1 00:00:00,270 --> 00:00:02,070 - [Instructor] We talk about fluid pressure all the time, 2 00:00:02,070 --> 00:00:05,970 for example, blood pressure or pressure inside our tires, 3 00:00:05,970 --> 00:00:08,040 but what exactly are these numbers? 4 00:00:08,040 --> 00:00:09,390 What does 120 mean over here? 5 00:00:09,390 --> 00:00:11,910 Or what does 40 mean over here? 6 00:00:11,910 --> 00:00:15,900 So the big question over here is what exactly is pressure? 7 00:00:15,900 --> 00:00:17,700 That's what we're gonna find out in this video, 8 00:00:17,700 --> 00:00:18,690 so let's begin. 9 00:00:18,690 --> 00:00:19,920 To make sense of this, 10 00:00:19,920 --> 00:00:22,560 let's hang a perfectly cubicle wooden box 11 00:00:22,560 --> 00:00:24,300 and think about all the forces. 12 00:00:24,300 --> 00:00:26,430 We know that there's gravitational force acting downwards, 13 00:00:26,430 --> 00:00:29,160 which is perfectly balanced by the tension force, 14 00:00:29,160 --> 00:00:32,400 but there is another set of force over here. 15 00:00:32,400 --> 00:00:34,920 Remember that there are air molecules in our room, 16 00:00:34,920 --> 00:00:37,650 and so these air molecules are constantly bumping 17 00:00:37,650 --> 00:00:40,260 and exerting tiny forces. 18 00:00:40,260 --> 00:00:42,300 But how do we model these forces? 19 00:00:42,300 --> 00:00:44,880 There are billions of air molecules around 20 00:00:44,880 --> 00:00:47,490 and they're constantly bumping, so how do we model this? 21 00:00:47,490 --> 00:00:49,920 Well, we can model them to be continuous, 22 00:00:49,920 --> 00:00:52,320 that simplifies things, but more importantly, 23 00:00:52,320 --> 00:00:54,270 if you think about the surface area, 24 00:00:54,270 --> 00:00:55,650 since this is the perfect cube, 25 00:00:55,650 --> 00:00:57,780 and the surface areas are exactly the same, 26 00:00:57,780 --> 00:01:01,110 that means the number of molecules bumping per second 27 00:01:01,110 --> 00:01:04,230 on each surface is pretty much the same, 28 00:01:04,230 --> 00:01:06,090 and therefore, we could model these forces 29 00:01:06,090 --> 00:01:09,480 to be pretty much the same from all the directions, 30 00:01:09,480 --> 00:01:13,350 and that's why these forces together cancel out 31 00:01:13,350 --> 00:01:16,470 and they do not accelerate the box or anything like that. 32 00:01:16,470 --> 00:01:20,520 But these forces, look, are pressing on the box 33 00:01:20,520 --> 00:01:22,230 from all the direction, 34 00:01:22,230 --> 00:01:26,220 so it's these forces that are related to pressure. 35 00:01:26,220 --> 00:01:28,140 But how exactly? 36 00:01:28,140 --> 00:01:30,690 Well, to answer that question, 37 00:01:30,690 --> 00:01:32,790 let's think of a bigger box. 38 00:01:32,790 --> 00:01:35,070 Let's imagine that the surface area over here 39 00:01:35,070 --> 00:01:39,720 was twice as much as the surface area over here. 40 00:01:39,720 --> 00:01:43,050 Then what would be the amount of force 41 00:01:43,050 --> 00:01:44,910 that the air molecules would be putting here 42 00:01:44,910 --> 00:01:46,470 compared to over here? 43 00:01:46,470 --> 00:01:49,110 Pause the video and think about this. 44 00:01:49,110 --> 00:01:52,080 Okay, since we have twice the area, 45 00:01:52,080 --> 00:01:53,040 if you take any surface, 46 00:01:53,040 --> 00:01:54,900 if you consider, for example, the top surface, 47 00:01:54,900 --> 00:01:57,540 the number of molecules bumping per second 48 00:01:57,540 --> 00:01:59,700 would be twice as much compared to over here, 49 00:01:59,700 --> 00:02:01,110 because you have twice the area, 50 00:02:01,110 --> 00:02:03,120 and there are air molecules everywhere. 51 00:02:03,120 --> 00:02:06,300 So can you see just from that logic, 52 00:02:06,300 --> 00:02:07,680 the amount of force over here 53 00:02:07,680 --> 00:02:09,960 must be twice as much as over here. 54 00:02:09,960 --> 00:02:12,210 If the surface area was three times as much, 55 00:02:12,210 --> 00:02:14,850 the amount of force must be thrice as much. 56 00:02:14,850 --> 00:02:16,200 In other words, you can see 57 00:02:16,200 --> 00:02:19,200 the force that the air molecules are exerting on the box 58 00:02:19,200 --> 00:02:22,050 is proportional to the surface area. 59 00:02:22,050 --> 00:02:27,050 Or in other words, the force per area is a constant. 60 00:02:27,150 --> 00:02:31,320 That is the key characteristics of the force exerted by air 61 00:02:31,320 --> 00:02:33,153 or, in general, any fluid. 62 00:02:34,020 --> 00:02:38,100 And that ratio, force per area, 63 00:02:38,100 --> 00:02:42,330 is what we call, in general, stress, okay? 64 00:02:42,330 --> 00:02:44,010 But what exactly is pressure? 65 00:02:44,010 --> 00:02:47,070 Well, in our example, notice that 66 00:02:47,070 --> 00:02:48,810 the forces are not in some random direction. 67 00:02:48,810 --> 00:02:52,530 All the forces are perpendicular to the surface area, 68 00:02:52,530 --> 00:02:54,690 but how can we be so sure, you ask? 69 00:02:54,690 --> 00:02:57,570 I mean, these are random molecules bumping into it, right? 70 00:02:57,570 --> 00:03:00,120 Well, if you zoom in, what do you notice? 71 00:03:00,120 --> 00:03:01,860 You notice that when the molecules bump 72 00:03:01,860 --> 00:03:03,570 and they sort of collide, 73 00:03:03,570 --> 00:03:07,590 and they reflect off of the walls of our cubicle box, 74 00:03:07,590 --> 00:03:09,480 what you notice is the acceleration 75 00:03:09,480 --> 00:03:11,550 is always perpendicular to the box, 76 00:03:11,550 --> 00:03:14,250 and therefore, the force that the box is exerting on them 77 00:03:14,250 --> 00:03:16,650 is also perpendicular to the box. 78 00:03:16,650 --> 00:03:18,240 And therefore, from Newton's Third Law, 79 00:03:18,240 --> 00:03:21,450 the force that the molecules are exerting on the box 80 00:03:21,450 --> 00:03:24,960 will also be equal and opposite perpendicular to the box. 81 00:03:24,960 --> 00:03:27,810 And that's why the forces over here 82 00:03:27,810 --> 00:03:29,370 are always perpendicular. 83 00:03:29,370 --> 00:03:33,150 And so look, there's a special kind of stress over here. 84 00:03:33,150 --> 00:03:34,560 It's not just any force, 85 00:03:34,560 --> 00:03:36,270 the force there are always perpendicular, 86 00:03:36,270 --> 00:03:39,030 and that particular special kind of stress 87 00:03:39,030 --> 00:03:42,330 where the forces are perpendicular, that is called pressure. 88 00:03:42,330 --> 00:03:45,180 So you can think of pressure as a special kind of stress 89 00:03:45,180 --> 00:03:48,090 where the forces are perpendicular to the area. 90 00:03:48,090 --> 00:03:50,040 But wait, this raises another question. 91 00:03:50,040 --> 00:03:52,920 Is it possible for fluids to exert forces 92 00:03:52,920 --> 00:03:55,500 which are paddle to the area as well? 93 00:03:55,500 --> 00:03:56,910 Yes. 94 00:03:56,910 --> 00:03:59,820 Consider the air molecules again, they're all moving, 95 00:03:59,820 --> 00:04:03,210 but they're all moving in random direction, isn't it? 96 00:04:03,210 --> 00:04:07,050 But now, imagine that there was some kind of wind 97 00:04:07,050 --> 00:04:09,150 that could happen if the block itself is moving down, 98 00:04:09,150 --> 00:04:11,580 or there is wind blowing upwards, whatever that is, 99 00:04:11,580 --> 00:04:14,070 there's some kind of a relative motion. 100 00:04:14,070 --> 00:04:15,510 Now, if there is a wind, 101 00:04:15,510 --> 00:04:17,850 let's say the air molecules are moving upwards 102 00:04:17,850 --> 00:04:18,960 along with the random motion 103 00:04:18,960 --> 00:04:20,850 that they're doing along with that, 104 00:04:20,850 --> 00:04:23,880 then because these molecules also interact 105 00:04:23,880 --> 00:04:25,290 with the molecules of the box, 106 00:04:25,290 --> 00:04:28,560 they will also exert force on them, 107 00:04:28,560 --> 00:04:31,200 pulling them upwards. 108 00:04:31,200 --> 00:04:33,690 This force is called the viscous force, 109 00:04:33,690 --> 00:04:35,460 and at the heart of it comes from the fact 110 00:04:35,460 --> 00:04:37,500 that molecules can interact with each other 111 00:04:37,500 --> 00:04:39,270 and as a result, look, 112 00:04:39,270 --> 00:04:42,840 this total viscous force is parallel 113 00:04:42,840 --> 00:04:45,420 to the surface area, 114 00:04:45,420 --> 00:04:49,230 and this force tends to shear our cube. 115 00:04:49,230 --> 00:04:50,790 It's kind of like this deck of cards. 116 00:04:50,790 --> 00:04:52,830 If you press it perpendicular to the surface, 117 00:04:52,830 --> 00:04:55,050 then the stress is just the pressure. 118 00:04:55,050 --> 00:04:57,810 But if you press at an angle, then there's a pedal component 119 00:04:57,810 --> 00:04:59,820 which shears the deck of cards. 120 00:04:59,820 --> 00:05:02,250 And shearing stresses can be quite complicated, 121 00:05:02,250 --> 00:05:03,240 but we don't have to worry about it 122 00:05:03,240 --> 00:05:06,930 because in our model, we are considering ideal fluids 123 00:05:06,930 --> 00:05:09,750 and ideal fluids do not have any viscosity. 124 00:05:09,750 --> 00:05:11,400 And we're also going to assume that we are dealing 125 00:05:11,400 --> 00:05:14,370 with static fluids, which means no viscus forces, 126 00:05:14,370 --> 00:05:16,530 no relative motion, and therefore, 127 00:05:16,530 --> 00:05:19,110 we can completely ignore shearing stress. 128 00:05:19,110 --> 00:05:23,070 And therefore, in our model, we only have pressure. 129 00:05:23,070 --> 00:05:25,320 Forces will always be only perpendicular 130 00:05:25,320 --> 00:05:26,670 to the surface area. 131 00:05:26,670 --> 00:05:28,410 All right, so let's try to understand pressure 132 00:05:28,410 --> 00:05:29,370 a little bit better. 133 00:05:29,370 --> 00:05:30,630 What about its units? 134 00:05:30,630 --> 00:05:32,820 Well, because it's force per area, 135 00:05:32,820 --> 00:05:33,930 the unit of force is Newtons 136 00:05:33,930 --> 00:05:36,120 and the unit of area is meters squared, 137 00:05:36,120 --> 00:05:37,170 so the unit of pressure, 138 00:05:37,170 --> 00:05:38,220 at least a standard unit of pressure, 139 00:05:38,220 --> 00:05:39,870 becomes Newtons per meters squared, 140 00:05:39,870 --> 00:05:42,120 which we also call pascals. 141 00:05:42,120 --> 00:05:43,680 So over an area of one meter squared, 142 00:05:43,680 --> 00:05:46,470 if there's one Newton of force acting perpendicular to it, 143 00:05:46,470 --> 00:05:49,440 then we would say that the pressure is one pascal. 144 00:05:49,440 --> 00:05:50,910 And that is a very tiny pressure. 145 00:05:50,910 --> 00:05:52,650 One Newton exerted over one meter squared 146 00:05:52,650 --> 00:05:54,270 is incredibly tiny. 147 00:05:54,270 --> 00:05:56,280 So the big question is, what is the pressure 148 00:05:56,280 --> 00:05:57,840 over here in a room? 149 00:05:57,840 --> 00:05:59,190 What is the atmospheric pressure? 150 00:05:59,190 --> 00:06:01,650 Pressure that the air molecules are pushing 151 00:06:01,650 --> 00:06:03,690 on the sides of this cube with? 152 00:06:03,690 --> 00:06:05,490 Well, turns out that pressure 153 00:06:05,490 --> 00:06:08,460 is about 10 to the power of five pascals. 154 00:06:08,460 --> 00:06:12,120 In other words, that is a 100,000 pascals. 155 00:06:12,120 --> 00:06:14,760 That is insanely huge. 156 00:06:14,760 --> 00:06:18,450 A 100,000 Newtons of force is exerted 157 00:06:18,450 --> 00:06:21,390 by the air molecules per square meter. 158 00:06:21,390 --> 00:06:24,300 That is insanely high. I wouldn't have expected. 159 00:06:24,300 --> 00:06:27,300 That is the amount of pressure we are all feeling 160 00:06:27,300 --> 00:06:28,680 just sitting in our room 161 00:06:28,680 --> 00:06:30,420 due to the air molecules over there. 162 00:06:30,420 --> 00:06:32,220 But that was an important question. 163 00:06:32,220 --> 00:06:35,670 Why don't things get crushed under that pressure? 164 00:06:35,670 --> 00:06:37,650 I mean, sure, this cube is not getting crushed 165 00:06:37,650 --> 00:06:40,260 because the internal forces are able to balance that out. 166 00:06:40,260 --> 00:06:42,720 But what if I take a plastic bag 167 00:06:42,720 --> 00:06:44,760 and there's nothing inside it, 168 00:06:44,760 --> 00:06:47,250 then because there's so much air pressure 169 00:06:47,250 --> 00:06:48,630 over from the outside, 170 00:06:48,630 --> 00:06:50,700 shouldn't the plastic bag just get crushed 171 00:06:50,700 --> 00:06:52,230 due to the air pressure? 172 00:06:52,230 --> 00:06:53,510 Well, the reason it doesn't get crushed 173 00:06:53,510 --> 00:06:55,470 is because there's air inside as well. 174 00:06:55,470 --> 00:06:58,530 And that air also has the exact same pressure, 175 00:06:58,530 --> 00:07:01,050 which means it is able to balance out the pressure 176 00:07:01,050 --> 00:07:01,950 from the outside. 177 00:07:01,950 --> 00:07:04,560 But what if you could somehow suck that air out? 178 00:07:04,560 --> 00:07:06,090 Ooh, then the balance will be lost 179 00:07:06,090 --> 00:07:07,170 because the pressure drops 180 00:07:07,170 --> 00:07:11,850 and then we would see the plastic bag collapsing, 181 00:07:11,850 --> 00:07:14,730 getting crushed under the pressure from the outside. 182 00:07:14,730 --> 00:07:16,320 That all makes sense, right? 183 00:07:16,320 --> 00:07:18,060 Okay, before moving forward, let's also quickly talk 184 00:07:18,060 --> 00:07:19,500 about a couple of other units of pressure 185 00:07:19,500 --> 00:07:21,270 that we usually use in our daily life. 186 00:07:21,270 --> 00:07:23,670 For example, when it comes to tire pressure, 187 00:07:23,670 --> 00:07:28,230 we usually talk in terms of pounds per square inch. 188 00:07:28,230 --> 00:07:31,020 And just to give some feeling for numbers, 189 00:07:31,020 --> 00:07:33,060 10 to the power of five pascals happens to be close 190 00:07:33,060 --> 00:07:36,360 to 14.7 pounds per square inch. 191 00:07:36,360 --> 00:07:38,130 So that is the atmospheric pressure 192 00:07:38,130 --> 00:07:39,540 in pounds per square inch. 193 00:07:39,540 --> 00:07:41,913 Another unit is millimeters of mercury, 194 00:07:41,913 --> 00:07:44,520 and 10 to the power of five pascals happens to be close 195 00:07:44,520 --> 00:07:46,560 to 760 millimeters of mercury. 196 00:07:46,560 --> 00:07:49,680 In other words, the atmospheric pressure can pull up 197 00:07:49,680 --> 00:07:54,120 the mercury up to 760 millimeters in a column. 198 00:07:54,120 --> 00:07:56,400 Anyways, let's focus on Pascals for now. 199 00:07:56,400 --> 00:07:59,370 And the big question now, is pressure a scalar 200 00:07:59,370 --> 00:08:01,890 or a vector quantity, what do you think? 201 00:08:01,890 --> 00:08:03,540 Well, my intuition says it's vector 202 00:08:03,540 --> 00:08:06,030 because there's force in one over here, 203 00:08:06,030 --> 00:08:07,260 but let's think a little bit more about it. 204 00:08:07,260 --> 00:08:11,970 In fact, think about pressure at a specific point. 205 00:08:11,970 --> 00:08:13,290 How do we do that? 206 00:08:13,290 --> 00:08:16,230 Well, one way to do that is you take this box 207 00:08:16,230 --> 00:08:17,910 and shrink it down. 208 00:08:17,910 --> 00:08:20,640 Let's say we shrink the size of the box. 209 00:08:20,640 --> 00:08:23,550 Well, now, the area has become, let's say half, 210 00:08:23,550 --> 00:08:27,360 but the number of air molecules will also become half, 211 00:08:27,360 --> 00:08:29,850 and therefore, the force also becomes half, 212 00:08:29,850 --> 00:08:31,980 making sure force per area stays the same. 213 00:08:31,980 --> 00:08:33,930 So the pressure stays the same. 214 00:08:33,930 --> 00:08:35,550 We'll keep shrinking it, keep shrinking, 215 00:08:35,550 --> 00:08:36,840 and keep shrinking it. 216 00:08:36,840 --> 00:08:41,190 Now shrink it all the way to an extremely tiny point. 217 00:08:41,190 --> 00:08:43,350 We call that as an infinitesimal. 218 00:08:43,350 --> 00:08:45,090 Now, the areas are incredibly tiny, 219 00:08:45,090 --> 00:08:47,040 the forces are incredibly tiny, 220 00:08:47,040 --> 00:08:50,040 yet the force per area stays the same. 221 00:08:50,040 --> 00:08:52,800 That is the pressure at a particular point. 222 00:08:52,800 --> 00:08:55,770 And now the question is, should we assign a direction 223 00:08:55,770 --> 00:08:58,440 to this number, to this pressure? 224 00:08:58,440 --> 00:09:01,590 Well, not really, because all I need is a number, 225 00:09:01,590 --> 00:09:04,020 because what this number is saying is that if you zoom in 226 00:09:04,020 --> 00:09:06,090 and if you have any area, 227 00:09:06,090 --> 00:09:09,360 then there will always be forces perpendicular to the area 228 00:09:09,360 --> 00:09:12,690 that is exerted by the fluid, and that force per area 229 00:09:12,690 --> 00:09:14,580 will be 10 to the power of five pascals. 230 00:09:14,580 --> 00:09:16,500 And that is valid from any direction. 231 00:09:16,500 --> 00:09:18,540 It doesn't matter how your area is oriented, 232 00:09:18,540 --> 00:09:21,300 that will always be the case. 233 00:09:21,300 --> 00:09:24,420 Which means, look, all I need is a number. 234 00:09:24,420 --> 00:09:26,670 I don't need a direction 235 00:09:26,670 --> 00:09:28,560 to communicate the idea of pressure, 236 00:09:28,560 --> 00:09:32,370 and therefore, pressure is a scalar quantity. 237 00:09:32,370 --> 00:09:33,840 And because the number of air molecules 238 00:09:33,840 --> 00:09:36,300 bumping per square meter is pretty much the same everywhere 239 00:09:36,300 --> 00:09:39,030 and their speeds are pretty much the same anywhere you take, 240 00:09:39,030 --> 00:09:43,440 therefore, the pressure now is the same everywhere. 241 00:09:43,440 --> 00:09:46,920 But if we zoom out and look at the entire atmosphere, 242 00:09:46,920 --> 00:09:49,140 for example, that's not the case. 243 00:09:49,140 --> 00:09:50,910 In fact, that 10 to the power of five we said 244 00:09:50,910 --> 00:09:53,160 is the pressure close to the sea level, 245 00:09:53,160 --> 00:09:54,630 but if you were to go a little bit above it, 246 00:09:54,630 --> 00:09:56,864 say at 10 kilometers, which is the cruising altitude 247 00:09:56,864 --> 00:09:59,670 of commercial airplanes, you would now find 248 00:09:59,670 --> 00:10:01,560 that the pressure is about one fourth 249 00:10:01,560 --> 00:10:03,030 is what we'd find over here. 250 00:10:03,030 --> 00:10:04,620 Why is the pressure different over here 251 00:10:04,620 --> 00:10:05,910 compared to over here? 252 00:10:05,910 --> 00:10:07,440 Well, that's because the molecules over here 253 00:10:07,440 --> 00:10:09,870 are carrying the weight of the entire atmosphere 254 00:10:09,870 --> 00:10:10,703 on top of it. 255 00:10:10,703 --> 00:10:12,780 That entire weight is pushing down, 256 00:10:12,780 --> 00:10:14,700 pressing the molecules over here. 257 00:10:14,700 --> 00:10:17,340 However, if you consider the molecules at this level, 258 00:10:17,340 --> 00:10:18,780 they're not carrying the entire weight, 259 00:10:18,780 --> 00:10:20,550 they're carrying the weight only on top of them. 260 00:10:20,550 --> 00:10:22,260 They're not carrying the weight of this amount 261 00:10:22,260 --> 00:10:23,093 of air molecules. 262 00:10:23,093 --> 00:10:25,290 And that's why the pressure over here is slightly lower, 263 00:10:25,290 --> 00:10:27,120 which means the pressure depends 264 00:10:27,120 --> 00:10:29,070 on the height if you zoom out. 265 00:10:29,070 --> 00:10:31,500 And so now, the next obvious question would be, 266 00:10:31,500 --> 00:10:33,750 is there a relationship between the pressure and the height? 267 00:10:33,750 --> 00:10:35,640 Well, there is, and it's harder to think about that 268 00:10:35,640 --> 00:10:37,770 for air molecules because air molecules 269 00:10:37,770 --> 00:10:40,260 are very compressible, so it's a little hard over there. 270 00:10:40,260 --> 00:10:42,660 But let's consider non-compressible fluids, 271 00:10:42,660 --> 00:10:45,180 like water, for example. 272 00:10:45,180 --> 00:10:46,770 So here's a specific question. 273 00:10:46,770 --> 00:10:49,200 If we know the pressure at some level over here, 274 00:10:49,200 --> 00:10:53,790 what is the pressure at some depth, say h, below? 275 00:10:53,790 --> 00:10:55,560 So let's say the pressure at the top is PT 276 00:10:55,560 --> 00:10:56,860 and the pressure at the bottom is PB. 277 00:10:56,860 --> 00:10:58,920 Well, we know that the pressure at the bottom is higher 278 00:10:58,920 --> 00:10:59,760 than the pressure at the top. 279 00:10:59,760 --> 00:11:00,840 So we could say pressure at the bottom 280 00:11:00,840 --> 00:11:04,530 equals pressure at the top, plus some additional pressure 281 00:11:04,530 --> 00:11:06,240 due to this weight of the water. 282 00:11:06,240 --> 00:11:07,830 But how do we figure that out? 283 00:11:07,830 --> 00:11:10,830 Well, for that, let's just draw a cuboidal surface, 284 00:11:10,830 --> 00:11:12,600 having the surface area A. 285 00:11:12,600 --> 00:11:14,910 Now, the additional pressure that we are getting over here 286 00:11:14,910 --> 00:11:17,190 is due to the weight of this cuboidal column. 287 00:11:17,190 --> 00:11:20,700 To be precise, it's going to be the weight per area. 288 00:11:20,700 --> 00:11:23,550 So this term over here is gonna be weight per area. 289 00:11:23,550 --> 00:11:26,700 But what exactly is the weight of this cubital column? 290 00:11:26,700 --> 00:11:28,110 Well, weight is just the force of gravity, 291 00:11:28,110 --> 00:11:30,900 so it's gonna be MG, where M is the mass of the water 292 00:11:30,900 --> 00:11:34,290 in this column divided by the area, which is A, 293 00:11:34,290 --> 00:11:36,060 but how do we figure out what is the mass 294 00:11:36,060 --> 00:11:38,430 of the column of this water? 295 00:11:38,430 --> 00:11:41,880 Well, we know that density is mass per volume. 296 00:11:41,880 --> 00:11:44,670 So mass can written as density times volume, 297 00:11:44,670 --> 00:11:47,220 and that's because we know the density of a fluid. 298 00:11:47,220 --> 00:11:51,720 So we can write this as density of the fluid, 299 00:11:51,720 --> 00:11:53,940 density of water, times the volume of the column 300 00:11:53,940 --> 00:11:55,710 times GD divided by A. 301 00:11:55,710 --> 00:11:58,200 But the final question is what is the volume of this column? 302 00:11:58,200 --> 00:11:59,970 Hey, we know the volume of the column. 303 00:11:59,970 --> 00:12:01,200 Volume of this cuboidal column 304 00:12:01,200 --> 00:12:03,540 is just going to be area times the height. 305 00:12:03,540 --> 00:12:05,970 And so we can plug that in over here, 306 00:12:05,970 --> 00:12:07,920 and if we cancel out the areas, 307 00:12:07,920 --> 00:12:10,230 we finally get our expression. 308 00:12:10,230 --> 00:12:11,640 The pressure at the bottom will equal 309 00:12:11,640 --> 00:12:15,750 the pressure at the top plus this additional pressure 310 00:12:15,750 --> 00:12:18,030 due to the weight of this column. 311 00:12:18,030 --> 00:12:20,100 And this equation will work anywhere 312 00:12:20,100 --> 00:12:22,590 as long as you're dealing with a non-compressible fluid 313 00:12:22,590 --> 00:12:25,170 because we are assuming the density to be the same. 314 00:12:25,170 --> 00:12:27,090 If you consider a compressible fluid, like air, 315 00:12:27,090 --> 00:12:30,120 the density varies and this calculation becomes harder, 316 00:12:30,120 --> 00:12:34,350 and so you'll get a considerably different expression. 317 00:12:34,350 --> 00:12:36,630 But what I find really surprising about this 318 00:12:36,630 --> 00:12:38,820 is that for a given non-compressible fluid, 319 00:12:38,820 --> 00:12:40,890 which means it has a specific density, 320 00:12:40,890 --> 00:12:42,900 the pressure difference between two points 321 00:12:42,900 --> 00:12:46,440 only depends on their height, nothing else. 322 00:12:46,440 --> 00:12:48,030 In other words, this means the pressure difference 323 00:12:48,030 --> 00:12:50,280 between two points, say 10 centimeter apart, 324 00:12:50,280 --> 00:12:52,770 whether you consider that in an ocean 325 00:12:52,770 --> 00:12:56,220 or a tiny test tube, it's the same. 326 00:12:56,220 --> 00:12:59,220 It doesn't matter how much water you're dealing with, 327 00:12:59,220 --> 00:13:02,400 it's just the height that matters. 328 00:13:02,400 --> 00:13:04,740 Anyways, now we can introduce two kinds of pressure. 329 00:13:04,740 --> 00:13:07,050 The pressure that we have over here, these two, 330 00:13:07,050 --> 00:13:09,180 they're called absolute pressures. 331 00:13:09,180 --> 00:13:11,460 For example, if this was the atmospheric pressure, 332 00:13:11,460 --> 00:13:13,590 then that is the absolute pressure. 333 00:13:13,590 --> 00:13:15,870 The absolute pressure of the atmosphere 334 00:13:15,870 --> 00:13:16,770 close to the sea level 335 00:13:16,770 --> 00:13:20,280 is about 10 to the power of five pascals. 336 00:13:20,280 --> 00:13:21,810 But now, look at this term. 337 00:13:21,810 --> 00:13:23,400 What does that term represent? 338 00:13:23,400 --> 00:13:26,130 That represents the extra pressure 339 00:13:26,130 --> 00:13:27,900 that you have at this point 340 00:13:27,900 --> 00:13:29,910 over and above the atmospheric pressure. 341 00:13:29,910 --> 00:13:34,110 That extra pressure is what we call the gauge pressure. 342 00:13:34,110 --> 00:13:36,150 And most of the time when we're talking about pressure 343 00:13:36,150 --> 00:13:37,290 in our day-today life, 344 00:13:37,290 --> 00:13:39,360 we are not talking about the absolute pressure, 345 00:13:39,360 --> 00:13:41,070 we're talking about the gauge pressure. 346 00:13:41,070 --> 00:13:43,530 So for example, when we talk about the blood pressure, 347 00:13:43,530 --> 00:13:46,020 we say it's 120 millimeters of mercury. 348 00:13:46,020 --> 00:13:47,490 What does that even mean? 349 00:13:47,490 --> 00:13:49,770 Well, remember that the atmospheric pressure 350 00:13:49,770 --> 00:13:52,800 is 760 millimeters of mercury. 351 00:13:52,800 --> 00:13:55,530 This is the pressure over and above that. 352 00:13:55,530 --> 00:13:59,190 So the pressure in the arteries, or veins, 353 00:13:59,190 --> 00:14:00,750 during a cysto for example, 354 00:14:00,750 --> 00:14:05,430 is 760 millimeters plus 120 millimeters of mercury. 355 00:14:05,430 --> 00:14:06,480 That's what it really means. 356 00:14:06,480 --> 00:14:08,400 So this is the additional pressure 357 00:14:08,400 --> 00:14:10,290 above the atmospheric pressure, 358 00:14:10,290 --> 00:14:12,090 so this is the gauge pressure. 359 00:14:12,090 --> 00:14:14,070 The same as the case with our tires. 360 00:14:14,070 --> 00:14:16,140 For example, if you look at the pressure inside the tire, 361 00:14:16,140 --> 00:14:19,440 you can see it's about 40 PSI, 362 00:14:19,440 --> 00:14:21,420 but that is a gauge pressure, 363 00:14:21,420 --> 00:14:23,460 meaning it's over and above the atmospheric pressure. 364 00:14:23,460 --> 00:14:26,340 Remember, the atmospheric pressure is 14.7 PSI. 365 00:14:26,340 --> 00:14:30,150 So the pressure in the tire is 40 PSI 366 00:14:30,150 --> 00:14:32,400 above the atmospheric pressure. 367 00:14:32,400 --> 00:14:35,520 So most of the time, we're dealing with gauge pressures. 368 00:14:35,520 --> 00:14:36,840 Okay, finally. coming back over here, 369 00:14:36,840 --> 00:14:39,360 suppose we were to draw a graph of the gauge pressure 370 00:14:39,360 --> 00:14:40,770 versus the depth. 371 00:14:40,770 --> 00:14:42,630 Okay, what do you think the graph would look like for, 372 00:14:42,630 --> 00:14:44,820 say, a lake and for the ocean? 373 00:14:44,820 --> 00:14:46,440 Why don't you pause it and have a think about this? 374 00:14:46,440 --> 00:14:47,790 Okay, let's consider the lake. 375 00:14:47,790 --> 00:14:49,860 Right at the surface, the gauge pressure is zero 376 00:14:49,860 --> 00:14:51,630 because the pressure over there is the same 377 00:14:51,630 --> 00:14:53,850 as the atmospheric pressure, so we start from zero, 378 00:14:53,850 --> 00:14:56,460 and then you can see that the gauge pressure is proportional 379 00:14:56,460 --> 00:14:58,020 to the height, it's proportional to it, 380 00:14:58,020 --> 00:14:59,280 so we get a straight line. 381 00:14:59,280 --> 00:15:02,880 And so we'd expect the pressure to increase linearly. 382 00:15:02,880 --> 00:15:05,580 That's what we would get for the lake. What about the ocean? 383 00:15:05,580 --> 00:15:08,010 Well, ocean is also water, so it has the same density, 384 00:15:08,010 --> 00:15:09,060 or does it? 385 00:15:09,060 --> 00:15:10,890 Remember, ocean has salt water, 386 00:15:10,890 --> 00:15:12,750 so the density is slightly higher. 387 00:15:12,750 --> 00:15:14,610 So for the ocean, we expect the line to be steeper, 388 00:15:14,610 --> 00:15:16,650 having slightly higher slope. 389 00:15:16,650 --> 00:15:18,330 Finally, before wrapping up the video, 390 00:15:18,330 --> 00:15:20,460 if you were to submerge a cube inside water, 391 00:15:20,460 --> 00:15:22,170 earlier, we said that the pressure is gonna be the same 392 00:15:22,170 --> 00:15:23,490 from all directions, 393 00:15:23,490 --> 00:15:25,080 but now we know that the pressure on the bottom 394 00:15:25,080 --> 00:15:27,840 is slightly higher than the pressure on the top, 395 00:15:27,840 --> 00:15:29,640 which means the forces on the bottom 396 00:15:29,640 --> 00:15:31,530 would be slightly higher than the force on the top 397 00:15:31,530 --> 00:15:33,030 because the area is the same. 398 00:15:33,030 --> 00:15:36,900 So wouldn't that produce a net upward force? 399 00:15:36,900 --> 00:15:40,290 Yes, it would, and that's called the buoyant force, 400 00:15:40,290 --> 00:15:43,080 which is responsible for making certain things float. 401 00:15:43,080 --> 00:15:45,880 And that's something we'll talk about in a future video.