1 00:00:00,000 --> 00:00:03,360 - [Instructor] I wanna talk to you about LOL diagrams. 2 00:00:03,360 --> 00:00:06,448 That's right, I said LOL diagrams. 3 00:00:06,448 --> 00:00:09,684 These are a great ways to visualize conservation of energy, 4 00:00:09,684 --> 00:00:12,339 and even better, they force you to think about 5 00:00:12,339 --> 00:00:14,308 what's part of your energy system 6 00:00:14,308 --> 00:00:17,060 and what isn't part of your energy system. 7 00:00:17,060 --> 00:00:19,407 And if you don't know what an energy system is, 8 00:00:19,407 --> 00:00:21,310 maybe that's where we should start. 9 00:00:21,310 --> 00:00:25,422 An energy system is an object or a collection of objects 10 00:00:25,422 --> 00:00:27,693 whose energies were gonna keep track of, 11 00:00:27,693 --> 00:00:30,094 and we're gonna keep track of them in these two charts. 12 00:00:30,094 --> 00:00:32,531 And notice, it looks like an L, 13 00:00:32,531 --> 00:00:34,125 and then an O, and then an L. 14 00:00:34,125 --> 00:00:35,579 That's where they get their name from. 15 00:00:35,579 --> 00:00:36,628 This isn't laugh out loud. 16 00:00:36,628 --> 00:00:38,083 This is an energy chart, 17 00:00:38,083 --> 00:00:40,423 a circle where we're gonna define our system, 18 00:00:40,423 --> 00:00:41,465 and then another chart. 19 00:00:41,465 --> 00:00:43,430 So to understand what these LOL diagrams mean, 20 00:00:43,430 --> 00:00:45,524 let's just look at an actually example. 21 00:00:45,524 --> 00:00:46,923 Let's say you took a mass M, 22 00:00:46,923 --> 00:00:49,423 and you released it from rest from a height H. 23 00:00:49,423 --> 00:00:51,071 And this mass falls down. 24 00:00:51,071 --> 00:00:52,132 The first thing we should do 25 00:00:52,132 --> 00:00:54,663 is choose what is gonna be part of our energy system, 26 00:00:54,663 --> 00:00:56,246 whose energies are we gonna keep track of, 27 00:00:56,246 --> 00:00:58,633 and whose are we not gonna keep track of. 28 00:00:58,633 --> 00:01:00,066 The way people typically do this problem 29 00:01:00,066 --> 00:01:01,266 when they're doing conservation of energy, 30 00:01:01,266 --> 00:01:03,857 they would say it starts with potential energy, 31 00:01:03,857 --> 00:01:05,306 and then it turns into kinetic energy. 32 00:01:05,306 --> 00:01:06,957 And if you're doing that, what you're really doing 33 00:01:06,957 --> 00:01:08,947 is you're saying, "I'm taking my mass. 34 00:01:08,947 --> 00:01:10,088 "That's gonna be part of my system." 35 00:01:10,088 --> 00:01:11,722 So I'm gonna take this mass here, 36 00:01:11,722 --> 00:01:13,305 and I'm gonna make that part of my system. 37 00:01:13,305 --> 00:01:16,103 So we'll keep track of the energy of this mass, 38 00:01:16,103 --> 00:01:16,966 and then if you're talking about 39 00:01:16,966 --> 00:01:18,587 gravitational potential energy, 40 00:01:18,587 --> 00:01:19,764 here's the weird thing, 41 00:01:19,764 --> 00:01:21,285 you're also talking about the earth. 42 00:01:21,285 --> 00:01:23,075 That earth is part of your system. 43 00:01:23,075 --> 00:01:24,636 So this is the earth right here. 44 00:01:24,636 --> 00:01:25,686 It's got continents, 45 00:01:25,686 --> 00:01:28,914 and it's got California, and Mexico, and South America, 46 00:01:28,914 --> 00:01:30,061 and there's Florida, 47 00:01:30,061 --> 00:01:31,645 and then there's all kinds of stuff here. 48 00:01:31,645 --> 00:01:33,583 So now that we've selected our system, 49 00:01:33,583 --> 00:01:34,895 we can go ahead and starting charting 50 00:01:34,895 --> 00:01:36,446 what the energies look like. 51 00:01:36,446 --> 00:01:38,221 So what kind of energy was there initially 52 00:01:38,221 --> 00:01:39,464 in this system? 53 00:01:39,464 --> 00:01:41,137 There was gravitational potential energy, 54 00:01:41,137 --> 00:01:44,723 because this mass M started at height H above the earth. 55 00:01:44,723 --> 00:01:46,698 So I'm gonna represent that with this bar. 56 00:01:46,698 --> 00:01:48,047 This is gonna be a bar chart. 57 00:01:48,047 --> 00:01:49,685 Let me just go up four units. 58 00:01:49,685 --> 00:01:51,224 So I'm gonna say there were four units 59 00:01:51,224 --> 00:01:53,230 of gravitational potential energy. 60 00:01:53,230 --> 00:01:55,352 You might be like, "How do you know there were four? 61 00:01:55,352 --> 00:01:57,281 "How do you know there weren't like three, or five, 62 00:01:57,281 --> 00:01:58,448 "or 4.2?" 63 00:01:58,448 --> 00:02:00,134 Well, it doesn't really matter too much. 64 00:02:00,134 --> 00:02:02,473 The real value in this LOL diagram 65 00:02:02,473 --> 00:02:05,343 is being able to conceptualize what's happening 66 00:02:05,343 --> 00:02:06,470 to the energies. 67 00:02:06,470 --> 00:02:07,969 No matter what you draw over here, 68 00:02:07,969 --> 00:02:09,270 you just wanna be consistent 69 00:02:09,270 --> 00:02:11,280 when you then draw this diagram over here, 70 00:02:11,280 --> 00:02:12,660 and I'll show you what that means in a minute. 71 00:02:12,660 --> 00:02:14,210 So for now let's just say there were four units 72 00:02:14,210 --> 00:02:16,854 of gravitational potential energy to start with. 73 00:02:16,854 --> 00:02:18,486 Was there any kinetic energy? 74 00:02:18,486 --> 00:02:20,579 No, because it was released from rest. 75 00:02:20,579 --> 00:02:21,979 So if we drop this mass, 76 00:02:21,979 --> 00:02:23,786 it starts with no kinetic energy. 77 00:02:23,786 --> 00:02:26,221 And this US here is gonna be any spring energy. 78 00:02:26,221 --> 00:02:28,146 We won't have that for this example, 79 00:02:28,146 --> 00:02:29,778 because there aren't any springs, 80 00:02:29,778 --> 00:02:32,216 so there won't be any spring or elastic energy. 81 00:02:32,216 --> 00:02:33,071 That's it, we just had 82 00:02:33,071 --> 00:02:35,191 gravitational potential energy to start. 83 00:02:35,191 --> 00:02:36,833 And then what did that energy turn into? 84 00:02:36,833 --> 00:02:39,038 If this was the initial position, 85 00:02:39,038 --> 00:02:40,461 what kind of energy do we have 86 00:02:40,461 --> 00:02:43,234 when the mass gets right above the surface of the earth? 87 00:02:43,234 --> 00:02:44,842 Right before it's gonna hit the earth, 88 00:02:44,842 --> 00:02:46,451 what kind of energy is there? 89 00:02:46,451 --> 00:02:47,749 There's gonna be kinetic energy, 90 00:02:47,749 --> 00:02:49,465 so this mass is gonna have kinetic energy, 91 00:02:49,465 --> 00:02:50,900 and we know it's gonna have kinetic energy 92 00:02:50,900 --> 00:02:52,317 because it's gonna be moving 93 00:02:52,317 --> 00:02:53,852 right before it hits the earth, 94 00:02:53,852 --> 00:02:56,273 and kinetic energy is the energy an object has 95 00:02:56,273 --> 00:02:57,969 because of its motion. 96 00:02:57,969 --> 00:02:59,401 So it's gonna have kinetic energy. 97 00:02:59,401 --> 00:03:01,751 How much kinetic energy is it gonna have? 98 00:03:01,751 --> 00:03:04,198 Well, energy is gonna be conserved in our system. 99 00:03:04,198 --> 00:03:06,087 And since the energy is gonna be conserved, 100 00:03:06,087 --> 00:03:07,871 if I started with four units 101 00:03:07,871 --> 00:03:09,946 of gravitational potential energy, 102 00:03:09,946 --> 00:03:13,047 I should end with four units of kinetic energy. 103 00:03:13,047 --> 00:03:13,970 Now, you might be confused. 104 00:03:13,970 --> 00:03:15,356 You might wonder why wasn't there any 105 00:03:15,356 --> 00:03:17,581 gravitational potential energy. 106 00:03:17,581 --> 00:03:19,465 Well, I was assuming that this ground 107 00:03:19,465 --> 00:03:22,080 was the H equals zero line. 108 00:03:22,080 --> 00:03:24,501 And we know that the gravitational potential energy 109 00:03:24,501 --> 00:03:25,334 is mgh. 110 00:03:26,887 --> 00:03:28,361 And where H equals zero, 111 00:03:28,361 --> 00:03:29,968 the potential energy is zero. 112 00:03:29,968 --> 00:03:32,080 So when this mass gets down to the final point, 113 00:03:32,080 --> 00:03:34,170 which I'm representing in this diagram, 114 00:03:34,170 --> 00:03:36,330 it no longer has any potential energy 115 00:03:36,330 --> 00:03:38,207 since the H would be zero. 116 00:03:38,207 --> 00:03:39,867 And since it has no potential energy, 117 00:03:39,867 --> 00:03:41,829 all of this gravitational potential energy 118 00:03:41,829 --> 00:03:42,795 it had initially 119 00:03:42,795 --> 00:03:44,720 has to turn into kinetic energy 120 00:03:44,720 --> 00:03:48,037 in order for the energy of our system to be conserved. 121 00:03:48,037 --> 00:03:49,662 Now, if I were you, I might object and say, 122 00:03:49,662 --> 00:03:50,495 "Wait a minute. 123 00:03:50,495 --> 00:03:51,492 "We learned earlier 124 00:03:51,492 --> 00:03:52,777 "that when work is done, 125 00:03:52,777 --> 00:03:54,592 "energy gets transferred, 126 00:03:54,592 --> 00:03:57,490 "and the total energy of a object might change. 127 00:03:57,490 --> 00:04:00,361 "And this earth is pulling down on this mass, right? 128 00:04:00,361 --> 00:04:03,718 "So the earth is exerting a gravitational force downward. 129 00:04:03,718 --> 00:04:06,646 "The objects moving downward through some displacement, 130 00:04:06,646 --> 00:04:08,568 "that means the earth is doing some positive 131 00:04:08,568 --> 00:04:11,482 "amount of work on this mass, giving it energy. 132 00:04:11,482 --> 00:04:12,322 "Doesn't that mean 133 00:04:12,322 --> 00:04:14,287 "the energy of the system is gonna change?" 134 00:04:14,287 --> 00:04:15,506 And the answer is no. 135 00:04:15,506 --> 00:04:16,571 And the reason is 136 00:04:16,571 --> 00:04:19,825 this earth is also part of our energy system. 137 00:04:19,825 --> 00:04:22,367 So the earth did do work on the mass, 138 00:04:22,367 --> 00:04:24,564 and this mass gained kinetic energy. 139 00:04:24,564 --> 00:04:26,171 But since the earth and the mass 140 00:04:26,171 --> 00:04:28,075 were both part of the system, 141 00:04:28,075 --> 00:04:30,539 this work was internal to the system, 142 00:04:30,539 --> 00:04:32,765 and internal work never changes 143 00:04:32,765 --> 00:04:34,686 the total energy of the system. 144 00:04:34,686 --> 00:04:36,238 And if that was confusing, think about it this way. 145 00:04:36,238 --> 00:04:37,938 Let's say you and your friend, 146 00:04:37,938 --> 00:04:39,226 instead of the earth and the mass, 147 00:04:39,226 --> 00:04:40,294 there is you and your friend. 148 00:04:40,294 --> 00:04:42,614 And instead of energy, we'll talk about money. 149 00:04:42,614 --> 00:04:45,188 You, the earth, give your friend, the mass, 150 00:04:45,188 --> 00:04:46,281 $10. 151 00:04:46,281 --> 00:04:48,851 Instead of 10 joules of energy, you give him $10, 152 00:04:48,851 --> 00:04:51,262 and ask you, "How much total energy is there 153 00:04:51,262 --> 00:04:52,822 "between you and your friend?" 154 00:04:52,822 --> 00:04:54,490 Well, there's still the total amount. 155 00:04:54,490 --> 00:04:56,156 You lost $10. 156 00:04:56,156 --> 00:04:57,656 Your friend gained $10. 157 00:04:57,656 --> 00:04:59,151 But between the two of you, 158 00:04:59,151 --> 00:05:00,938 you still have the total amount of money 159 00:05:00,938 --> 00:05:02,255 you had when you started. 160 00:05:02,255 --> 00:05:03,648 And the same is true with energy. 161 00:05:03,648 --> 00:05:05,670 Whenever work is done internally, 162 00:05:05,670 --> 00:05:07,792 between objects in your system, 163 00:05:07,792 --> 00:05:09,922 there's no change in the total energy 164 00:05:09,922 --> 00:05:11,648 from one chart to another. 165 00:05:11,648 --> 00:05:13,644 That's why I had to draw the kinetic energy 166 00:05:13,644 --> 00:05:14,705 as four units, 167 00:05:14,705 --> 00:05:17,850 because no energy entered or exited our system. 168 00:05:17,850 --> 00:05:19,003 And something that's really useful 169 00:05:19,003 --> 00:05:20,371 about these LOL diagrams 170 00:05:20,371 --> 00:05:21,809 is that you can translate them 171 00:05:21,809 --> 00:05:24,599 straight into a conservation of energy equation. 172 00:05:24,599 --> 00:05:27,409 Because whenever initial energy that you start with, 173 00:05:27,409 --> 00:05:30,164 plus any external work that's done 174 00:05:30,164 --> 00:05:33,076 has to equal the final energy that you end with, 175 00:05:33,076 --> 00:05:34,639 and the reason is external work 176 00:05:34,639 --> 00:05:38,048 is how much energy gets transferred into the system. 177 00:05:38,048 --> 00:05:40,031 So if you start with 10 joules of energy, 178 00:05:40,031 --> 00:05:42,578 when you transfer five joules of energy into the system, 179 00:05:42,578 --> 00:05:44,806 you've gotta end with 15 joules of energy. 180 00:05:44,806 --> 00:05:46,218 So we can just plug straight into this. 181 00:05:46,218 --> 00:05:48,366 What do we have for our particular scenario? 182 00:05:48,366 --> 00:05:50,086 We started with potential energy. 183 00:05:50,086 --> 00:05:54,169 So our initial energy was mgh plus external work. 184 00:05:55,134 --> 00:05:56,509 There was no external work. 185 00:05:56,509 --> 00:05:57,870 There's only internal work, 186 00:05:57,870 --> 00:05:59,328 so this would be zero, 187 00:05:59,328 --> 00:06:01,348 and that's gotta equal the final energy. 188 00:06:01,348 --> 00:06:03,394 The only type of energy we had to end with 189 00:06:03,394 --> 00:06:06,554 was kinetic energy, 1/2 mv squared. 190 00:06:06,554 --> 00:06:07,387 And then if we had numbers, 191 00:06:07,387 --> 00:06:08,891 we can plug in to this formula 192 00:06:08,891 --> 00:06:10,885 and just solve for the height, or the speed, 193 00:06:10,885 --> 00:06:12,567 or whatever we wanted to solve for. 194 00:06:12,567 --> 00:06:14,247 So what would an example look like 195 00:06:14,247 --> 00:06:16,107 where energy wasn't conserved, 196 00:06:16,107 --> 00:06:17,300 in other words, where there was 197 00:06:17,300 --> 00:06:19,299 external work done on the system? 198 00:06:19,299 --> 00:06:20,986 Well, we could use this same example. 199 00:06:20,986 --> 00:06:22,133 I just need to take the earth. 200 00:06:22,133 --> 00:06:25,222 I wanna stick this earth outside of our system. 201 00:06:25,222 --> 00:06:26,055 So now our system 202 00:06:26,055 --> 00:06:27,461 is just the mass M. 203 00:06:27,461 --> 00:06:29,072 What that means is that the work done 204 00:06:29,072 --> 00:06:31,284 by the earth on the mass 205 00:06:31,284 --> 00:06:33,326 is now gonna be external work. 206 00:06:33,326 --> 00:06:36,354 And since there's external work done on this system, 207 00:06:36,354 --> 00:06:38,650 this system's energy has gotta change now. 208 00:06:38,650 --> 00:06:39,876 It's not gonna stay the same. 209 00:06:39,876 --> 00:06:42,149 You're not gonna see the energy of this system 210 00:06:42,149 --> 00:06:45,248 remained conserved, since something is giving it energy. 211 00:06:45,248 --> 00:06:47,753 The earth is giving this mass energy this time 212 00:06:47,753 --> 00:06:49,579 because it's doing external work. 213 00:06:49,579 --> 00:06:51,831 So how would that change our LOL diagram? 214 00:06:51,831 --> 00:06:54,755 This mass M still ends with kinetic energy. 215 00:06:54,755 --> 00:06:56,385 I mean just by imagining the earth 216 00:06:56,385 --> 00:06:57,958 as not part of the system, 217 00:06:57,958 --> 00:06:59,903 that doesn't change what actually happened. 218 00:06:59,903 --> 00:07:01,383 The mass still has to get down here 219 00:07:01,383 --> 00:07:03,384 with four units of kinetic energy. 220 00:07:03,384 --> 00:07:05,890 It's still moving just as fast as it did before. 221 00:07:05,890 --> 00:07:07,188 But our system, which is this mass, 222 00:07:07,188 --> 00:07:08,889 has to be gaining energy. 223 00:07:08,889 --> 00:07:10,804 And a way that's possible is that 224 00:07:10,804 --> 00:07:12,228 the mass does not start 225 00:07:12,228 --> 00:07:14,652 with any gravitational potential energy. 226 00:07:14,652 --> 00:07:15,579 So, people don't like this. 227 00:07:15,579 --> 00:07:17,398 People are like, "Wait, what? 228 00:07:17,398 --> 00:07:18,968 "Of course this mass starts with 229 00:07:18,968 --> 00:07:20,958 "gravitational potential energy. 230 00:07:20,958 --> 00:07:22,185 "It started up here." 231 00:07:22,185 --> 00:07:25,179 But technically speaking, gravitational potential energy 232 00:07:25,179 --> 00:07:28,478 is an energy that exists between two masses. 233 00:07:28,478 --> 00:07:29,677 It takes two to tango, 234 00:07:29,677 --> 00:07:31,061 and it takes two masses 235 00:07:31,061 --> 00:07:33,150 to have gravitational potential energy. 236 00:07:33,150 --> 00:07:35,164 If I stick one of these masses out here, 237 00:07:35,164 --> 00:07:36,993 it can do work on the other mass. 238 00:07:36,993 --> 00:07:38,457 We wouldn't say that it started 239 00:07:38,457 --> 00:07:40,690 with any gravitational potential energy, 240 00:07:40,690 --> 00:07:42,519 so this would now be zero. 241 00:07:42,519 --> 00:07:45,624 But we would say that there was external work done, 242 00:07:45,624 --> 00:07:47,377 so that's where this energy comes from. 243 00:07:47,377 --> 00:07:48,991 The earth is now doing 244 00:07:48,991 --> 00:07:50,135 how much work? 245 00:07:50,135 --> 00:07:52,121 Well, if the mass gained four units 246 00:07:52,121 --> 00:07:53,223 of kinetic energy? 247 00:07:53,223 --> 00:07:55,869 Then the work done had to be four units of work. 248 00:07:55,869 --> 00:07:58,222 So if you stick the earth outside of the system, 249 00:07:58,222 --> 00:08:01,129 the system no longer has gravitational potential energy. 250 00:08:01,129 --> 00:08:04,213 The earth is doing external work on the mass, 251 00:08:04,213 --> 00:08:05,648 and this might bother you. 252 00:08:05,648 --> 00:08:06,481 You might be like, "Wait. 253 00:08:06,481 --> 00:08:09,014 "But it's the same situation that's happening. 254 00:08:09,014 --> 00:08:10,378 "All we did was change 255 00:08:10,378 --> 00:08:12,582 "whether we considered the earth part of our system, 256 00:08:12,582 --> 00:08:14,313 "so how can that change the math?" 257 00:08:14,313 --> 00:08:15,609 It didn't really. 258 00:08:15,609 --> 00:08:16,830 I mean if you think about it, 259 00:08:16,830 --> 00:08:18,988 instead of the four units of energy 260 00:08:18,988 --> 00:08:21,521 existing initially as this potential energy 261 00:08:21,521 --> 00:08:22,687 and having now work, 262 00:08:22,687 --> 00:08:23,520 we're just saying 263 00:08:23,520 --> 00:08:25,686 that there was no potential energy to start, 264 00:08:25,686 --> 00:08:28,174 and there were for units of external work done. 265 00:08:28,174 --> 00:08:29,820 It's just a different story. 266 00:08:29,820 --> 00:08:31,418 The numbers will come out the same, 267 00:08:31,418 --> 00:08:32,789 and it's all gonna be consistent. 268 00:08:32,789 --> 00:08:34,412 But depending on whether you stick that earth 269 00:08:34,412 --> 00:08:36,780 outside of your system or inside of your system 270 00:08:36,780 --> 00:08:38,123 will determine whether you say 271 00:08:38,124 --> 00:08:41,164 that there was initially gravitation potential energy, 272 00:08:41,164 --> 00:08:42,207 or whether you say there was 273 00:08:42,207 --> 00:08:44,798 external work done on the mass. 274 00:08:44,798 --> 00:08:46,172 So let's look at another example. 275 00:08:46,172 --> 00:08:47,371 Let me get rid of this. 276 00:08:47,371 --> 00:08:49,322 Let's instead consider this example, 277 00:08:49,322 --> 00:08:51,064 where a mass starts compressed 278 00:08:51,064 --> 00:08:52,970 against the spring from rest. 279 00:08:52,970 --> 00:08:54,483 That'll be our initial point. 280 00:08:54,483 --> 00:08:55,572 And it fires the mass, 281 00:08:55,572 --> 00:08:56,969 and the mass goes up this incline, 282 00:08:56,969 --> 00:08:57,945 and comes over to here, 283 00:08:57,945 --> 00:08:59,695 and then it's moving with some velocity over here. 284 00:08:59,695 --> 00:09:01,541 We'll make this our final point. 285 00:09:01,541 --> 00:09:02,854 So if we wanna make a LOL diagram, 286 00:09:02,854 --> 00:09:05,833 we first have to pick what's part of our energy system. 287 00:09:05,833 --> 00:09:07,427 Because if we don't know what's part of our system, 288 00:09:07,427 --> 00:09:09,395 we don't even know what's gonna be eligible 289 00:09:09,395 --> 00:09:11,109 to have an energy in here. 290 00:09:11,109 --> 00:09:13,614 So again, let's make the mass part of our system. 291 00:09:13,614 --> 00:09:15,075 So the mass will be part of our system. 292 00:09:15,075 --> 00:09:17,272 Let's make the spring part of our system, 293 00:09:17,272 --> 00:09:19,699 so we'll put the spring in here. 294 00:09:19,699 --> 00:09:22,352 And let's again put the earth into our system. 295 00:09:22,352 --> 00:09:24,250 So what would this LOL diagram look like 296 00:09:24,250 --> 00:09:25,565 for this scenario? 297 00:09:25,565 --> 00:09:27,206 Well, since we start from rest, 298 00:09:27,206 --> 00:09:29,225 we don't start with any kinetic energy. 299 00:09:29,225 --> 00:09:31,541 And even though the earth is part of our system, 300 00:09:31,541 --> 00:09:32,502 which means that our system 301 00:09:32,502 --> 00:09:34,910 could have gravitational potential energy. 302 00:09:34,910 --> 00:09:36,618 I'm just gonna assume that the mass 303 00:09:36,618 --> 00:09:38,212 starts at H equals zero, 304 00:09:38,212 --> 00:09:40,733 so let's make the ground H equals zero. 305 00:09:40,733 --> 00:09:43,113 That way this potential energy term mgh, 306 00:09:43,113 --> 00:09:44,915 starts as zero, 307 00:09:44,915 --> 00:09:45,748 so there'd be no 308 00:09:45,748 --> 00:09:47,692 gravitational potential energy to start with. 309 00:09:47,692 --> 00:09:49,833 And the only energy we would start with 310 00:09:49,833 --> 00:09:52,042 is the fact that this spring is coiled up, 311 00:09:52,042 --> 00:09:55,252 ready to explode and fire this mass forward. 312 00:09:55,252 --> 00:09:57,279 All this energy is stored up as spring 313 00:09:57,279 --> 00:09:58,515 or elastic energy, 314 00:09:58,515 --> 00:10:01,963 so we will have elastic or spring energy to start with. 315 00:10:01,963 --> 00:10:05,797 Let's just say we started with five units of spring energy. 316 00:10:05,797 --> 00:10:07,583 Again, this is somewhat random. 317 00:10:07,583 --> 00:10:09,753 For a particular problem with numbers, 318 00:10:09,753 --> 00:10:12,262 you could solve for exactly how much spring energy 319 00:10:12,262 --> 00:10:13,153 you started with, 320 00:10:13,153 --> 00:10:15,912 because remember, spring energy can be found 321 00:10:15,912 --> 00:10:19,354 using the formula 1/2 k x squared, 322 00:10:19,354 --> 00:10:21,436 where x is the compression of the spring, 323 00:10:21,436 --> 00:10:23,187 and k is the spring constant. 324 00:10:23,187 --> 00:10:24,262 But to keep things simple, 325 00:10:24,262 --> 00:10:26,376 let's just say we start with five units 326 00:10:26,376 --> 00:10:27,412 of spring energy. 327 00:10:27,412 --> 00:10:29,147 So what would our final energies look like? 328 00:10:29,147 --> 00:10:31,018 If this mass is up here moving, 329 00:10:31,018 --> 00:10:33,257 if it's moving, we know it's got kinetic energy. 330 00:10:33,257 --> 00:10:34,525 So I can say there's kinetic energy. 331 00:10:34,525 --> 00:10:35,417 How much? 332 00:10:35,417 --> 00:10:37,309 Again, to figure out exactly how much, 333 00:10:37,309 --> 00:10:38,900 you'd have to solve for your particular problem, 334 00:10:38,900 --> 00:10:39,883 and know this height. 335 00:10:39,883 --> 00:10:42,189 But to give you an idea of what conservation of energy says, 336 00:10:42,189 --> 00:10:45,598 let's just say there were three units of kinetic energy 337 00:10:45,598 --> 00:10:47,226 that you ended with up there 338 00:10:47,226 --> 00:10:49,236 What are the kinds of energy do you have up here? 339 00:10:49,236 --> 00:10:51,264 We're gonna have gravitational potential energy 340 00:10:51,264 --> 00:10:54,798 because this mass starts above the H equals zero line. 341 00:10:54,798 --> 00:10:57,155 Since the mass is higher up in the air, 342 00:10:57,155 --> 00:10:59,604 it's got gravitational potential energy. 343 00:10:59,604 --> 00:11:02,313 So now, if I say there's three units of kinetic energy, 344 00:11:02,313 --> 00:11:04,911 and I started with five units of spring energy, 345 00:11:04,911 --> 00:11:06,459 if energy is conserved, 346 00:11:06,459 --> 00:11:08,280 I better end up with two units 347 00:11:08,280 --> 00:11:10,159 of gravitational potential energy. 348 00:11:10,159 --> 00:11:11,600 So I could draw two unit, 349 00:11:11,600 --> 00:11:12,649 and now I'd make sure that 350 00:11:12,649 --> 00:11:14,762 if I started with five units of energy, 351 00:11:14,762 --> 00:11:17,422 I've ended with a total amount of five units. 352 00:11:17,422 --> 00:11:19,409 That would say energy is conserved. 353 00:11:19,409 --> 00:11:20,330 Let's check. 354 00:11:20,330 --> 00:11:22,557 Is energy of our system gonna be conserved? 355 00:11:22,557 --> 00:11:23,390 We can determine that 356 00:11:23,390 --> 00:11:25,036 by just figuring out whether there were any 357 00:11:25,036 --> 00:11:26,882 external sources of work. 358 00:11:26,882 --> 00:11:30,570 Did anything external to our system do any work? 359 00:11:30,570 --> 00:11:32,966 If not, the total energy has got to stay the same. 360 00:11:32,966 --> 00:11:36,263 This five units has got to turn into five units over here. 361 00:11:36,263 --> 00:11:39,255 And if we assume that these surfaces here are frictionless, 362 00:11:39,255 --> 00:11:42,572 and there was no air resistance or no loss in energy 363 00:11:42,572 --> 00:11:44,527 due to dissipative forces, 364 00:11:44,527 --> 00:11:46,245 there will be no external work done 365 00:11:46,245 --> 00:11:48,379 because everything is part of our system. 366 00:11:48,379 --> 00:11:50,280 So, like, work was done in here, 367 00:11:50,280 --> 00:11:53,439 the spring did work on the mass, giving it energy. 368 00:11:53,439 --> 00:11:55,292 There was internal work done. 369 00:11:55,292 --> 00:11:57,200 But since there was no external work, 370 00:11:57,200 --> 00:11:58,769 the energy had to be conserved. 371 00:11:58,769 --> 00:12:00,648 We can write this in an equation again 372 00:12:00,648 --> 00:12:02,873 by saying that the total energy we start with, 373 00:12:02,873 --> 00:12:06,339 the spring energy, which would be 1/2 k x squared, 374 00:12:06,339 --> 00:12:08,268 plus any external work done, 375 00:12:08,268 --> 00:12:10,045 but the external work done was zero, 376 00:12:10,045 --> 00:12:11,712 since everything is internal, 377 00:12:11,712 --> 00:12:13,412 has to equal the final energy. 378 00:12:13,412 --> 00:12:15,569 And the final energy here is the kinetic energy, 379 00:12:15,569 --> 00:12:18,457 which is 1/2 m v squared, 380 00:12:18,457 --> 00:12:20,241 plus the potential energy, 381 00:12:20,241 --> 00:12:22,858 which is gonna be plus mgh. 382 00:12:22,858 --> 00:12:24,939 And now if you had numbers for a particular problem, 383 00:12:24,939 --> 00:12:26,503 you could just plug those into here, 384 00:12:26,503 --> 00:12:29,016 and solve for whatever variable you were looking for. 385 00:12:29,016 --> 00:12:29,996 But you might be wondering, 386 00:12:29,996 --> 00:12:33,090 "What if there is friction between the mass and the ramp? 387 00:12:33,090 --> 00:12:34,648 "What if there is friction over here? 388 00:12:34,648 --> 00:12:35,894 "What would that do?" 389 00:12:35,894 --> 00:12:37,856 Well, there's two ways we can handle it now. 390 00:12:37,856 --> 00:12:38,783 You know there's two ways. 391 00:12:38,783 --> 00:12:42,336 We can include these surfaces as part of our system, 392 00:12:42,336 --> 00:12:43,726 or we could say that those surfaces 393 00:12:43,726 --> 00:12:45,017 are outside of our system. 394 00:12:45,017 --> 00:12:47,351 If the surfaces are outside of our system, 395 00:12:47,351 --> 00:12:48,376 we would represent that 396 00:12:48,376 --> 00:12:50,218 as them taking energy away. 397 00:12:50,218 --> 00:12:51,773 Now there's external work done, 398 00:12:51,773 --> 00:12:55,438 but instead of the external work giving our system energy, 399 00:12:55,438 --> 00:12:57,517 the external work is taking energy away. 400 00:12:57,517 --> 00:12:59,486 This friction is taking energy 401 00:12:59,486 --> 00:13:00,828 and turning it into thermal energy. 402 00:13:00,828 --> 00:13:04,360 That's what this extra spot is here, for thermal energy. 403 00:13:04,360 --> 00:13:07,167 But since the surfaces are now not part of our system, 404 00:13:07,167 --> 00:13:08,926 we wouldn't show that over here. 405 00:13:08,926 --> 00:13:09,858 We would just include that 406 00:13:09,858 --> 00:13:11,784 as part of the external work done, 407 00:13:11,784 --> 00:13:14,916 so there would be some negative amount of work done. 408 00:13:14,916 --> 00:13:18,942 Maybe the surfaces did negative one unit of external work. 409 00:13:18,942 --> 00:13:19,775 What that would mean is that 410 00:13:19,775 --> 00:13:21,780 if we started with five units of energy, 411 00:13:21,780 --> 00:13:24,673 and there was negative one unit of external work done, 412 00:13:24,673 --> 00:13:28,147 we've got to only end up with four units of energy, 413 00:13:28,147 --> 00:13:29,647 but potential energy has got to be the same. 414 00:13:29,647 --> 00:13:31,011 I mean I ended up this high, 415 00:13:31,011 --> 00:13:32,512 so that can't really change. 416 00:13:32,512 --> 00:13:34,286 So that means my kinetic energy now is gonna be smaller, 417 00:13:34,286 --> 00:13:35,375 and that makes sense. 418 00:13:35,375 --> 00:13:37,704 The friction slowed this mass down. 419 00:13:37,704 --> 00:13:40,033 I end with four units of energy, 420 00:13:40,033 --> 00:13:41,577 even though is started with five, 421 00:13:41,577 --> 00:13:44,436 because there was negative one unit of external work done 422 00:13:44,436 --> 00:13:46,339 by the friction on these surfaces. 423 00:13:46,339 --> 00:13:48,112 That's one way to handle this problem, 424 00:13:48,112 --> 00:13:49,556 and that's all consistent. 425 00:13:49,556 --> 00:13:50,569 The other way to do it, 426 00:13:50,569 --> 00:13:52,343 we could just say that our surfaces, 427 00:13:52,343 --> 00:13:56,666 which we said were external, are now internal to our system. 428 00:13:56,666 --> 00:14:00,410 That way, this external work done is no longer external. 429 00:14:00,410 --> 00:14:02,621 This work done is now gonna be internal. 430 00:14:02,621 --> 00:14:04,034 The surface took the energy. 431 00:14:04,034 --> 00:14:05,647 It still did negative work, 432 00:14:05,647 --> 00:14:07,423 but since it's inside of our system, 433 00:14:07,423 --> 00:14:10,053 that's now gonna be part of our energy system. 434 00:14:10,053 --> 00:14:11,376 So how would we represent that? 435 00:14:11,376 --> 00:14:13,117 It'd no longer be a negative one here. 436 00:14:13,117 --> 00:14:14,906 We wouldn't say that this is done. 437 00:14:14,906 --> 00:14:16,618 We'd essentially, conceptually 438 00:14:16,618 --> 00:14:19,853 move that negative one over by adding one to both sides. 439 00:14:19,853 --> 00:14:22,589 And that means we would gain some thermal energy. 440 00:14:22,589 --> 00:14:24,949 I'm gonna write that as delta E thermal, 441 00:14:24,949 --> 00:14:26,438 because there's gonna be some gain 442 00:14:26,438 --> 00:14:28,655 in the thermal energy of our system. 443 00:14:28,655 --> 00:14:31,494 And I would represent that now not as an external work done, 444 00:14:31,494 --> 00:14:34,916 but as one unit of thermal energy generated 445 00:14:34,916 --> 00:14:36,476 during this process. 446 00:14:36,476 --> 00:14:39,925 So recapping, LOL diagrams are a great way to visualize 447 00:14:39,925 --> 00:14:41,023 what we're talking about 448 00:14:41,023 --> 00:14:43,153 when we say conservation of energy. 449 00:14:43,153 --> 00:14:45,098 And since we define our system, 450 00:14:45,098 --> 00:14:47,318 they're a nice way to visualize what we mean 451 00:14:47,318 --> 00:14:50,859 when we say that the total initial energy of our system, 452 00:14:50,859 --> 00:14:54,345 plus any external work done on our system 453 00:14:54,345 --> 00:00:00,000 has to equal the total final energy of our system.