1 00:00:03,240 --> 00:00:09,400 Here you see MIT's Walter Lewin, about to swing on a pendulum. We can write a function 2 00:00:09,400 --> 00:00:15,440 for how the angle theta 1 depends on time, which completely describes his motion. 3 00:00:15,440 --> 00:00:20,810 But what if Walter were to start swinging from here, instead of there? We'd need a different 4 00:00:20,810 --> 00:00:22,810 function, theta 2. 5 00:00:22,810 --> 00:00:26,650 And what if one of his students were to give him a little push? That's right—we'd need 6 00:00:26,650 --> 00:00:28,929 a third function, theta 3. 7 00:00:28,929 --> 00:00:33,830 It would be great if we could write a single rule, based on physical principles, that governs 8 00:00:33,830 --> 00:00:38,840 the behavior of the pendulum. And every function that describes pendulum motion would have 9 00:00:38,840 --> 00:00:41,430 to satisfy this rule. 10 00:00:41,430 --> 00:00:46,470 This video will show you how to write rules like these using the language of differential 11 00:00:46,470 --> 00:00:47,720 equations. 12 00:00:47,720 --> 00:00:52,820 This video is part of the Differential Equations video series. Laws that govern a system's 13 00:00:52,820 --> 00:00:56,690 properties can be modeled using differential equations. 14 00:00:56,690 --> 00:01:03,690 Hi, my name is Peter Dourmashkin. And I am a senior physics lecturer at MIT. 15 00:01:05,440 --> 00:01:10,300 Differential equations are very important in modeling physical phenomena. They describe 16 00:01:10,310 --> 00:01:16,009 rules that constrain the behavior of many complex physical and social systems. 17 00:01:16,009 --> 00:01:21,530 Before watching this video, you should be familiar with drawing free body force diagrams 18 00:01:21,530 --> 00:01:27,310 and applying Newton's second law in polar coordinates. You should remember how to write 19 00:01:27,310 --> 00:01:34,299 a Taylor series expansion of a function about a point, and understand what it means. 20 00:01:34,299 --> 00:01:39,679 After watching this video, you should be able to do the following 3 things: 21 00:01:39,679 --> 00:01:43,350 Understand that the physical laws governing a system's properties can be modeled using 22 00:01:43,350 --> 00:01:45,759 differential equations, 23 00:01:45,759 --> 00:01:51,630 Explain that the solution to a differential equation is a family of functions, 24 00:01:51,630 --> 00:01:57,490 And recognize that specifying initial conditions determines a particular solution function 25 00:01:57,490 --> 00:02:01,229 to a differential equation. 26 00:02:01,229 --> 00:02:06,670 Let's see how differential equations arise as a natural language to model the pendulum. 27 00:02:06,670 --> 00:02:10,819 To develop this model, we will use our knowledge of physics. 28 00:02:10,819 --> 00:02:17,819 First, what is a pendulum? A pendulum is typically an object, called a bob, which hangs from 29 00:02:18,030 --> 00:02:23,060 one end of a string. The other end is fixed to a pivot point. 30 00:02:23,060 --> 00:02:27,580 Let's pause the video. Try to describe what you think are the important properties of 31 00:02:27,580 --> 00:02:29,380 pendulum motion. 32 00:02:36,100 --> 00:02:40,060 You might have said that the behavior is periodic, that the same motion 33 00:02:40,250 --> 00:02:47,250 repeats over and over. We say that the pendulum exhibits oscillations, or harmonic motion. 34 00:02:48,550 --> 00:02:53,660 Maybe you also noted that the motion is rotational, moving in a circular path about the pivot 35 00:02:53,660 --> 00:02:55,160 point. 36 00:02:55,160 --> 00:03:01,900 Finally, you may know that the motion eventually stops, that is, the oscillations experience 37 00:03:01,900 --> 00:03:08,900 a damping force. Overall we describe the motion as "damped harmonic motion." 38 00:03:09,070 --> 00:03:16,070 Now, let's develop a model for the pendulum. A good first step is to choose a coordinate 39 00:03:16,510 --> 00:03:21,870 system. Because the pendulum rotates about a fixed pivot point, it makes sense to use 40 00:03:21,870 --> 00:03:28,870 polar coordinates. Theta is the angle the bob makes with the vertical, and r is the 41 00:03:29,120 --> 00:03:32,570 distance of the bob from the pivot point. 42 00:03:32,570 --> 00:03:38,580 Remember that when using vector in polar coordinates, r-hat is the unit vector that points radially 43 00:03:38,580 --> 00:03:43,940 outward from the pivot point, and theta-hat is the unit vector that points perpendicular 44 00:03:43,940 --> 00:03:49,200 to r-hat, tangential to the motion of the pendulum. 45 00:03:49,200 --> 00:03:53,600 Whenever you model a system, it's important to recognize that you are making some simplifying 46 00:03:53,600 --> 00:03:58,560 assumptions. We are going to start by making three: 47 00:03:58,720 --> 00:03:59,600 1. 48 00:03:59,600 --> 00:04:06,600 The string has no mass. 2. The mass, m, of the bob is uniformly distributed. 49 00:04:08,140 --> 00:04:09,400 3. 50 00:04:09,410 --> 00:04:13,820 The string has length L and does not stretch or shrink. 51 00:04:13,820 --> 00:04:20,820 This means that all motion happens in the tangential direction; none in the radial direction. 52 00:04:21,829 --> 00:04:27,120 Now let's figure out what we need to know to describe the motion of the pendulum. 53 00:04:27,120 --> 00:04:32,009 At the speeds we are dealing with, we can ignore relativistic effects; there are no 54 00:04:32,009 --> 00:04:38,870 electromagnetic forces, and the scale is large enough that quantum mechanics isn't necessary. 55 00:04:38,870 --> 00:04:43,650 So we'll use Newton's second law to describe the motion of the pendulum. 56 00:04:43,650 --> 00:04:48,090 What forces do you think we need to consider in this problem? 57 00:04:54,039 --> 00:04:59,229 Let's draw a free body diagram for an instant where the bob is to the right of vertical, 58 00:04:59,229 --> 00:05:01,620 and moving downwards. 59 00:05:01,620 --> 00:05:06,139 There is the gravitational force acting on the bob, due to the interaction between the 60 00:05:06,139 --> 00:05:12,800 Earth and the bob, which points straight down with magnitude mg. We treat this force as 61 00:05:12,800 --> 00:05:17,189 acting on the center of mass of the bob. 62 00:05:17,189 --> 00:05:21,699 The string exerts a force on the bob, which you know as tension. We don't know what the 63 00:05:21,699 --> 00:05:26,860 magnitude of this force is, so we'll just write it as T, pointing inwards towards the 64 00:05:26,860 --> 00:05:28,539 pivot point. 65 00:05:28,539 --> 00:05:33,629 Finally, the motion will eventually stop, so there must be damping forces acting on 66 00:05:33,629 --> 00:05:39,460 the system. The damping forces are complicated. For example, there is friction at the pivot 67 00:05:39,460 --> 00:05:45,259 point, and air resistance on the bob. Let's make an assumption that the damping force 68 00:05:45,259 --> 00:05:51,830 due to friction at the pivot point is negligible because the string is small and lubricated. 69 00:05:51,830 --> 00:05:58,830 Then the damping force is almost entirely from the drag due to air resistance. 70 00:05:58,830 --> 00:06:05,259 The bob doesn't experience any drag when it is stationary, and experiences larger amounts 71 00:06:05,259 --> 00:06:11,569 of drag the faster it is moving. Because the speeds of the pendulum are relatively slow, 72 00:06:11,569 --> 00:06:18,069 that is we do not expect there to be turbulence, we can model the drag force as being linearly 73 00:06:18,069 --> 00:06:24,729 proportional to velocity, and opposing the direction of motion. Because we assume that 74 00:06:24,729 --> 00:06:31,729 all motion occurs in tangential direction, we can draw the damping force like so. 75 00:06:32,249 --> 00:06:38,809 Recall that Newton's second law is a vector equation. Because force and acceleration are 76 00:06:38,809 --> 00:06:44,979 both vector quantities, we need to decompose them into r-hat and theta-hat components. 77 00:06:44,979 --> 00:06:51,539 Remember, acceleration can be decomposed into two component vectors, one pointing radially 78 00:06:51,539 --> 00:06:57,839 inward called centripetal acceleration, and a tangential component vector called angular 79 00:06:57,839 --> 00:07:04,839 acceleration. We also need to decompose our force vectors. Notice that the tension force 80 00:07:05,169 --> 00:07:10,800 is already pointing radially inward, in the negative r-hat direction. And the damping 81 00:07:10,800 --> 00:07:16,419 force is pointing in the positive theta-hat direction, opposite the direction of the angular velocity 82 00:07:16,419 --> 00:07:19,409 vector. 83 00:07:19,409 --> 00:07:26,120 So all we have to do is decompose the gravitational force. We project that vector onto the r-hat 84 00:07:26,120 --> 00:07:34,340 and theta-hat component vectors. In the r-hat direction the magnitude is mg cos(theta) and 85 00:07:34,349 --> 00:07:40,770 points outwards. In the theta-hat direction, the magnitude is mg sin(theta) and points 86 00:07:40,770 --> 00:07:44,110 in the negative theta-hat direction. 87 00:07:44,110 --> 00:07:49,769 Newton's second law can now be written as two component equations. For the radial 88 00:07:49,769 --> 00:07:52,531 direction, we get this. 89 00:07:52,550 --> 00:07:54,849 And in the tangential direction, we get this. 90 00:07:56,160 --> 00:07:59,960 This system of two equations has two unknowns. 91 00:07:59,960 --> 00:08:03,000 The angle theta, and the tension. 92 00:08:03,040 --> 00:08:09,480 Remember, that we are trying to find an expression for how the angle theta depends on time. This 93 00:08:09,490 --> 00:08:16,490 second equation is more useful to us, because we can express both the angular velocity and 94 00:08:16,779 --> 00:08:22,669 the angular acceleration in terms of theta. Let's see how. 95 00:08:22,669 --> 00:08:29,660 The circular motion of the pendulum is a 1-dimensional motion, parameterized by time t. 96 00:08:30,800 --> 00:08:37,779 The displacement of the pendulum away from center is the arc length, which is L times theta. Because 97 00:08:37,779 --> 00:08:42,849 the angular velocity is tangent to the circle of motion, it can be found simply by differentiating 98 00:08:42,849 --> 00:08:49,259 the displacement with respect to the parameter t, which gives us that the angular velocity 99 00:08:49,259 --> 00:08:56,999 is L times the time derivative of theta. The angular acceleration is just the time derivative 100 00:08:57,000 --> 00:09:04,000 of the angular velocity, and is given by L times the second time derivative of theta. 101 00:09:04,970 --> 00:09:09,769 Observe that this works because we are only looking for the angular velocity and angular 102 00:09:09,769 --> 00:09:13,750 acceleration which point tangent to our 1-dimensional motion. 103 00:09:14,980 --> 00:09:21,860 So now we've found expressions for the magnitudes of the angular velocity and angular acceleration: 104 00:09:21,860 --> 00:09:23,000 Here they are. 105 00:09:24,610 --> 00:09:29,480 We can substitute these equations into Newton's Second Law in the theta-hat direction; and we 106 00:09:29,480 --> 00:09:36,480 obtain the following equation. 107 00:09:42,610 --> 00:09:49,610 This is a second order ordinary differential equation or 2nd order ODE. 108 00:09:51,340 --> 00:09:58,160 It is ordinary because there is only one independent variable, time, that the angle theta depends on. It 109 00:09:58,160 --> 00:10:03,079 is second order because the highest degree derivative that appears in the equation is 110 00:10:03,079 --> 00:10:05,940 a second derivative. 111 00:10:05,940 --> 00:10:11,639 Any function describing the pendulum motion must satisfy this differential equation. Keep 112 00:10:11,639 --> 00:10:17,060 in mind that there are other forces we ignored, such as friction at the pivot point. This 113 00:10:17,060 --> 00:10:23,899 is a very simple but nontrivial model for the non-linear pendulum. 114 00:10:23,899 --> 00:10:30,130 Notice that the first derivative term is our velocity dependent drag force. If we decide 115 00:10:30,130 --> 00:10:37,130 that this term is negligible, we would get this equation. 116 00:10:37,960 --> 00:10:43,740 It is still complicated, but you might notice that it looks somewhat familiar. 117 00:10:43,750 --> 00:10:49,149 If we assume that the displacement of the angle is very small, we can approximate sin 118 00:10:49,149 --> 00:10:56,149 theta as the angle theta, which gives us this equation. 119 00:10:57,260 --> 00:11:05,259 This is the equation of the simple harmonic oscillator. So you see that this piece connecting the term sin theta and the 120 00:11:05,259 --> 00:11:11,740 second derivative of theta produces the oscillatory behavior, and the first derivative term is 121 00:11:11,740 --> 00:11:15,690 responsible for the damping of the motion. 122 00:11:15,690 --> 00:11:20,500 We've seen that we can apply our knowledge of physical laws to model a non-linear pendulum 123 00:11:20,500 --> 00:11:22,820 with a differential equation. 124 00:11:27,400 --> 00:11:31,680 How does this differential equation describe the motion of a pendulum? 125 00:11:31,680 --> 00:11:36,069 A differential equation is an equation where the unknown element is a function, rather 126 00:11:36,069 --> 00:11:42,720 than a number or a single value. The differential equation is a rule that describes how the 127 00:11:42,720 --> 00:11:48,920 system evolves from any time t to a time t plus Delta t. 128 00:11:49,320 --> 00:11:52,740 Where Delta t is a small time interval. 129 00:11:52,740 --> 00:12:00,200 In particular, it describes the relationship between the angle theta and its first and second derivatives as the system 130 00:12:00,209 --> 00:12:03,220 evolves over time. 131 00:12:03,220 --> 00:12:08,540 Remember that to describe the motion of the pendulum, we need an expression for the angle 132 00:12:08,540 --> 00:12:15,209 theta as a function of time. Do you think there is a unique solution to this differential 133 00:12:15,209 --> 00:12:19,569 equation? Pause the video here and discuss. 134 00:12:25,160 --> 00:12:30,230 You probably have some sense that the motion of the pendulum depends on how and where you 135 00:12:30,230 --> 00:12:36,350 release it—conditions we refer to as initial conditions. 136 00:12:36,350 --> 00:12:41,889 For example, if we barely move the pendulum away from center, and release it from rest, 137 00:12:41,889 --> 00:12:48,550 the oscillation will be small. Larger displacements will give larger oscillations, even though 138 00:12:48,550 --> 00:12:55,129 eventually they all damp down to zero. All of these pendulum motions satisfy the same 139 00:12:55,129 --> 00:12:58,069 differential equation! 140 00:12:58,069 --> 00:13:02,430 Since every starting position will give a different dependence of theta on time, the 141 00:13:02,430 --> 00:13:09,430 solution is not unique. We say that the solution to a differential equation is an infinite 142 00:13:09,480 --> 00:13:12,589 family of functions. 143 00:13:12,589 --> 00:13:17,199 But what if we were are interested in finding one solution in particular? 144 00:13:17,199 --> 00:13:21,680 In the examples you just saw, we specified the initial angle from which the pendulum 145 00:13:21,680 --> 00:13:27,430 was released. Is this the only piece of information we need to pick out a specific solution from 146 00:13:27,430 --> 00:13:29,629 the infinite family? 147 00:13:29,629 --> 00:13:36,449 Let's test that. Here you see two identical pendulums, each starting from the same initial 148 00:13:36,449 --> 00:13:40,069 angle. Are we guaranteed that the motion will be the same? 149 00:13:40,069 --> 00:13:42,689 Pause the video here to make a prediction. 150 00:13:42,689 --> 00:13:47,049 Clearly not. What was the difference between the initial conditions for these two pendulums? 151 00:13:52,629 --> 00:13:57,939 The initial angles of both pendulums were the same. However, one was released from rest, 152 00:13:57,939 --> 00:14:03,529 and the other had an initial push. In other words, the initial angular velocities were 153 00:14:03,529 --> 00:14:10,189 different. Remember, our differential equation only holds after the pendulum is released 154 00:14:10,189 --> 00:14:14,860 because it doesn't include the action of the pushing force. 155 00:14:14,860 --> 00:14:20,870 Now we see the importance of specifying two initial conditions—the initial angle and 156 00:14:20,870 --> 00:14:26,910 the initial angular velocity. Remember, the differential equation is the rule for how 157 00:14:26,910 --> 00:14:32,819 the system evolves over time. In order to solve this differential equation, we are going 158 00:14:32,819 --> 00:14:39,389 to use the Taylor series expansion for the function theta about t=0. 159 00:14:39,389 --> 00:14:45,189 Recall that this is a polynomial series, whose coefficients are determined by all of the 160 00:14:45,189 --> 00:14:52,189 derivatives of the function theta at time t=0. So how can we find all of these derivatives 161 00:14:52,959 --> 00:14:55,540 of theta? 162 00:14:55,540 --> 00:15:01,350 Let's use the differential equation to find the second derivative at time t=0. Notice 163 00:15:01,350 --> 00:15:08,350 that it depends on both of our initial conditions: initial angle and initial angular velocity. 164 00:15:08,970 --> 00:15:12,949 This is why we need those initial conditions. 165 00:15:12,949 --> 00:15:19,550 We can differentiate the differential equation to find the third derivative at time t=0. 166 00:15:19,550 --> 00:15:25,910 Notice again that the third derivative depends on the initial angle, the initial angular 167 00:15:25,910 --> 00:15:31,509 velocity, and the second derivative at time t=0. 168 00:15:31,509 --> 00:15:37,850 We can continue this process and find recursive formulas for the all of the higher order derivatives 169 00:15:37,850 --> 00:15:46,680 of theta at time t=0! That's how the two initial conditions allow us to pick out a solution, 170 00:15:46,680 --> 00:15:53,680 by specifying all of the coefficients for the Taylor series at t=0. 171 00:15:53,680 --> 00:16:01,399 Our Taylor series solution completely describes the function theta. But it only works on the interval of 172 00:16:01,399 --> 00:16:08,399 convergence. If the series converges everywhere, we're done! But if it doesn't converge everywhere, 173 00:16:08,580 --> 00:16:10,860 it must converge on some interval. 174 00:16:11,480 --> 00:16:16,620 A differential equation determines a specific function if we have enough pieces 175 00:16:16,620 --> 00:16:23,449 of initial data to determine all the Taylor coefficients for the function. In our case, 176 00:16:23,449 --> 00:16:28,959 with a second order ordinary differential equation, we just needed two initial conditions. 177 00:16:28,959 --> 00:16:34,120 Pause the video. If you had a third order differential equation, how many initial conditions 178 00:16:34,120 --> 00:16:38,540 would you need to specify one solution? Pause the video. 179 00:16:45,140 --> 00:16:46,680 That's right, three! 180 00:16:51,019 --> 00:16:55,439 We wanted to describe the motion of a pendulum. We made a model, where we made 181 00:16:55,439 --> 00:17:01,339 some simplifying assumptions then we applied Newton's Second Law. 182 00:17:01,339 --> 00:17:05,310 This gave us the differential equation, which is the rule that angle theta 183 00:17:05,310 --> 00:17:09,279 and its derivatives must satisfy. 184 00:17:09,279 --> 00:17:15,520 We saw there is an infinite family of functions that solve our differential equation. 185 00:17:15,520 --> 00:17:20,949 Because the differential equation was second order, we needed 2 initial conditions to specify 186 00:17:20,949 --> 00:17:24,169 one unique solution. 187 00:17:24,169 --> 00:17:29,460 The initial conditions allowed us to determine the coefficients of the Taylor series for 188 00:17:29,460 --> 00:17:35,840 the angle theta as a function of time, which solved the differential equation. 189 00:17:35,840 --> 00:17:40,580 I hope you found this video helpful in understanding differential equations. 190 00:17:40,580 --> 00:17:45,520 Now we challenge you to model other systems using differential equations.