1 00:00:02,919 --> 00:00:09,139 On the evening of August 5, 2012 Pacific Daylight Time, NASA's Mars Rover, named Curiosity, 2 00:00:09,139 --> 00:00:19,439 entered Mars' atmosphere at 20,000km/h. Drag slowed it down to around 1600km/hr, at which 3 00:00:19,449 --> 00:00:26,449 point a parachute opened. This parachute slowed the Rover more, to about 320km/hr, or 90 m/s. 4 00:00:28,769 --> 00:00:34,699 Finally, after rockets decelerated it completely, the rover was lowered to the surface of Mars. 5 00:00:34,699 --> 00:00:40,780 Every step of this dance was carefully choreographed and rehearsed in many experiments here on 6 00:00:40,780 --> 00:00:47,129 Earth. But how could NASA engineers be sure that their designs would work on a totally 7 00:00:47,129 --> 00:00:54,129 different planet? The answer is a problem-solving method called dimensional analysis. 8 00:00:54,780 --> 00:00:59,299 This video is part of the Problem Solving video series. 9 00:00:59,299 --> 00:01:03,319 Problem-solving skills, in combination with an understanding of the natural and human-made 10 00:01:03,319 --> 00:01:07,009 world, are critical to the design and optimization of systems and processes. 11 00:01:07,009 --> 00:01:13,899 Hi, my name is Ken Kamrin, and I am a professor of mechanical engineering at MIT. Dimensional 12 00:01:13,899 --> 00:01:20,600 analysis is a powerful tool; I use it, NASA uses it, and you will too. 13 00:01:20,600 --> 00:01:24,960 Before watching this video, you should be familiar with unit analysis, and understand 14 00:01:24,960 --> 00:01:28,280 the difference between dependent and independent variables. 15 00:01:28,280 --> 00:01:32,109 By the end of this video, you will be able to use dimensional analysis to estimate the 16 00:01:32,109 --> 00:01:39,109 size of a parachute canopy that can slow the Rover down to 90 m/s on its descent to Mars. 17 00:01:44,060 --> 00:01:49,630 Before we talk about dimensional analysis, we need to know what dimension is. Dimensions 18 00:01:49,630 --> 00:01:56,100 and units are related, but different, concepts. Physical quantities are measured in units. 19 00:01:56,100 --> 00:02:01,090 The dimension of the physical quantity is independent of the particular units choosen. 20 00:02:01,090 --> 00:02:05,569 For example: Both grams and kilograms are units, but they 21 00:02:05,569 --> 00:02:10,810 are units of mass.And mass is what we'll call the dimension. 22 00:02:10,810 --> 00:02:19,319 There are 5 fundamental dimensions that we commonly deal with: length, mass, time, temperature, 23 00:02:19,319 --> 00:02:21,260 and charge. 24 00:02:21,260 --> 00:02:27,510 All other dimensions are obtained by taking products and powers of these fundamental dimensions. 25 00:02:27,510 --> 00:02:31,620 In this video, we'll be dealing with length, which we denote by the letter 26 00:02:31,620 --> 00:02:41,140 L, mass, which we denote M, and time, T. For example, no matter how you measure the physical 27 00:02:41,150 --> 00:02:47,450 quantity velocity, it has the dimension, which we denote with square brackets, of length 28 00:02:47,450 --> 00:02:52,950 divided by time, or length times time to the negative 1 power. 29 00:02:52,950 --> 00:02:57,530 Pause the video here and determine the dimension of energy. 30 00:03:03,260 --> 00:03:06,120 Energy has dimension Mass times 31 00:03:06,129 --> 00:03:10,060 Length squared over Time squared. 32 00:03:10,060 --> 00:03:17,480 Okay, great, so what's the big deal? How is this useful? 33 00:03:17,480 --> 00:03:22,720 Remember NASA's rover? Part of the landing sequence calls for a parachute to slow the 34 00:03:22,720 --> 00:03:28,620 vehicle down. Suppose it is our job to design the parachute to slow the Rover to exactly 35 00:03:28,620 --> 00:03:34,680 90m/s. The terminal velocity of the Rover depends on the mass of the rover itself and 36 00:03:34,680 --> 00:03:40,480 its heat shield, and several different variables related to the parachute design: the material 37 00:03:40,480 --> 00:03:45,939 of the canopy, the diameter of the hemispherical parachute canopy, the number of suspension 38 00:03:45,939 --> 00:03:48,540 lines, etc. 39 00:03:48,540 --> 00:03:52,920 For simplicity, let's suppose that all parachute parameters other than the diameter of the 40 00:03:52,920 --> 00:03:59,319 canopy have already been determined. Our goal is to find the canopy diameter that is as 41 00:03:59,319 --> 00:04:06,319 small as possible, but will correspond to the desired terminal velocity of 90 m/s. 42 00:04:06,599 --> 00:04:13,439 Clearly, we can't test our designs on Mars. The question is: how can we get meaningful 43 00:04:13,439 --> 00:04:20,899 data here on Earth that will allow us to appropriately size the parachute for use on Mars? How do 44 00:04:20,910 --> 00:04:25,790 we predict the behavior of a parachute on Mars based on an Earth experiment? And what 45 00:04:25,790 --> 00:04:31,300 variables do we need to consider in designing our experiment on Earth? This is where a problem 46 00:04:31,300 --> 00:04:36,880 solving method called dimensional analysis can help us. 47 00:04:40,720 --> 00:04:45,780 Before we get started, we must first determine what the dependent and independent variables 48 00:04:45,780 --> 00:04:48,380 are in our system. 49 00:04:48,380 --> 00:04:52,440 The dependent variable is terminal velocity. 50 00:04:52,440 --> 00:04:58,350 This is the quantity that we wish to constrain by our parachute design. So what variables 51 00:04:58,350 --> 00:05:02,700 affect the terminal velocity of the parachute and rover system? 52 00:05:02,700 --> 00:05:08,080 The diameter of the parachute canopy is one such independent variable. Take a moment to 53 00:05:08,080 --> 00:05:10,680 pause the video and identify others. 54 00:05:16,500 --> 00:05:24,220 Ok. Here's our list: canopy diameter, mass of the Rover (we assume the mass of the parachute 55 00:05:24,220 --> 00:05:32,520 to be negligible), acceleration due to gravity, and the density and viscosity of the atmosphere. 56 00:05:33,370 --> 00:05:38,440 For this problem, we can assume that the dependence of the terminal velocity on atmospheric viscosity 57 00:05:38,440 --> 00:05:45,440 is negligible, because the atmosphere on Mars like the atmosphere on Earth is not very viscous. 58 00:05:46,120 --> 00:05:51,060 If we wanted to derive a functional relationship that would work, for example, underwater, 59 00:05:51,060 --> 00:05:56,020 it would be important that we include viscosity as an independent variable. 60 00:05:56,020 --> 00:06:00,090 Why didn't we include the surface area of the parachute canopy in our list of independent 61 00:06:00,090 --> 00:06:01,850 variables? 62 00:06:01,850 --> 00:06:04,850 Pause the video and take a moment to discuss with a classmate. 63 00:06:11,260 --> 00:06:15,520 We didn't include the surface area of the canopy, because it is not independent from 64 00:06:15,520 --> 00:06:16,920 the diameter of the canopy. 65 00:06:16,920 --> 00:06:23,150 In fact, we can determine the area as a function of diameter. So we don't need both! 66 00:06:23,150 --> 00:06:30,150 Question: Could we use the surface area instead of the diameter? Absolutely. We need the variables 67 00:06:30,770 --> 00:06:37,090 to be independent, but it doesn't matter which variables we use! The key is to have identified 68 00:06:37,090 --> 00:06:43,330 all of the correct variables to begin with. This is where human error can come into play. 69 00:06:43,330 --> 00:06:47,430 If our list of variables isn't exhaustive, the relationship we develop through dimensional 70 00:06:47,430 --> 00:06:51,750 analysis may not be correct! 71 00:06:51,750 --> 00:06:56,159 Once we have the full list of independent variables, we can express the terminal velocity 72 00:06:56,159 --> 00:07:00,780 as some function of these independent variables. 73 00:07:00,780 --> 00:07:05,880 In order to find the function that describes the relationship, we need do several experiments 74 00:07:05,880 --> 00:07:12,620 involving 4 independent variables, and fit the data. Phew, that's a lot of work! Especially 75 00:07:12,620 --> 00:07:16,970 because we don't know what the function might look like. 76 00:07:16,970 --> 00:07:23,640 But whenever you have an equation, all terms in the equation must have the same dimension. 77 00:07:23,640 --> 00:07:28,590 Multiplying two terms multiplies the dimensions. 78 00:07:28,590 --> 00:07:33,240 This restricts the possible form that a function describing the terminal velocity in terms 79 00:07:33,240 --> 00:07:38,860 of our 4 other variables can take, because the function must combine the variables in 80 00:07:38,860 --> 00:07:44,550 some way that has the same dimension as velocity. 81 00:07:44,550 --> 00:07:51,270 And many functions—exponential, logarithmic, trigonometric --cannot have input variables 82 00:07:51,270 --> 00:07:58,270 that have dimension. What would e to the 1kg mean? What units could it possibly have? 83 00:07:59,370 --> 00:08:03,780 We're going to show you a problem solving method that will allow you to find the most 84 00:08:03,780 --> 00:08:10,780 general form of such a function. This method is called dimensional analysis. 85 00:08:14,210 --> 00:08:20,360 We begin this process by creating dimensionless versions of the variables in our system. We 86 00:08:20,360 --> 00:08:24,390 create these dimensionless expressions out of the variables in our system, so we don't 87 00:08:24,390 --> 00:08:28,510 introduce any new physical parameters. 88 00:08:28,510 --> 00:08:32,719 The first step is to take our list of variables, and distill them down to their fundamental 89 00:08:32,719 --> 00:08:34,360 dimensions. 90 00:08:34,360 --> 00:08:39,000 Remember our fundamental dimensions are length, mass, and time. 91 00:08:39,000 --> 00:08:46,000 Distilling gravity to its fundamental dimension, we get length per time squared. Now you distill 92 00:08:46,500 --> 00:08:53,500 the remaining variables of velocity, diameter, mass, and density into their fundamental dimensions. 93 00:08:54,170 --> 00:08:56,470 Pause the video here. 94 00:09:02,040 --> 00:09:08,880 Velocity is length per time; diameter is length; mass is mass; and density is mass per length 95 00:09:08,889 --> 00:09:11,290 cubed. 96 00:09:11,290 --> 00:09:15,910 The second step is to express the fundamental dimensions of mass, length, and time in terms 97 00:09:15,910 --> 00:09:18,040 of our independent variables. 98 00:09:18,040 --> 00:09:20,920 We can write M as little m. 99 00:09:20,920 --> 00:09:26,819 We can write L as mass divided by density to the 1/3 power. 100 00:09:26,819 --> 00:09:32,439 We can write T as velocity divided by gravity. 101 00:09:32,439 --> 00:09:37,980 We had many choices as to how to write these fundamental dimensions in terms of our variables. 102 00:09:37,980 --> 00:09:42,769 In the end, it doesn't matter which expressions you choose. 103 00:09:42,769 --> 00:09:47,980 The third step is to use these fundamental dimensions to turn all of the variables involved 104 00:09:47,980 --> 00:09:54,980 into dimensionless quantities. For example, the terminal velocity v has dimension of length 105 00:09:55,339 --> 00:10:02,339 over time. So we multiply v by the dimension of time, and divide by the dimension of length 106 00:10:09,639 --> 00:10:15,689 to get a dimension of one. 107 00:10:15,689 --> 00:10:22,689 We define a new dimensionless variable vbar as this dimensionless version of v. 108 00:10:26,040 --> 00:10:36,120 Now you try; find dbar, mbar, gbar, and rhobar. Pause the video here. 109 00:10:42,480 --> 00:10:48,840 You should have found that dbar is d times rho over m to the 1/3. 110 00:10:48,840 --> 00:10:56,339 Mbar is 1. Gbar is v squared over g times rho over 111 00:10:56,339 --> 00:11:01,240 m to the 1/3., and rho bar is 1. 112 00:11:01,240 --> 00:11:07,110 Now we can rewrite the equation for velocity in terms of the new dimensionless variables. 113 00:11:07,110 --> 00:11:12,709 It is a new function, because the variables have been modified. Notice that vbar is equal 114 00:11:12,709 --> 00:11:21,339 to gbar. This means that vbar and gbar are not independent! So our function for vbar 115 00:11:21,339 --> 00:11:28,999 cannot depend on gbar. Also, notice that mbar and rho bar are both equal to one, so our 116 00:11:29,009 --> 00:11:32,939 function doesn't depend on them either. 117 00:11:32,939 --> 00:11:39,939 This has simplified our relationship: vbar is a function of only one variable, dbar. 118 00:11:40,790 --> 00:11:46,819 And remember that dbar is dimensionless, so it is just a real number. This means that, 119 00:11:46,819 --> 00:11:50,949 phi can be any function. 120 00:11:50,949 --> 00:11:56,709 The forth and final step is to rearrange to find a formula for the terminal velocity. 121 00:11:56,709 --> 00:12:03,709 The key here is that this equation for the terminal velocity has the correct units. 122 00:12:04,029 --> 00:12:09,879 And this formula is so general, that any expression with dimension of Length over Time can be 123 00:12:09,879 --> 00:12:18,079 written in terms of this formula by defining phi in different ways. Let's see how. First, 124 00:12:18,079 --> 00:12:23,559 create an expression from the independent variables that has the same dimension as velocity. 125 00:12:23,559 --> 00:12:30,550 One such expression is the square root of g times d. By setting the formula for v equal 126 00:12:30,550 --> 00:12:38,579 to the square root of gd, we see that by setting phi equal to the identity function phi(x) 127 00:12:38,580 --> 00:12:44,040 = x, the two sides of the equation can be made equal. 128 00:12:44,050 --> 00:12:49,220 And in fact, we claim that any expression with the correct dimension of Length over 129 00:12:49,220 --> 00:12:54,800 time created using these variables can be written in terms of this formula by simply 130 00:12:54,800 --> 00:12:57,949 changing the definition of phi! 131 00:12:57,949 --> 00:13:03,339 It can be fun to try this. Come up with different formulas that have the correct dimension. 132 00:13:03,339 --> 00:13:08,069 You can even add them together. Then see if you can find a way to define phi so that our 133 00:13:08,069 --> 00:13:13,209 formula is equal to the expression you wrote. Pause the video here. 134 00:13:18,699 --> 00:13:23,850 Now you may be concerned because this formula is not unique. We made some choices about 135 00:13:23,850 --> 00:13:30,850 how to represent our fundamental dimensions. What happens if we make different choices? 136 00:13:31,490 --> 00:13:44,069 Here we chose M, L, and T this way: M was little m, L was d, and T was v over g. Running through 137 00:13:44,069 --> 00:13:48,779 the dimensional analysis process with this choice of fundamental dimensions, we obtain 138 00:13:48,779 --> 00:13:51,980 an equation for v that looks like this. 139 00:13:51,980 --> 00:13:57,499 To see that these two formulas are equivalent, we set the arguments under the square root 140 00:13:57,500 --> 00:14:03,040 equal, and find that we can express phi as a function of psi. 141 00:14:15,840 --> 00:14:18,040 So any formula with the 142 00:14:18,040 --> 00:14:24,170 correct dimension can be expressed by this general formula. 143 00:14:24,170 --> 00:14:30,559 And this general formula works for any rover on any planet whose terminal velocity through 144 00:14:30,559 --> 00:14:35,899 the atmosphere depends on the same variables. Because it is a general law! 145 00:14:41,410 --> 00:14:47,660 Of course, we still don't know what this function phi is! In order to find phi, we can fit experimental 146 00:14:47,660 --> 00:14:55,600 data from any planet, for example, Earth. On Earth, we know the gravity and atmospheric 147 00:14:55,610 --> 00:15:02,129 density. We can specify the mass of a test rover to be 10kg. 148 00:15:02,129 --> 00:15:06,559 Then we might set up Earth bound experiments by varying the canopy diameter of a parachute 149 00:15:06,559 --> 00:15:13,719 between 1m to 20m and measuring the terminal velocity. For example, suppose we obtained 150 00:15:13,720 --> 00:15:18,300 the following data on Earth. 151 00:15:18,300 --> 00:15:24,220 Then we could convert this data to the dbar, vbar axes, by scaling the variables v and 152 00:15:24,220 --> 00:15:32,680 d according to the Earth values for the mass, gravity, and atmospheric density. We can fit 153 00:15:32,689 --> 00:15:37,899 this data to some best-fit curve. And this best fit curve is our best approximation to 154 00:15:37,899 --> 00:15:40,999 the function phi. 155 00:15:40,999 --> 00:15:47,170 Now that we have phi, we can transform the axes again to represent the canopy diameter 156 00:15:47,170 --> 00:15:55,860 and terminal velocity on Mars. This is done by converting vbar and dbar to v and d by 157 00:15:55,860 --> 00:16:01,759 scaling according to known values of the gravity, atmospheric density, and mass of the rover 158 00:16:01,759 --> 00:16:04,579 on Mars! 159 00:16:04,579 --> 00:16:10,209 To find the diameter, we find the point on the v-axis that corresponds to a terminal 160 00:16:10,209 --> 00:16:17,939 velocity of 90m/s, and use our curve to determine the diameter that corresponds to this terminal 161 00:16:17,939 --> 00:16:24,249 velocity! Now we have the specification we need to design the size of our parachute to 162 00:16:24,249 --> 00:16:27,569 be used on the descent to Mars! 163 00:16:31,490 --> 00:16:36,040 In this example, we used dimensional analysis to restrict the possible form of a function 164 00:16:36,040 --> 00:16:42,550 describing the terminal velocity of the Mars rover as a function of parachute canopy diameter, 165 00:16:42,550 --> 00:16:48,220 gravitational acceleration, atmospheric density, and the mass of the rover. 166 00:16:48,220 --> 00:16:54,600 This allowed us to design a parachute for use on Mars based on Earth bound experiments. 167 00:16:55,620 --> 00:17:00,360 In general, the process of dimensional analysis involves... 1. 168 00:17:00,360 --> 00:17:05,079 Identifying the dependent variable and independent variables, 169 00:17:05,079 --> 00:17:11,880 2. Expressing the relevant fundamental dimensions in terms of the variables found in step (1), 170 00:17:11,880 --> 00:17:17,400 3. Generating dimensionless expressions for all of the variables using expressions from step 171 00:17:17,400 --> 00:17:21,329 (2). 4. Producing a functional relationship between 172 00:17:21,329 --> 00:17:26,069 the dimensionless dependent variable in terms of the remaining independent dimensionless 173 00:17:26,069 --> 00:17:31,970 expressions. 5. Rearranging to determine a formula for the 174 00:17:31,970 --> 00:17:37,880 variable of interest. And 6. Performing experiments to determine the form 175 00:17:37,880 --> 00:17:42,880 of the general real valued function that appears in the formula. 176 00:17:42,880 --> 00:17:48,380 We've just shown you a powerful tool which can save you a lot of time. So the next time you 177 00:17:48,390 --> 00:17:54,190 encounter a difficult challenge, you might just want to try... Dimensional Analysis.