We're just talking about regular
displacement would be zero.

00:02:05.226 --> 00:02:08.570
The angular displacement
was two pi radians,

00:02:08.570 --> 00:02:10.072
but what about the distance.

00:02:10.072 --> 00:02:12.463
We'll denote the distance by S.

00:02:12.463 --> 00:02:14.091
As we can see, as we'll see,

00:02:14.091 --> 00:02:15.870
we can also view that as arc length.

00:02:15.870 --> 00:02:18.119
Here the arc is the entire circle.

00:02:18.119 --> 00:02:20.432
What is that going to be equal to?

00:02:20.432 --> 00:02:24.562
Well, we know from
earlier geometry classes

00:02:24.562 --> 00:02:25.757
that this is just going to be

00:02:25.757 --> 00:02:27.025
the circumference of the circle

00:02:27.025 --> 00:02:31.558
which is going to be equal
to two pi times the radius.

00:02:31.558 --> 00:02:35.641
It's going to be equal to
two pi times six meters

00:02:38.082 --> 00:02:41.415
which in this case is going to be 12 pi.

00:02:43.036 --> 00:02:47.287
Our units, in this case,
is going to be 12 pi meters

00:02:47.287 --> 00:02:49.236
would be the distance that we've traveled.

00:02:49.236 --> 00:02:51.573
Now what's interesting right over here is

00:02:51.573 --> 00:02:54.888
at least for this particular
case to figure out

00:02:54.888 --> 00:02:57.273
the distance traveled to
figure out that arc length,

00:02:57.273 --> 00:02:59.660
it looks like what we did is

00:02:59.660 --> 00:03:02.410
we took our angular displacement,

00:03:04.121 --> 00:03:06.792
in fact we took the magnitude
of our angular displacement,

00:03:06.792 --> 00:03:09.113
you could just view that as
the absolute value of it,

00:03:09.113 --> 00:03:12.913
and we multiplied it times
the radius of our circle.

00:03:12.913 --> 00:03:15.310
If you view that as the
length of that string as R,

00:03:15.310 --> 00:03:17.810
we just multiply that times R.

00:03:19.012 --> 00:03:21.726
We said our arc length,
in this case, was equal

00:03:21.726 --> 00:03:24.630
to the magnitude of our
change in displacement

00:03:24.630 --> 00:03:26.283
times our radius.

00:03:26.283 --> 00:03:28.830
Let's see if that is always true.

00:03:28.830 --> 00:03:31.789
In this situation, I have
a ball, and let's say

00:03:31.789 --> 00:03:33.079
this is a shorter string.

00:03:33.079 --> 00:03:36.402
Let's say this string
is only three meters,

00:03:36.402 --> 00:03:40.569
and its initial angle, theta
initial, is pi radians.

00:03:41.490 --> 00:03:44.563
We see that as measured
from the positive x-axis.

00:03:44.563 --> 00:03:48.378
Let's say we were to rotate it clockwise.

00:03:48.378 --> 00:03:52.461
Let's say our theta final
is pi over two radians.

00:03:53.297 --> 00:03:56.964
Theta final is equal
to pi over two radians.

00:03:57.999 --> 00:03:59.447
Pause this video, and see if you can

00:03:59.447 --> 00:04:03.127
figure out the angular displacement.

00:04:03.127 --> 00:04:06.680
The angular displacement
in this situation is going

00:04:06.680 --> 00:04:09.950
to be equal to theta
final which is pi over two

00:04:09.950 --> 00:04:13.748
minus theta initial which
is pi which is going

00:04:13.748 --> 00:04:16.136
to be equal to negative pi over two.

00:04:16.136 --> 00:04:17.162
Does this make sense?

00:04:17.162 --> 00:04:20.263
Yes, because we went clockwise.

00:04:20.263 --> 00:04:24.347
Clockwise rotations by convention
are going to be negative.

00:04:24.347 --> 00:04:25.265
That makes sense.

00:04:25.265 --> 00:04:29.716
We have a clockwise rotation
of pi over two radians.

00:04:29.716 --> 00:04:32.404
Now based on the information
we've just figured out,

00:04:32.404 --> 00:04:34.626
see if you can figure out the arc length

00:04:34.626 --> 00:04:37.667
or the distance that this tennis ball

00:04:37.667 --> 00:04:39.823
at then end of the
string actually travels.

00:04:39.823 --> 00:04:42.613
This tennis ball at the end
of the three meter string.

00:04:42.613 --> 00:04:46.023
What is this distance
actually going to be?

00:04:46.023 --> 00:04:48.146
Well, there's a couple of
ways you could think about it.

00:04:48.146 --> 00:04:50.151
You could say, hey
look, this is one fourth

00:04:50.151 --> 00:04:52.458
of the circumference of the circle.

00:04:52.458 --> 00:04:56.357
You could just say,
hey, this arc length, S,

00:04:56.357 --> 00:04:59.774
is just gonna be one fourth times two pi

00:05:04.064 --> 00:05:08.657
times the radius of three
meters, times three.

00:05:08.657 --> 00:05:10.954
This indeed would give
you the right answer.

00:05:10.954 --> 00:05:14.469
This would be, what, two
over four is one half,

00:05:14.469 --> 00:05:19.223
so you get three pi over two,
and we're dealing with meters,

00:05:19.223 --> 00:05:22.589
so this would be three pi over two.

00:05:22.589 --> 00:05:25.014
But let's see if this is
consistent with this formula

00:05:25.014 --> 00:05:26.624
we just had over here.

00:05:26.624 --> 00:05:29.047
Is this the same thing if we were to take

00:05:29.047 --> 00:05:31.738
the absolute value of our displacement.

00:05:31.738 --> 00:05:33.437
Let's do that.

00:05:33.437 --> 00:05:35.615
If we were to take the
absolute value of our,

00:05:35.615 --> 00:05:37.635
I'd should say our angular displacement,

00:05:37.635 --> 00:05:40.997
so we take the absolute value
of angular displacement,

00:05:40.997 --> 00:05:43.561
and we were to multiply it by our radius.

00:05:43.561 --> 00:05:46.573
Well, our radius is three meters.

00:05:46.573 --> 00:05:49.886
These would indeed be equal because this

00:05:49.886 --> 00:05:52.507
is just going to be positive
pi over two times three

00:05:52.507 --> 00:05:55.423
which is indeed three pi over two.

00:05:55.423 --> 00:05:59.422
It looks like this formula
is holding up pretty well.

00:05:59.422 --> 00:06:02.815
It makes sense because
what you're really doing

00:06:02.815 --> 00:06:04.007
is you're saying, look,

00:06:04.007 --> 00:06:08.174
you have your traditional
circumference of a circle.

00:06:09.267 --> 00:06:11.123
Then, you're thinking
about, well, what proportion

00:06:11.123 --> 00:06:13.148
of the circle is this arc length?

00:06:13.148 --> 00:06:16.484
If you were to say, well,
the proportion is going to be

00:06:16.484 --> 00:06:20.734
the magnitude of your
angular displacement.

00:06:20.734 --> 00:06:24.154
This is the proportion of the
circumference of the circle

00:06:24.154 --> 00:06:25.239
that you're going over.

00:06:25.239 --> 00:06:27.336
If this was two pi, you'd
be the entire circle.

00:06:27.336 --> 00:06:30.770
If it's pi, then you're
going half of the circle.

00:06:30.770 --> 00:06:32.832
Notice, these two things cancel out.

00:06:32.832 --> 00:06:36.282
This would actually give
you your arc length.

00:06:36.282 --> 00:06:38.248
Let's do one more example.

00:06:38.248 --> 00:06:40.552
Let's say in this situation, the string

00:06:40.552 --> 00:06:45.116
that is tethering our
ball is five meters long,

00:06:45.116 --> 00:06:50.053
and let's say our initial
angle is pi over six radians

00:06:50.053 --> 00:06:51.035
right over here.

00:06:51.035 --> 00:06:53.868
Let's say that our final angle is,

00:06:55.653 --> 00:06:57.430
we end up right over there.

00:06:57.430 --> 00:07:01.190
Our theta final is equal to pi over three.

00:07:01.190 --> 00:07:03.331
Based on everything
we've just talked about,

00:07:03.331 --> 00:07:07.057
what is going to be the
distance that the ball travels?

00:07:07.057 --> 00:07:09.974
What is going to be the arc length?

00:07:11.594 --> 00:07:15.193
Well, we could, first, figure
out our angular displacement.

00:07:15.193 --> 00:07:19.360
Our angular displacement is
going to be equal to theta final

00:07:22.127 --> 00:07:25.778
which is pi over three minus theta initial

00:07:25.778 --> 00:07:28.808
which is pi over six which is going to be

00:07:28.808 --> 00:07:31.400
equal to pi over six,
and that makes sense.

00:07:31.400 --> 00:07:33.391
This angle right over
here, we just went through

00:07:33.391 --> 00:07:36.354
a positive pi over six radian rotation.

00:07:36.354 --> 00:07:38.879
We went in the counterclockwise direction.

00:07:38.879 --> 00:07:41.336
Then, we just want to
figure out the arc length.

00:07:41.336 --> 00:07:44.516
We just multiply that times the radius.

00:07:44.516 --> 00:07:49.298
Our arc length is going
to be equal to pi over six

00:07:49.298 --> 00:07:52.215
times our radius times five meters.

00:07:53.219 --> 00:07:56.969
This would get us to
five pi over six meters,

00:08:01.083 --> 00:08:02.831
and we are done.

00:08:02.831 --> 00:08:04.224
Once again, no magic here.

00:08:04.224 --> 00:08:05.713
It comes straight out of the idea

00:08:05.713 --> 00:08:07.777
of the circumference of a circle.


Angular motion variables

WEBVTT
Kind: captions
Language: en

00:00:00.802 --> 00:00:02.007
- [Instructor] Things in the universe

00:00:02.007 --> 00:00:04.001
don't just shift around,

00:00:04.001 --> 00:00:06.017
they also rotate.

00:00:06.017 --> 00:00:07.664
And so what we're going
to do in this video

00:00:07.664 --> 00:00:12.035
is start to think about
rotations and rotational motion.

00:00:12.035 --> 00:00:14.439
And I'm intentionally
continuing to spin this

00:00:14.439 --> 00:00:16.896
because I find it hypnotic.

00:00:16.896 --> 00:00:17.933
But the question is,

00:00:17.933 --> 00:00:19.909
well, how do we start to
quantify these things,

00:00:19.909 --> 00:00:21.453
measure them, and describe them

00:00:21.453 --> 00:00:23.965
so that we can predict what will happen?

00:00:23.965 --> 00:00:28.629
Well, what if we introduce a
notion of angular displacement?

00:00:28.629 --> 00:00:31.826
We have displacement, which
is a shift in position.

00:00:31.826 --> 00:00:34.153
But what if you have a change in angle?

00:00:34.153 --> 00:00:35.729
That seems like it would
describe something.

00:00:35.729 --> 00:00:39.163
For example, right here, my
angle would be zero radians.

00:00:39.163 --> 00:00:42.205
And then if I move counterclockwise

00:00:42.205 --> 00:00:44.521
pi over two radians,

00:00:44.521 --> 00:00:46.357
now my angle is pi over two.

00:00:46.357 --> 00:00:47.774
So why not define

00:00:49.516 --> 00:00:52.178
going from there to there

00:00:52.178 --> 00:00:55.646
as an angular displacement
of pi over two radians?

00:00:55.646 --> 00:00:58.253
And if I were to displace
by pi over two again,

00:00:58.253 --> 00:01:00.003
I would go over here.

00:01:01.336 --> 00:01:03.232
And if I were to go negative pi radians,

00:01:03.232 --> 00:01:05.419
well, that would be just like that.

00:01:05.419 --> 00:01:07.948
Negative pi over two from there

00:01:07.948 --> 00:01:10.743
would get me right over there.

00:01:10.743 --> 00:01:13.137
That seems like a fairly intuitive thing.

00:01:13.137 --> 00:01:17.197
And you might be saying y is
positive counterclockwise,

00:01:17.197 --> 00:01:19.502
and y is negative clockwise?

00:01:19.502 --> 00:01:21.185
Well, that just tends
to be the conventions

00:01:21.185 --> 00:01:23.052
that we use for angles.

00:01:23.052 --> 00:01:23.885
So there you have it.

00:01:23.885 --> 00:01:25.631
We've already built our foundation

00:01:25.631 --> 00:01:28.385
for describing rotational motion.

00:01:28.385 --> 00:01:32.730
We could have an idea
of angular displacement,

00:01:32.730 --> 00:01:34.480
angular displacement,

00:01:36.262 --> 00:01:39.017
which we can define more formally

00:01:39.017 --> 00:01:40.666
just the way we just described it.

00:01:40.666 --> 00:01:42.239
We could say, well, look,

00:01:42.239 --> 00:01:44.788
this is going to be our change in angle,

00:01:44.788 --> 00:01:47.337
so change in, Greek letter delta,

00:01:47.337 --> 00:01:49.980
and angle, we use the Greek letter theta,

00:01:49.980 --> 00:01:53.585
which we've been using since
geometry or trigonometry class.

00:01:53.585 --> 00:01:56.730
And so our angular
displacement you can define as

00:01:56.730 --> 00:02:00.817
your final angle minus your initial angle.

00:02:00.817 --> 00:02:03.354
And we could say that
this is a vector quantity

00:02:03.354 --> 00:02:06.383
because you can either
go counterclockwise,

00:02:06.383 --> 00:02:09.880
in which case this is going
to be a positive quantity.

00:02:09.880 --> 00:02:11.829
Or you could go clockwise,

00:02:11.829 --> 00:02:15.079
in which case this is going
to be a negative quantity.

00:02:15.079 --> 00:02:16.941
Well, that's all nice and fair,

00:02:16.941 --> 00:02:19.102
but I know what some of you are thinking.

00:02:19.102 --> 00:02:23.083
It's nice to be able to see how
much your angle has changed,

00:02:23.083 --> 00:02:26.378
but isn't it much more
interesting to also describe

00:02:26.378 --> 00:02:28.413
how fast that angle is changing?

00:02:28.413 --> 00:02:31.477
For example, here it's
changing quite slowly.

00:02:31.477 --> 00:02:34.020
Let's say it takes us three seconds

00:02:34.020 --> 00:02:37.070
to have an angular displacement of two pi.

00:02:37.070 --> 00:02:41.237
One Mississippi, two
Mississippi, three Mississippi.

00:02:42.383 --> 00:02:43.987
That feels very different,

00:02:43.987 --> 00:02:46.635
a lot slower than if I
did that in one second.

00:02:46.635 --> 00:02:49.971
One Mississippi, two Mississippi.

00:02:49.971 --> 00:02:52.299
So what if we were to have something

00:02:52.299 --> 00:02:56.951
that would describe our
rate of change of angle?

00:02:56.951 --> 00:02:59.227
Well, we could borrow some terminology

00:02:59.227 --> 00:03:00.561
that we've used in the past.

00:03:00.561 --> 00:03:02.394
Instead of calling that velocity,

00:03:02.394 --> 00:03:05.160
we could call that angular velocity,

00:03:05.160 --> 00:03:05.993
angular

00:03:07.113 --> 00:03:07.946
velocity.

00:03:08.794 --> 00:03:10.439
And how would you define that?

00:03:10.439 --> 00:03:12.505
Pause this video, think about that.

00:03:12.505 --> 00:03:16.804
Well, angular velocity, you
could just say that is our

00:03:16.804 --> 00:03:18.054
change in angle

00:03:19.474 --> 00:03:22.360
over a certain amount of time.

00:03:22.360 --> 00:03:25.201
If I change two pi radians

00:03:25.201 --> 00:03:26.818
in one second,

00:03:26.818 --> 00:03:28.799
well, that's going to be a lot slower

00:03:28.799 --> 00:03:32.185
than if I change two pi
radians in half a second.

00:03:32.185 --> 00:03:35.046
And the letter we use
for angular velocity,

00:03:35.046 --> 00:03:38.183
the convention, is the Greek letter omega,

00:03:38.183 --> 00:03:41.141
which looks like a fancy lowercase w.

00:03:41.141 --> 00:03:43.591
Or at least this right over
here is the lowercase omega,

00:03:43.591 --> 00:03:46.628
which looks like a fancy lowercase w,

00:03:46.628 --> 00:03:49.916
which we use for angular velocity.

00:03:49.916 --> 00:03:51.866
And this, too, is a vector quantity

00:03:51.866 --> 00:03:56.033
because it's measuring a rate
of another vector quantity.

00:03:57.059 --> 00:03:59.321
And so just to hit the point home,

00:03:59.321 --> 00:04:03.014
just to review it all, make
it all jell in your mind,

00:04:03.014 --> 00:04:04.375
angular displacement,

00:04:04.375 --> 00:04:07.407
it's nothing more than
our change in angle.

00:04:07.407 --> 00:04:11.332
If our change in angle is
counterclockwise, like this,

00:04:11.332 --> 00:04:12.408
it's going to be positive.

00:04:12.408 --> 00:04:13.432
So let's say,

00:04:13.432 --> 00:04:15.209
so an angular displacement,
we might start here.

00:04:15.209 --> 00:04:17.024
We don't always start at zero radians.

00:04:17.024 --> 00:04:18.756
So let's say we start at pi over two.

00:04:18.756 --> 00:04:20.896
An angular displacement of pi over four

00:04:20.896 --> 00:04:22.705
would take us right over there.

00:04:22.705 --> 00:04:24.419
An angular displacement from there

00:04:24.419 --> 00:04:25.835
of negative pi over two,

00:04:25.835 --> 00:04:28.418
might take us right over there.

00:04:29.382 --> 00:04:31.825
And if we cared about angular velocity,

00:04:31.825 --> 00:04:34.882
if we said, hey, we are going to go

00:04:34.882 --> 00:04:37.602
pi radians every second,

00:04:37.602 --> 00:04:39.016
well, then that would be like this.

00:04:39.016 --> 00:04:41.973
It would be one second, two seconds,

00:04:41.973 --> 00:04:44.954
three seconds, four seconds.

00:04:44.954 --> 00:04:47.469
If we were to say we'd do
two pi radians every second,

00:04:47.469 --> 00:04:49.019
then that would be twice as fast

00:04:49.019 --> 00:04:51.120
in terms of angular velocity.

00:04:51.120 --> 00:04:54.765
One Mississippi, two
Mississippi, three Mississippi,

00:04:54.765 --> 00:04:58.432
four Mississippi, five Mississippi.

00:04:58.432 --> 00:04:59.634
So there you have it.

00:04:59.634 --> 00:05:00.960
And in the next few videos

00:05:00.960 --> 00:05:03.391
and in the practice
exercise on Khan Academy,

00:05:03.391 --> 00:05:06.556
we will get even more
familiar with these ideas

00:05:06.556 --> 00:05:08.510
and see how powerful they are

00:05:08.510 --> 00:05:12.593
for describing rotational
motion in the universe.


Percent from fraction models

WEBVTT
Kind: captions
Language: en

00:00:00.532 --> 00:00:01.707
- [Instructor] So we're told the square

00:00:01.707 --> 00:00:03.578
below represents one whole,

00:00:03.578 --> 00:00:05.824
so this entire square is a whole.

00:00:05.824 --> 00:00:08.358
And then they ask us what
percent is represented

00:00:08.358 --> 00:00:11.775
by the shaded area, so why
don't you pause this video

00:00:11.775 --> 00:00:13.902
and see if you can figure that out.

00:00:13.902 --> 00:00:14.904
So let's see.

00:00:14.904 --> 00:00:18.704
The whole is divided into
one, two, three, four,

00:00:18.704 --> 00:00:23.137
five, six, seven, eight,
nine, 10 equal sections,

00:00:23.137 --> 00:00:26.136
of which one, two, three, four, five,

00:00:26.136 --> 00:00:28.792
six, seven are actually filled in.

00:00:28.792 --> 00:00:30.723
That's the shaded area,
so one way to think

00:00:30.723 --> 00:00:34.616
about it is 7/10 are shaded in,

00:00:34.616 --> 00:00:37.312
but how do we express this
fraction as a percent?

00:00:37.312 --> 00:00:39.010
They're asking for a percent.

00:00:39.010 --> 00:00:42.613
Well remember, percent, it
literally means per hundred.

00:00:42.613 --> 00:00:44.557
Cent, same root as the word 100.

00:00:44.557 --> 00:00:48.010
You see it cents or century,
and so can we write this

00:00:48.010 --> 00:00:50.905
as per 100 instead of per 10?

00:00:50.905 --> 00:00:55.072
Well, seven per 10 is the same
thing as 70 per 100, or 70%.

00:01:00.472 --> 00:01:03.455
And how did I go from 7/10 to 70 over 100?

00:01:03.455 --> 00:01:05.400
Well, I just multiply both the numerator

00:01:05.400 --> 00:01:07.567
and the denominator by 10.

00:01:08.652 --> 00:01:10.099
And once you do more and more percents,

00:01:10.099 --> 00:01:11.336
you'll get a hang of it.

00:01:11.336 --> 00:01:12.400
You'll say, "Oh, 7/10.

00:01:12.400 --> 00:01:16.928
"That's the same thing as
70 per 100, which is 70%."

00:01:16.928 --> 00:01:19.600
Let's do another example.

00:01:19.600 --> 00:01:22.493
Here, we're told 100% is shown

00:01:22.493 --> 00:01:24.618
on the following tape diagram,

00:01:24.618 --> 00:01:28.827
so just this amount
right over here is 100%,

00:01:28.827 --> 00:01:30.843
and then they ask us what
percent is represented

00:01:30.843 --> 00:01:32.823
by the entire tape diagram,

00:01:32.823 --> 00:01:36.940
so by this entire thing right over here.

00:01:36.940 --> 00:01:39.964
Pause this video and see
if you can answer that.

00:01:39.964 --> 00:01:41.746
Well, one way to think about 100%,

00:01:41.746 --> 00:01:44.770
100% is equivalent to a whole,
and now we have three times

00:01:44.770 --> 00:01:47.657
as much as that for the
entire tape diagram,

00:01:47.657 --> 00:01:49.291
so you could view this as three wholes,

00:01:49.291 --> 00:01:51.184
or you could say that's 100%.

00:01:51.184 --> 00:01:53.893
We have another 100% right over here.

00:01:53.893 --> 00:01:57.403
And then we have another
100% right over here,

00:01:57.403 --> 00:02:01.570
so the whole tape diagram,
that would be 300%.

00:02:03.125 --> 00:02:04.003
Let's do another example.

00:02:04.003 --> 00:02:06.648
This is strangely fun. (laughs)

00:02:06.648 --> 00:02:08.522
Okay, and I'll see.

00:02:08.522 --> 00:02:11.639
It says the large rectangle
below represents one whole.

00:02:11.639 --> 00:02:15.719
All right, so that's this
whole thing is one whole.

00:02:15.719 --> 00:02:19.010
What percentage is represented
by the shaded area?

00:02:19.010 --> 00:02:22.590
So pause the video and see if
you can figure that out again.

00:02:22.590 --> 00:02:24.597
So let's just express
it as a fraction first,

00:02:24.597 --> 00:02:28.457
so we have a total of one,
two, three, four, five, six,

00:02:28.457 --> 00:02:30.394
seven, eight, nine, 10, 11, 12,

00:02:30.394 --> 00:02:33.644
13, 14, 15, 16, 17, 18, 19, 20 squares.

00:02:35.135 --> 00:02:37.662
So out of those 20
squares, we see that six

00:02:37.662 --> 00:02:39.953
of them are actually shaded in.

00:02:39.953 --> 00:02:43.946
So 6/20, could we write that as per 100?

00:02:43.946 --> 00:02:45.716
Well, let's see.

00:02:45.716 --> 00:02:48.940
To go from 20 to 100, I multiply by five,

00:02:48.940 --> 00:02:52.344
and so if I multiply
the numerator by five,

00:02:52.344 --> 00:02:53.437
I'll get the same value.

00:02:53.437 --> 00:02:57.582
Six times five is 30, so
six per 20 is the same thing

00:02:57.582 --> 00:03:01.749
as 30 per 100, which is
the same thing as 30%,

00:03:03.697 --> 00:03:08.568
which literally means
per 100, so this is 30%.

00:03:08.568 --> 00:03:11.467
Let's do one last example.

00:03:11.467 --> 00:03:14.547
Here we are told each large rectangle

00:03:14.547 --> 00:03:17.762
below represents one
whole, so this is a whole,

00:03:17.762 --> 00:03:20.331
and then this whole thing right
over here is another whole.

00:03:20.331 --> 00:03:23.221
What percentage is represented
by the shaded area?

00:03:23.221 --> 00:03:24.596
Again, pause the video.

00:03:24.596 --> 00:03:26.611
See if you can answer that.

00:03:26.611 --> 00:03:30.958
So this one, we've shaded
in a whole, so that is 100%,

00:03:30.958 --> 00:03:33.181
and then over here, we have shaded in one,

00:03:33.181 --> 00:03:36.747
two, three, 4/5 of the whole.

00:03:36.747 --> 00:03:39.186
So 4/5, if I wanted to express it

00:03:39.186 --> 00:03:42.416
as per 100, what would it be?

00:03:42.416 --> 00:03:46.583
Well, five time 20 is 100,
so four times 20 is 80,

00:03:48.629 --> 00:03:51.993
so 4/5, or 80/100, is filled out here.

00:03:51.993 --> 00:03:56.462
We could say 80 per 100, which
is the same thing as 80%,

00:03:56.462 --> 00:04:00.076
so this right over here is 80%.

00:04:00.076 --> 00:04:02.868
So what percent is represented
by the shaded area?

00:04:02.868 --> 00:04:07.035
Well, we have 100%, and then
we have 80%, so we have 180%.

00:04:08.805 --> 00:04:10.295
It's more than a whole.

00:04:10.295 --> 00:04:13.478
If you have a percentage
that is larger than 100%,

00:04:13.478 --> 00:04:14.665
you're talking about
something that is more

00:04:14.665 --> 00:04:16.341
than a whole, and then we see that.

00:04:16.341 --> 00:04:18.109
We have a whole right over here,

00:04:18.109 --> 00:04:21.406
and then we have 80% more than that.