1
00:00:00,450 --> 00:00:03,630
Let's say we have some object that's moving in a circular path

2
00:00:03,630 --> 00:00:08,200
Let's say this is the center of the object path, the center of the circle

3
00:00:08,390 --> 00:00:12,880
So the object is moving in a circular path that looks something like that

4
00:00:12,880 --> 00:00:16,600
counterclockwise circular path--you could do that with clockwise as well

5
00:00:16,870 --> 00:00:23,010
I want to think about how fast it is spinning or orbiting around this center

6
00:00:23,070 --> 00:00:25,930
how that relates to its velocity?

7
00:00:25,950 --> 00:00:33,290
So let's say that this thing right over here is making five revolutions every second

8
00:00:33,310 --> 00:00:40,880
So in 1 second, 1 2 3 4 5. Every second it's making 5 revolutions

9
00:00:40,900 --> 00:00:45,700
So how could we relate that to how many radians it is doing per second?

10
00:00:45,740 --> 00:00:48,660
Remember radians is just one way to measure angles

11
00:00:48,660 --> 00:00:50,710
You could do with how degrees per second

12
00:00:54,550 --> 00:01:02,590
If we do it with radians, we know that each revolution is 2 pi radians

13
00:01:02,610 --> 00:01:10,340
If you go all the way around a circle, you have gone 2 pi radians

14
00:01:10,340 --> 00:01:16,800
which is really just you say you've gone 2 pi radii, whatever the radius of the circle is

15
00:01:16,820 --> 00:01:20,120
and that's where actually the definition of the radian comes from

16
00:01:20,140 --> 00:01:30,000
So if you go 5 revolutions per second and they're 2 pi per revolution

17
00:01:30,000 --> 00:01:33,350
then you can do a little bit of dimensional analysis. These cancel out

18
00:01:33,490 --> 00:01:41,450
and you get 5 times 2 pi which gets us to 5 times 2 pi gets us 10 pi

19
00:01:41,450 --> 00:01:47,760
radians per second

20
00:01:47,770 --> 00:01:52,200
And it works out the dimensional analysis and hopefully it also makes sense to you intuitively

21
00:01:52,350 --> 00:01:56,640
If you're doing five revolution a second, each of those revolutions is 2 pi radians

22
00:01:56,710 --> 00:02:00,030
so you're doing 10 pi radians per second. You're going

23
00:02:00,310 --> 00:02:04,150
1 2 3 4 5, so that gives us 10, or 2 pi 2 pi 2 pi 2 pi 2 pi radians

24
00:02:04,160 --> 00:02:08,550
every time, you're doing it five times a second. So you're doing it 10 pi radians per second

25
00:02:08,810 --> 00:02:12,730
So this right here, either five revs per second or 10 pi radians per second

26
00:02:12,780 --> 00:02:14,990
they're both essentially measuring the same thing

27
00:02:14,990 --> 00:02:18,990
how fast are you orbiting around this central point?

28
00:02:19,310 --> 00:02:23,020
And this measure of how fast you're orbiting around a central point

29
00:02:23,020 --> 00:02:29,830
is called angular velocity

30
00:02:29,970 --> 00:02:32,460
It's called angular velocity because if you think about it

31
00:02:32,540 --> 00:02:40,360
this is telling us how fast is our angle changing, or speed of angle changing

32
00:02:40,550 --> 00:02:43,280
When you're dealing with it in two dimensions and this is

33
00:02:43,430 --> 00:02:47,260
typically when in a recent early physics course how we do deal with it

34
00:02:47,260 --> 00:02:49,230
Even though it's called the angular velocity

35
00:02:49,310 --> 00:02:51,020
it tends to be treated as angular speed

36
00:02:51,020 --> 00:02:52,870
It actually is a vector quantity

37
00:02:53,020 --> 00:02:56,360
and it's a little unintuitive that the vector's actually popping out of the page

38
00:02:56,460 --> 00:02:59,860
for this. It's actually a pseudo-vector and we'll talk more about that in the future

39
00:02:59,920 --> 00:03:03,580
So it is a vector quantity and the direction of the vector

40
00:03:03,580 --> 00:03:06,580
is dependant on which way it's spinning. So for example

41
00:03:06,580 --> 00:03:09,200
when it's spinning in a counterclockwise direction

42
00:03:09,280 --> 00:03:13,530
there is a vector, the real angular vector does pop out of the page

43
00:03:13,530 --> 00:03:16,700
We start thinking about operating in three dimensions

44
00:03:16,730 --> 00:03:22,760
And if it's going clockwise, the angular velocity vector would pop into the page

45
00:03:22,800 --> 00:03:25,220
The way you think about that, right-hand rule

46
00:03:25,230 --> 00:03:28,260
Curl your fingers of your right hand in the direction that it's spinning

47
00:03:28,260 --> 00:03:33,380
and then your thumb is essentially pointing in the direction that

48
00:03:33,380 --> 00:03:36,610
the actual vector or the pseudo-vector's gonna going

49
00:03:36,610 --> 00:03:38,480
We'll not think too much about that

50
00:03:38,510 --> 00:03:42,500
For our purposes, when we're just thinking about two-dimensional plane right over here

51
00:03:42,950 --> 00:03:49,320
we can really think of an angular velocity as a--the official term is a pseudo-scaler

52
00:03:49,320 --> 00:03:55,820
but we can include that as a scaler quantity, as long as we do specify which way it is rotating

53
00:03:55,830 --> 00:03:59,040
So this right over here, this 10 pi radians per second

54
00:03:59,040 --> 00:04:01,810
we could call this its angular velocity

55
00:04:02,030 --> 00:04:06,300
And this tends to be denoted by an omega

56
00:04:06,330 --> 00:04:08,370
a lower case omega right there

57
00:04:08,380 --> 00:04:10,250
Upper case omega looks like this

58
00:04:14,390 --> 00:04:16,430
So there's a couple of ways you could think about it

59
00:04:16,450 --> 00:04:25,220
You could say angular velocity is equal to change in angle over a change in time

60
00:04:25,320 --> 00:04:32,290
So for example, this is telling us 10 pi radians per second

61
00:04:32,450 --> 00:04:41,820
Or if you want to do in the calculus sense and take instantaneous angular velocity

62
00:04:41,950 --> 00:04:47,510
it would be the derivative of your angle with respect to time

63
00:04:47,590 --> 00:04:50,950
How the angle is changing with respect to time

64
00:04:50,960 --> 00:04:56,100
With that out of the way, I want to see if we can see how this relates to speed

65
00:04:56,440 --> 00:05:00,460
How does this relate to the actual speed of the object?

66
00:05:00,660 --> 00:05:05,910
So to get the speed of the object, we just have to think about how far is this object traveling

67
00:05:05,910 --> 00:05:08,200
every revolution that it's doing

68
00:05:08,860 --> 00:05:15,000
And what we can do right over here--let's say that this radius is r

69
00:05:15,270 --> 00:05:19,760
So in every revolution, it is traveling 2 pi r

70
00:05:19,780 --> 00:05:24,080
Let's say this is r meters. Give ourselves some units right over there

71
00:05:24,420 --> 00:05:33,270
So the circumference over here is going to be 2 pi r meters

72
00:05:33,290 --> 00:05:55,530
Let's say that the angular velocity is equal to omega radians per second

73
00:05:56,070 --> 00:06:00,760
And so how many revolutions is that per second?

74
00:06:00,760 --> 00:06:03,450
We can go backwards from what we did over here

75
00:06:03,470 --> 00:06:10,710
We have one revolution is equal to 2 pi radians

76
00:06:10,860 --> 00:06:14,940
Just to be clear, sometimes angular velocity is actually measured in revolutions per second

77
00:06:14,940 --> 00:06:17,640
but the SI unit is in radians per second

78
00:06:17,660 --> 00:06:22,040
So here I want to convert omega from radians per second into revolutions per second

79
00:06:22,280 --> 00:06:32,670
Radians cancel out. We are left with--we get omega over 2 pi revolutions per second

80
00:06:32,920 --> 00:06:35,600
We know how many meters we get for a revolution

81
00:06:35,770 --> 00:06:40,520
We have 2 pi r meters per revolutions

82
00:06:40,830 --> 00:06:42,980
So we copy and paste this

83
00:06:43,360 --> 00:06:46,780
So our angular velocity, if we want revolutions per second

84
00:06:46,780 --> 00:06:50,970
it's going to be omega over 2 pi revolutions per second

85
00:06:51,540 --> 00:06:55,960
Omega is in radians per second if we put it into revolutions per second

86
00:06:55,960 --> 00:06:58,560
omega / 2 pi revolutions per second

87
00:06:58,900 --> 00:07:05,310
And then let's multiply that times--we want to convert this into meters per second

88
00:07:05,310 --> 00:07:09,040
So how many meters do we have per revolution?

89
00:07:09,320 --> 00:07:17,720
Well, we're gonna travel a whole circumference per revolution, so 2 pi r meters per revolution

90
00:07:17,740 --> 00:07:19,680
So these two cancel out

91
00:07:19,700 --> 00:07:21,990
2 pi cancels out with the 2 pi

92
00:07:21,990 --> 00:07:36,130
So you end up getting omega times r meters per second

93
00:07:36,500 --> 00:07:40,720
And just like that, we have the magnitude of the velocity

94
00:07:40,720 --> 00:07:44,750
or we could say the speed of the object as it goes around in a circle

95
00:07:45,010 --> 00:07:48,720
So what we can say is the magnitude of the velocity--

96
00:07:48,720 --> 00:07:51,460
I'll specify that by v--I want to be clear. This is not vector

97
00:07:51,460 --> 00:07:54,320
quantity. It's not the velocity. It's the magnitude of velocity

98
00:07:54,560 --> 00:07:59,910
Or we can say this is the speed. It's going to be equal to omega times r

99
00:08:00,460 --> 00:08:08,070
So the speed is equal to the angular velocity times r

100
00:08:08,070 --> 00:08:14,030
I guess we could say the magnitude of the angular velocity times the radius

101
00:08:14,220 --> 00:08:17,150
I don't want you to be confused. I am not saying that this is a vector quantity

102
00:08:17,150 --> 00:08:21,460
If this was a vector, I would put an arrow right over there

103
00:08:21,680 --> 00:08:24,240
And if this was a vector, I would put an arrow over there

104
00:08:24,240 --> 00:08:26,560
then I'll be referring to the thing that's popping out of the page

105
00:08:26,560 --> 00:08:29,450
but here I'm talking about the magnitude of the angular velocity

106
00:08:29,710 --> 00:08:42,669
and so writing in words, you get speed is equal to angular velocity--

107
00:08:42,669 --> 00:08:49,610
if you want to be particular, this is the magnitude of the angular velocity--

108
00:08:49,610 --> 00:08:54,870
times the radius of the circle that you are going around

109
00:08:54,870 --> 00:08:57,640
and if you want to solve for angular velocity

110
00:08:57,640 --> 00:09:01,540
you divide both sides by radius and you get angular velocity

111
00:09:01,760 --> 00:09:14,880
Omega is equal to speed which we're using v for, divided by the radius

112
00:09:14,900 --> 00:09:19,020
So we can actually use this information to do other interesting things later on

113
00:09:19,030 --> 00:00:00,000
But hopefully this gives you a sense of how all of this stuff is related