1 00:00:02,970 --> 00:00:08,980 Here you see footage of hurricanes that formed in the Atlantic Ocean during 2009. Do you 2 00:00:08,980 --> 00:00:14,860 see that every single hurricane rotates counterclockwise? All hurricanes formed in the Northern hemisphere 3 00:00:14,860 --> 00:00:16,830 rotate counterclockwise. 4 00:00:16,830 --> 00:00:21,320 In this video, we'll provide you with the tools to explain why hurricanes rotate the 5 00:00:21,320 --> 00:00:23,060 way that they do. 6 00:00:23,060 --> 00:00:27,759 This video is part of the Representations video series. Information can be represented 7 00:00:27,759 --> 00:00:34,440 in words, through mathematical symbols, graphically, or in 3-D models. Representations are used 8 00:00:34,440 --> 00:00:40,590 to develop a deeper and more flexible understanding of objects, systems, and processes. 9 00:00:40,590 --> 00:00:46,640 Hi, my name is Sanjay Sarma, and I am a professor of Mechanical Engineering at MIT. Today we 10 00:00:46,640 --> 00:00:53,610 are going to demystify the origins of the forces that appear to act on objects in rotating 11 00:00:53,610 --> 00:00:55,040 frames. 12 00:00:55,040 --> 00:01:00,120 Before watching this video, you should be familiar with how to define basis vectors; 13 00:01:00,120 --> 00:01:05,720 inertial and non-inertial reference frames; and the representation of rotation rates as 14 00:01:05,720 --> 00:01:07,610 a vector cross products. 15 00:01:07,610 --> 00:01:13,780 After watching this video, you will be able to explain why centrifugal and Coriolis forces 16 00:01:13,780 --> 00:01:19,539 arise in rotating frames of reference, and apply your understanding of the Coriolis force 17 00:01:19,539 --> 00:01:26,539 to determine the direction of rotation of hurricanes. 18 00:01:27,000 --> 00:01:32,720 A frame of reference is a choice of coordinate frame, a set of orthonormal basis vectors. 19 00:01:32,720 --> 00:01:35,940 The frame is allowed to undergo rigid body motions. 20 00:01:35,940 --> 00:01:42,350 Rigid body motions include translation, rotation, or a combination of translation and rotation. 21 00:01:42,350 --> 00:01:47,590 As a frame of reference undergoes a rigid body motion, the 3 basis vectors retain their 22 00:01:47,590 --> 00:01:51,509 unit length and remain mutually orthogonal. 23 00:01:51,509 --> 00:01:57,090 In this video, we are going to focus on rotating frames of reference. In particular, we want 24 00:01:57,090 --> 00:02:02,240 to think about frames that are rotating with constant angular velocity and that aren't 25 00:02:02,240 --> 00:02:04,180 translating. 26 00:02:04,180 --> 00:02:09,590 Rotating frames of reference are non-inertial, thus we detect fictitious forces in them. 27 00:02:09,590 --> 00:02:14,470 We are going to explain how these so called fictitious forces arise. 28 00:02:14,470 --> 00:02:19,170 This is a turntable. We say that we are in the turntable frame of reference because the 29 00:02:19,170 --> 00:02:25,610 camera is mounted to the turntable. From this frame of reference, it appears that the turntable 30 00:02:25,610 --> 00:02:31,300 is stationary as the world spins around us, despite the fact that our experience tells 31 00:02:31,300 --> 00:02:35,150 us it is the turntable that is spinning. 32 00:02:35,150 --> 00:02:40,840 Here you see an orange disk attached to the turntable by a string. If we rotate the turntable 33 00:02:40,840 --> 00:02:46,760 quickly enough to overcome the friction between the disk and the turntable, you notice that 34 00:02:46,760 --> 00:02:49,020 the string becomes taut. 35 00:02:49,020 --> 00:02:53,910 Newton's second law implies that there must be a force equal and opposite to this tension 36 00:02:53,910 --> 00:02:56,819 force for the disk to remain stationary! 37 00:02:56,819 --> 00:03:02,520 This apparent force is what we call a fictitious force. Let's rotate the turntable with two 38 00:03:02,520 --> 00:03:07,840 disks—one is attached to the table by a string, while the other is unattached. The 39 00:03:07,840 --> 00:03:12,780 fictitious force causes the unattached disk to fly off of the Turntable. 40 00:03:12,780 --> 00:03:19,780 Is there really a force? If so, where does it come from? Pause the video and discuss. 41 00:03:26,420 --> 00:03:30,540 In addition to the Turntable frame, there is another frame of reference that will be 42 00:03:30,540 --> 00:03:32,420 useful in our analysis. 43 00:03:32,420 --> 00:03:37,230 This is the ground frame, G, which is any coordinate frame that appears to be stationary 44 00:03:37,230 --> 00:03:39,129 while standing on the ground. 45 00:03:39,129 --> 00:03:45,099 The Turntable frame, T, refers to any coordinate frame that appears stationary while standing 46 00:03:45,099 --> 00:03:47,280 on the Turntable. 47 00:03:47,280 --> 00:03:53,000 From the ground frame, the T frame is rotating counterclockwise with constant angular velocity, 48 00:03:53,000 --> 00:03:56,090 or almost constant angular velocity. 49 00:03:56,090 --> 00:04:01,209 It is important to be explicit about the particular frame of reference used to describe a velocity 50 00:04:01,209 --> 00:04:03,700 or acceleration vector. 51 00:04:03,700 --> 00:04:08,550 The velocity of the stationary vector is zero in the G frame, but the velocity of the rotating 52 00:04:08,550 --> 00:04:11,159 vector is zero in the T frame. 53 00:04:11,159 --> 00:04:17,159 We will denote the frame of such a vector by using a left superscript G or T to designate 54 00:04:17,159 --> 00:04:22,810 if we are considering the vector as an object in the G frame or the T frame. 55 00:04:22,810 --> 00:04:27,840 In the T frame, the position of the disk is fixed, so its velocity and acceleration are 56 00:04:27,840 --> 00:04:29,370 both zero. 57 00:04:29,370 --> 00:04:36,370 Pause the video here to determine the velocity of the disk as seen from the ground frame. 58 00:04:41,740 --> 00:04:46,590 According to the G frame, the T frame is rotating with some angular velocity, represented by 59 00:04:46,590 --> 00:04:53,060 the vector omega. Omega points along the axis of rotation with magnitude equal to the angular 60 00:04:53,060 --> 00:04:58,940 velocity. The velocity of the disk according to the G frame is entirely due to the rotation 61 00:04:58,940 --> 00:05:05,120 of the T frame, and thus can be represented by omega cross r. 62 00:05:05,120 --> 00:05:10,770 Recall that in general, any object that is moving on the turntable will have a velocity 63 00:05:10,770 --> 00:05:17,150 that can be defined in the T-frame. The velocity in the G-frame can be found as the sum of 64 00:05:17,150 --> 00:05:23,270 the velocity in the T-frame and the velocity that arises due to the rotation of the T-frame 65 00:05:23,270 --> 00:05:26,150 with respect to the G-frame. 66 00:05:26,150 --> 00:05:30,900 We can think of this as a rule for taking the time derivative of the position vector 67 00:05:30,900 --> 00:05:34,280 r in both the G-frame and the T-frame. 68 00:05:34,280 --> 00:05:39,259 We can generalize this as a formula for how to take the time derivative in the G frame 69 00:05:39,259 --> 00:05:45,680 of any vector, x, in terms of its T-frame derivative. This formula holds when the T 70 00:05:45,680 --> 00:05:50,960 frame is rotating, but not translating with respect to the G frame. 71 00:05:50,960 --> 00:05:56,370 Use this formula to take the time derivative of velocity. Try to find the general formula 72 00:05:56,370 --> 00:06:01,729 for the acceleration in the ground frame in terms of the acceleration in the turntable 73 00:06:01,729 --> 00:06:08,729 frame and various other terms. Pause the video while you carry out the computation. 74 00:06:23,580 --> 00:06:28,440 Looking at this formula, the first term is the object's acceleration as observed in the 75 00:06:28,440 --> 00:06:34,300 T frame. The remaining terms can be thought of as giving rise to the fictitious forces, 76 00:06:34,300 --> 00:06:40,500 which cause the acceleration observed in the T frame. The second term is the due to the 77 00:06:40,500 --> 00:06:46,680 angular acceleration of the T frame with respect to the G frame. This third term is the Coriolis 78 00:06:46,680 --> 00:06:53,360 acceleration—note that it depends on the velocity vector of the object in the T frame. 79 00:06:53,360 --> 00:06:59,990 And this last term is the centripetal acceleration—observe that it depends on how far the object is from 80 00:06:59,990 --> 00:07:02,919 the axis of rotation. 81 00:07:02,919 --> 00:07:08,330 Given this information, pause the video and determine the acceleration of the orange disk 82 00:07:08,330 --> 00:07:15,330 in the ground frame? 83 00:07:17,169 --> 00:07:22,650 The velocity and acceleration of the disk are zero in the T frame of reference. 84 00:07:22,650 --> 00:07:27,560 We did our best to rotate the turntable in this video with constant angular velocity. 85 00:07:27,560 --> 00:07:31,330 So we will assume that the angular acceleration is negligible. 86 00:07:31,330 --> 00:07:37,460 Thus the acceleration in the G-frame is given completely by the centripetal acceleration 87 00:07:37,460 --> 00:07:38,889 term. 88 00:07:38,889 --> 00:07:45,009 This vector points in the negative r direction with a magnitude given by the distance from 89 00:07:45,009 --> 00:07:50,020 the axis multiplied by the angular velocity squared. 90 00:07:50,020 --> 00:07:55,479 In the G frame, which is inertial, we do not observe effects of the so-called "fictitious 91 00:07:55,479 --> 00:08:02,130 forces". Remember, the disk wants to move in a straight line. It doesn't want to turn. 92 00:08:02,130 --> 00:08:06,940 The string imparts a tension force upon the orange disk, which provides the centripetal 93 00:08:06,940 --> 00:08:11,910 acceleration needed for the disk to rotate with the Turntable. 94 00:08:11,910 --> 00:08:17,650 But in the non-inertial T frame, we may think we observe a "fictitious force". The only 95 00:08:17,650 --> 00:08:22,889 thing that is fictitious is your perception that the disk is not accelerating. It is a 96 00:08:22,889 --> 00:08:27,860 "physical illusion" created by the fact that from the turntable frame of reference, you 97 00:08:27,860 --> 00:08:34,860 don't observe the centripetal acceleration of the turntable. 98 00:08:36,209 --> 00:08:41,929 In this video clip, we are rolling tennis balls through a plastic tube. In the T frame, 99 00:08:41,929 --> 00:08:47,819 we observe that the ball moves along a curved path. The curved motion observed in a rotating 100 00:08:47,819 --> 00:08:51,769 frame of reference is called the "Coriolis Effect." 101 00:08:51,769 --> 00:08:58,769 Pause the video and explain what you think is causing this curved motion. 102 00:09:03,550 --> 00:09:07,949 To better understand what is happening, let's look at the same motion from the inertial 103 00:09:07,949 --> 00:09:09,860 ground frame. 104 00:09:09,860 --> 00:09:14,029 But first, what do you think the motion of the ball will be when observed from the Ground 105 00:09:14,029 --> 00:09:21,029 frame? Pause the video and make a prediction. 106 00:09:24,989 --> 00:09:30,110 That's right. The path is a rather straight line along the initial trajectory! 107 00:09:30,110 --> 00:09:36,790 It is the motion of the turntable that is curved as the turntable rotates counterclockwise. 108 00:09:36,790 --> 00:09:41,699 But from the turntable frame, points on the turntable appear stationary, because you are 109 00:09:41,699 --> 00:09:47,819 rotating with the frame. So you perceive that the ball is curving to the right, even though 110 00:09:47,819 --> 00:09:51,050 it is you that is moving in a circular path. 111 00:09:51,050 --> 00:09:56,230 You might try to explain the curvature using fictitious forces. But we know that these 112 00:09:56,230 --> 00:10:02,790 forces are really just accelerations of the turntable frame that we do not perceive. 113 00:10:02,790 --> 00:10:07,790 Let's use what we know about the G frame to understand what these accelerations are in 114 00:10:07,790 --> 00:10:09,559 the T frame. 115 00:10:09,559 --> 00:10:15,309 Using the general formula for acceleration in a rotating frame that we found earlier, 116 00:10:15,309 --> 00:10:22,309 pause here and determine the acceleration in the T frame. 117 00:10:24,860 --> 00:10:30,199 Because the velocity is essentially constant in the G frame, the G frame acceleration is 118 00:10:30,199 --> 00:10:31,799 zero. 119 00:10:31,799 --> 00:10:37,110 We can rearrange the remaining nonzero terms to find an expression for the acceleration 120 00:10:37,110 --> 00:10:44,029 of the ball that we observe in the T frame. We find it is equal and opposite the sum of 121 00:10:44,029 --> 00:10:48,739 the Coriolis acceleration and the Centripetal acceleration. 122 00:10:48,739 --> 00:10:53,720 This term is the negative of the Centripetal acceleration, which we saw in the previous 123 00:10:53,720 --> 00:10:59,209 example. This acceleration is always pointing outwards from the center of rotation. 124 00:10:59,209 --> 00:11:06,059 This second term is the negative Coriolis acceleration. This acceleration is perpendicular 125 00:11:06,059 --> 00:11:12,220 to the velocity of the ball in the T frame, creating the curvature of the ball's path. 126 00:11:12,220 --> 00:11:17,360 We've said the accelerations are created by the rotation of the turntable. How does this 127 00:11:17,360 --> 00:11:18,489 happen? 128 00:11:18,489 --> 00:11:24,149 For an object to rotate with the turntable, a centripetal acceleration is required. Without 129 00:11:24,149 --> 00:11:29,339 it, an object appears to move outwards from the center of rotation. 130 00:11:29,339 --> 00:11:34,149 Even more is happening though. Because of the rigid body rotation, the velocity of a 131 00:11:34,149 --> 00:11:38,689 point near the outside edge of the turntable is greater than the velocity of point near 132 00:11:38,689 --> 00:11:40,889 the center of rotation. 133 00:11:40,889 --> 00:11:46,290 Moving objects accelerate due to this velocity differential. The acceleration is perpendicular 134 00:11:46,290 --> 00:11:51,449 to the velocity of the object. But you don't realize this acceleration exists from the 135 00:11:51,449 --> 00:11:54,819 turntable frame, because you don't perceive your own rotation. 136 00:11:54,819 --> 00:11:55,309 The acceleration created by this velocity differential between points on the turntable 137 00:11:55,309 --> 00:11:55,809 is responsible for the Coriolis effect, curving the paths of moving objects. 138 00:11:55,809 --> 00:12:00,049 You can feel these accelerations yourself if you walk around on a carousel. Or you can 139 00:12:00,049 --> 00:12:06,720 try waving your arms or legs on some other rotating theme park ride. 140 00:12:06,720 --> 00:12:13,720 Remember, the Earth is a rotating reference frame. Even though we are used to considering 141 00:12:14,920 --> 00:12:20,819 the Earth as a fixed frame, some phenomena, such as hurricanes, are created by the rotation 142 00:12:20,819 --> 00:12:26,239 of the Earth. We are going to use what we learned earlier to understand why hurricanes 143 00:12:26,239 --> 00:12:28,459 rotate the way they do. 144 00:12:28,459 --> 00:12:33,739 We can model regions of the Earth quite easily. For points that are in the Northern or Southern 145 00:12:33,739 --> 00:12:38,509 hemisphere, but sufficiently far from the equator, we can model the hemisphere by projecting 146 00:12:38,509 --> 00:12:40,970 it onto a disk. 147 00:12:40,970 --> 00:12:45,929 From this perspective, the northern hemisphere is a counterclockwise rotating disk, and the 148 00:12:45,929 --> 00:12:49,589 southern hemisphere is a clockwise rotating disk. 149 00:12:49,589 --> 00:12:54,799 Near the equator, we can't model the Earth as a disk. Instead, a better model would be 150 00:12:54,799 --> 00:12:58,359 the outside surface of a rotating cylinder. 151 00:12:58,359 --> 00:13:03,429 Our understanding of the Coriolis Effect from the turntable will directly apply to this 152 00:13:03,429 --> 00:13:05,399 model of the hemispheres. 153 00:13:05,399 --> 00:13:10,549 Now let's start by thinking about hurricanes. Hurricanes are formed when there are small 154 00:13:10,549 --> 00:13:17,549 regions of very low air pressure. The existence of this low-pressure region causes air from 155 00:13:17,980 --> 00:13:21,980 all directions to move towards the low-pressure zone. 156 00:13:21,980 --> 00:13:27,109 With this information and what you know about the Coriolis effect, explain why hurricanes 157 00:13:27,109 --> 00:13:34,109 in the northern hemisphere rotate counterclockwise. 158 00:13:38,290 --> 00:13:43,519 As air moves towards the low-pressure zone, the air moving from the south veers to the 159 00:13:43,519 --> 00:13:50,169 right due to the Coriolis effect. Similarly, air moving from the north veers to the left 160 00:13:50,169 --> 00:13:56,259 of the low-pressure zone. At the same time, air is constantly pulled in towards the low-pressure 161 00:13:56,259 --> 00:14:03,259 zone. The combination of these processes creates a region of counterclockwise rotating air. 162 00:14:03,910 --> 00:14:10,910 This is why hurricanes in the Northern hemisphere rotate counterclockwise. 163 00:14:12,139 --> 00:14:17,699 In this video, we saw that in rotating frames, an apparent force pulling away from the axis 164 00:14:17,699 --> 00:14:23,439 of rotation is really the objects tendency to move in a straight line. 165 00:14:23,439 --> 00:14:28,730 In a rotating frame of reference, objects require a centripetal acceleration to remain 166 00:14:28,730 --> 00:14:31,739 stationary in that frame. 167 00:14:31,739 --> 00:14:37,339 Moving objects in rotating frames move in curved paths due to the Coriolis acceleration, 168 00:14:37,339 --> 00:14:43,449 created by the fact that the velocity of points in the frame are greater further from the 169 00:14:43,449 --> 00:14:44,819 axis of rotation. 170 00:14:44,819 --> 00:14:50,989 On Earth, we can observe the Coriolis Effect in the counterclockwise rotation of hurricanes 171 00:14:50,989 --> 00:14:53,399 in the northern hemispheres. 172 00:14:53,399 --> 00:15:00,399 This leaves 2 questions for you. What direction do hurricanes rotate in the southern hemisphere? 173 00:15:00,529 --> 00:15:06,609 And why can't hurricanes form at the equator?