1 00:00:00,322 --> 00:00:02,076 - [Teacher] Not only are there many different kinds 2 00:00:02,076 --> 00:00:03,922 of energies, but both objects, 3 00:00:03,922 --> 00:00:06,696 and systems of objects, can have energy. 4 00:00:06,696 --> 00:00:08,891 Once they have that energy, they can transfer it 5 00:00:08,891 --> 00:00:10,446 to another system or object 6 00:00:10,446 --> 00:00:13,140 or that energy could transform to a different type 7 00:00:13,140 --> 00:00:15,218 of energy inside that system. 8 00:00:15,218 --> 00:00:18,051 When energy gets transferred, we call that work, 9 00:00:18,051 --> 00:00:20,059 and the amount of work that's done is the amount 10 00:00:20,059 --> 00:00:22,265 of energy that was transferred. 11 00:00:22,265 --> 00:00:24,390 You often hear people say, "Energy is conserved," 12 00:00:24,390 --> 00:00:26,480 which really just means that you can't create 13 00:00:26,480 --> 00:00:27,989 or destroy energy. 14 00:00:27,989 --> 00:00:31,588 You can simply transfer it between objects or systems. 15 00:00:31,588 --> 00:00:33,550 So, what are all the different types of energy? 16 00:00:33,550 --> 00:00:35,234 There's kinetic energy which is the energy 17 00:00:35,234 --> 00:00:37,602 due to something moving, and the formula's 18 00:00:37,602 --> 00:00:40,458 one-half the mass times the speed squared. 19 00:00:40,458 --> 00:00:42,339 There's gravitational potential energy 20 00:00:42,339 --> 00:00:44,754 which is the energy something has due to its height, 21 00:00:44,754 --> 00:00:46,019 and the formula's the mass 22 00:00:46,019 --> 00:00:48,469 times the magnitude of acceleration due to gravity 23 00:00:48,469 --> 00:00:50,234 times the height of the object. 24 00:00:50,234 --> 00:00:51,325 Height above what? 25 00:00:51,325 --> 00:00:52,834 Height above whatever you're choosing 26 00:00:52,834 --> 00:00:55,423 as the H equals zero reference line. 27 00:00:55,423 --> 00:00:56,256 Is that cheating? 28 00:00:56,256 --> 00:00:58,361 No, because all that really matters is the change 29 00:00:58,361 --> 00:01:00,411 in gravitational potential energy, 30 00:01:00,411 --> 00:01:02,610 not the actual value itself. 31 00:01:02,610 --> 00:01:04,084 There's also spring potential energy 32 00:01:04,084 --> 00:01:07,010 which has to do with a compressed or stretched spring, 33 00:01:07,010 --> 00:01:09,623 and the formula's one-half times the spring constant 34 00:01:09,623 --> 00:01:12,455 times x, which is not the length of the spring. 35 00:01:12,455 --> 00:01:14,882 X is the amount that the spring has been compressed 36 00:01:14,882 --> 00:01:16,112 or stretched. 37 00:01:16,112 --> 00:01:18,527 These three types of kinetic and potential energy 38 00:01:18,527 --> 00:01:21,720 constitute what we call mechanical energy. 39 00:01:21,720 --> 00:01:24,246 Mechanical energy's another word for the kinetic energy 40 00:01:24,246 --> 00:01:26,132 plus gravitational potential energy 41 00:01:26,132 --> 00:01:28,547 plus spring potential energy in a system, 42 00:01:28,547 --> 00:01:30,442 and it's important to know that mechanical energy 43 00:01:30,442 --> 00:01:32,808 does not include thermal energy. 44 00:01:32,808 --> 00:01:35,025 Thermal energy's the heat energy generated 45 00:01:35,025 --> 00:01:38,322 by dissipative forces like friction and air resistance, 46 00:01:38,322 --> 00:01:40,377 and you can find the amount of thermal energy generated 47 00:01:40,377 --> 00:01:42,917 by taking the size of the dissipated force 48 00:01:42,917 --> 00:01:45,996 times the distance through which that force was acting. 49 00:01:45,996 --> 00:01:49,955 The unit of energy is Joules and energy is not a vector. 50 00:01:49,955 --> 00:01:52,591 But maybe the most important thing to remember about energy 51 00:01:52,591 --> 00:01:55,575 is if there's no external work done on a system, 52 00:01:55,575 --> 00:01:58,291 then there's no change in the energy of that system. 53 00:01:58,291 --> 00:02:00,254 In other words, if there's no external work done 54 00:02:00,254 --> 00:02:02,854 on a system, the initial energy of that system 55 00:02:02,854 --> 00:02:05,432 will equal the final energy of that system, 56 00:02:05,432 --> 00:02:06,500 which is the way you solve 57 00:02:06,500 --> 00:02:09,100 many conservation of energy problems. 58 00:02:09,100 --> 00:02:11,852 So, what's an example problem involving energy look like? 59 00:02:11,852 --> 00:02:13,826 Let's say a box started with an initial speed 60 00:02:13,826 --> 00:02:17,494 and slides from one platform up to another platform. 61 00:02:17,494 --> 00:02:19,014 We'll assume that frictional forces 62 00:02:19,014 --> 00:02:20,675 and air resistance are negligible. 63 00:02:20,675 --> 00:02:22,475 And for the system that's consisting 64 00:02:22,475 --> 00:02:24,425 of the mass and the Earth, 65 00:02:24,425 --> 00:02:26,794 what's happening to the total mechanical energy 66 00:02:26,794 --> 00:02:28,350 in this system? 67 00:02:28,350 --> 00:02:29,778 So, you've got to pay special attention 68 00:02:29,778 --> 00:02:31,322 to what is in your system. 69 00:02:31,322 --> 00:02:33,040 Since my system includes the mass, 70 00:02:33,040 --> 00:02:34,235 which is going to be moving, 71 00:02:34,235 --> 00:02:36,070 my system's gonna have kinetic energy, 72 00:02:36,070 --> 00:02:37,580 and since my system has two objects 73 00:02:37,580 --> 00:02:39,669 that are interacting gravitationally, 74 00:02:39,669 --> 00:02:41,643 the mass and the Earth, my system's also going 75 00:02:41,643 --> 00:02:43,837 to have gravitational potential energy. 76 00:02:43,837 --> 00:02:45,990 So, when I asked about the total mechanical energy 77 00:02:45,990 --> 00:02:47,727 of the system, that's really just code 78 00:02:47,727 --> 00:02:50,838 for the total kinetic and potential energy of the system. 79 00:02:50,838 --> 00:02:53,218 So, as this mass slides up to a higher point 80 00:02:53,218 --> 00:02:56,786 on the ramp, the gravitational potential energy increases, 81 00:02:56,786 --> 00:02:58,396 but the mass is gonna slow down, 82 00:02:58,396 --> 00:03:00,602 so the kinetic energy's gonna decrease. 83 00:03:00,602 --> 00:03:03,272 However, since the Earth and the mass are in our system, 84 00:03:03,272 --> 00:03:05,188 and there's no dissipative forces, 85 00:03:05,188 --> 00:03:08,334 there's no external work done on our system. 86 00:03:08,334 --> 00:03:10,684 Yes, the Earth is doing work on the box, 87 00:03:10,684 --> 00:03:12,502 but the Earth is part of our system 88 00:03:12,502 --> 00:03:14,383 so it can't do external work, 89 00:03:14,383 --> 00:03:16,380 and that means energy just gets transferred 90 00:03:16,380 --> 00:03:19,120 from one form to another within our system, 91 00:03:19,120 --> 00:03:20,873 and the total mechanical energy, here, 92 00:03:20,873 --> 00:03:23,451 is gonna remain the same for the entire trip. 93 00:03:23,451 --> 00:03:24,763 Now, what if we asked this same question 94 00:03:24,763 --> 00:03:28,304 but we consider a system that consists only of the box. 95 00:03:28,304 --> 00:03:30,684 In that case, our system has a box that's moving, 96 00:03:30,684 --> 00:03:31,914 so it'll have kinetic energy. 97 00:03:31,914 --> 00:03:34,132 But, our system no longer includes two objects 98 00:03:34,132 --> 00:03:37,441 interacting gravitationally so our system will have no 99 00:03:37,441 --> 00:03:39,519 gravitational potential energy. 100 00:03:39,519 --> 00:03:42,259 What happens to the total mechanical energy in this case? 101 00:03:42,259 --> 00:03:44,511 Well, the only energy that I've got in my system, now, 102 00:03:44,511 --> 00:03:48,191 is kinetic energy and since that kinetic energy decreased, 103 00:03:48,191 --> 00:03:50,026 the total mechanical energy of the box, 104 00:03:50,026 --> 00:03:52,232 as a system, decreases. 105 00:03:52,232 --> 00:03:53,253 How does it decrease? 106 00:03:53,253 --> 00:03:55,361 It decreases because now the Earth is outside 107 00:03:55,361 --> 00:03:57,456 of our system and the work that it is doing 108 00:03:57,456 --> 00:03:59,662 on the box is external work 109 00:03:59,662 --> 00:04:02,855 and it's taking away energy from the box. 110 00:04:02,855 --> 00:04:04,086 What does work mean? 111 00:04:04,086 --> 00:04:06,663 In physics, work is the amount of energy transferred 112 00:04:06,663 --> 00:04:09,240 from one system, or object, to another. 113 00:04:09,240 --> 00:04:10,877 In other words, if a person lifted a box 114 00:04:10,877 --> 00:04:13,977 and gave it 10 Joules of gravitational potential energy, 115 00:04:13,977 --> 00:04:16,961 we'd say that person did positive 10 Joules of work 116 00:04:16,961 --> 00:04:19,411 on the box since that person gave the box 117 00:04:19,411 --> 00:04:20,966 10 Joules of energy. 118 00:04:20,966 --> 00:04:23,010 But since the box took 10 Joules of energy 119 00:04:23,010 --> 00:04:24,972 from that person, we'd say that the box 120 00:04:24,972 --> 00:04:27,514 did negative 10 Joules of work on the person 121 00:04:27,514 --> 00:04:29,929 since the box took 10 Joules of energy. 122 00:04:29,929 --> 00:04:31,125 So, you can find the work done 123 00:04:31,125 --> 00:04:32,878 if you can determine the amount of energy 124 00:04:32,878 --> 00:04:34,318 that was transferred. 125 00:04:34,318 --> 00:04:36,524 But, there's an alternative formula to find the work done. 126 00:04:36,524 --> 00:04:38,289 If something's having work done to it, 127 00:04:38,289 --> 00:04:40,135 there's got to be a force on that object, 128 00:04:40,135 --> 00:04:42,793 and that object has to be displaced. 129 00:04:42,793 --> 00:04:44,418 So, if you take the force on the object 130 00:04:44,418 --> 00:04:46,775 times the displacement of the object, 131 00:04:46,775 --> 00:04:48,656 and multiply by the cosine of the angle 132 00:04:48,656 --> 00:04:51,443 between the force and the displacement, 133 00:04:51,443 --> 00:04:52,835 you'll also get the work done. 134 00:04:52,835 --> 00:04:54,879 In other words, one way to find the work done 135 00:04:54,879 --> 00:04:56,423 is by finding the amount of energy 136 00:04:56,423 --> 00:04:57,654 that was transferred. 137 00:04:57,654 --> 00:04:58,838 But, another way to find the work done is 138 00:04:58,838 --> 00:05:02,083 by taking the magnitude of force exerted on an object 139 00:05:02,083 --> 00:05:04,087 times the displacement of the object 140 00:05:04,087 --> 00:05:05,688 and then times cosine of the angle 141 00:05:05,688 --> 00:05:08,556 between the displacement and the force. 142 00:05:08,556 --> 00:05:10,030 Since work is a transfer of energy, 143 00:05:10,030 --> 00:05:11,911 it also has units of Joules. 144 00:05:11,911 --> 00:05:13,954 And even though work is not a vector, 145 00:05:13,954 --> 00:05:16,044 it can be positive or negative. 146 00:05:16,044 --> 00:05:17,844 If the force on an object has a component 147 00:05:17,844 --> 00:05:19,608 in the direction of motion, 148 00:05:19,608 --> 00:05:21,930 that force will do positive work on the object 149 00:05:21,930 --> 00:05:23,776 and give the object energy. 150 00:05:23,776 --> 00:05:25,541 If the force on the object has a component 151 00:05:25,541 --> 00:05:27,503 in the opposite direction of the motion, 152 00:05:27,503 --> 00:05:29,465 the work done by that force would be negative 153 00:05:29,465 --> 00:05:31,845 and it would take away the object's energy. 154 00:05:31,845 --> 00:05:33,970 And, if the force on an object is perpendicular 155 00:05:33,970 --> 00:05:35,607 to the motion of the object, 156 00:05:35,607 --> 00:05:37,952 that force does zero work on the object. 157 00:05:37,952 --> 00:05:39,624 It neither gives the object energy 158 00:05:39,624 --> 00:05:41,714 nor takes away the object's energy. 159 00:05:41,714 --> 00:05:44,280 So, what's an example problem involving work look like? 160 00:05:44,280 --> 00:05:46,532 Let's say a box of mass M slides down 161 00:05:46,532 --> 00:05:49,794 a frictionless ramp of height, H, and angle two-theta, 162 00:05:49,794 --> 00:05:51,931 as seen in this diagram, here, 163 00:05:51,931 --> 00:05:54,415 and a separate box of mass two-M slides down 164 00:05:54,415 --> 00:05:56,215 another frictionless ramp of height, H, 165 00:05:56,215 --> 00:05:59,772 and angle theta, as seen in this diagram, here. 166 00:05:59,772 --> 00:06:01,822 And, we want to know how work done on the object 167 00:06:01,822 --> 00:06:04,725 by the Earth compares for each case? 168 00:06:04,725 --> 00:06:06,362 The easiest way to find the work done here 169 00:06:06,362 --> 00:06:08,405 is by finding the change in energy. 170 00:06:08,405 --> 00:06:10,205 The box will gain an amount of kinetic energy 171 00:06:10,205 --> 00:06:13,305 equal to the amount of potential energy that it loses. 172 00:06:13,305 --> 00:06:15,459 So, the work done by the Earth is just gonna equal 173 00:06:15,459 --> 00:06:17,496 positive m-g-h. 174 00:06:17,496 --> 00:06:18,935 Both heights are the same. 175 00:06:18,935 --> 00:06:22,726 So, the H's are equivalent, but one box has twice the mass. 176 00:06:22,726 --> 00:06:24,984 So, the work done by gravity on the mass, two-M, 177 00:06:24,984 --> 00:06:26,168 is gonna be twice as great 178 00:06:26,168 --> 00:06:28,409 as the work done on the mass one-M. 179 00:06:28,409 --> 00:06:30,661 What's the work energy principle mean? 180 00:06:30,661 --> 00:06:32,368 The work energy principle states 181 00:06:32,368 --> 00:06:35,980 that the total work, or the net work done on an object, 182 00:06:35,980 --> 00:06:38,417 is gonna be equal to the change in kinetic energy 183 00:06:38,417 --> 00:06:40,042 of that object. 184 00:06:40,042 --> 00:06:42,562 So, if you add up all the work done by all forces 185 00:06:42,562 --> 00:06:45,569 on an object, that's got to be equal to the change 186 00:06:45,569 --> 00:06:47,322 in the kinetic energy of that object. 187 00:06:47,322 --> 00:06:50,247 In other words, one-half m-v-final, squared, 188 00:06:50,247 --> 00:06:52,477 minus one-half m-v-initial, squared. 189 00:06:52,477 --> 00:06:54,857 So, this is a really handy way to find how the speed 190 00:06:54,857 --> 00:06:57,657 of an object changes if you can determine the net work 191 00:06:57,657 --> 00:06:58,781 on an object. 192 00:06:58,781 --> 00:07:01,277 In other words, if there's multiple forces on an object, 193 00:07:01,277 --> 00:07:04,063 and you can find the work done by each of those forces, 194 00:07:04,063 --> 00:07:06,200 you can determine how much kinetic energy 195 00:07:06,200 --> 00:07:08,278 that object gained or lost. 196 00:07:08,278 --> 00:07:10,333 So, what's an example of the work energy principle? 197 00:07:10,333 --> 00:07:12,179 Let's say a four kilogram box started 198 00:07:12,179 --> 00:07:15,174 with a velocity of six meters per second to the left. 199 00:07:15,174 --> 00:07:17,670 Some net amount of work is done on that box, 200 00:07:17,670 --> 00:07:20,097 and it's now moving with a velocity of four meters 201 00:07:20,097 --> 00:07:21,653 per second to the right. 202 00:07:21,653 --> 00:07:24,346 We want to know what was the amount of new work done 203 00:07:24,346 --> 00:07:25,368 on the box? 204 00:07:25,368 --> 00:07:26,923 Without even solving it, we can say 205 00:07:26,923 --> 00:07:30,662 since this object's slowed down, energy was taken from it, 206 00:07:30,662 --> 00:07:32,903 so the amount of net work had to be negative 207 00:07:32,903 --> 00:07:35,317 which means it's either B or D. 208 00:07:35,317 --> 00:07:36,548 To figure out which one exactly, 209 00:07:36,548 --> 00:07:38,023 we could use the work energy principle 210 00:07:38,023 --> 00:07:40,217 which says that the net work done is equal to 211 00:07:40,217 --> 00:07:41,784 the change in kinetic energy. 212 00:07:41,784 --> 00:07:44,028 So, if we take the final kinetic energy, 213 00:07:44,028 --> 00:07:45,952 which is one-half times four kilograms 214 00:07:45,952 --> 00:07:47,949 times the final speed squared, 215 00:07:47,949 --> 00:07:50,329 and we subtract the initial kinetic energy, 216 00:07:50,329 --> 00:07:53,812 one-half times four kilograms times the initial speed, 217 00:07:53,812 --> 00:07:55,809 squared, six meters per second, 218 00:07:55,809 --> 00:07:59,333 we get negative 40 Joules of net work. 219 00:07:59,333 --> 00:08:01,556 If you get a force versus position graph, 220 00:08:01,556 --> 00:08:05,120 the area under that graph will represent the work done. 221 00:08:05,120 --> 00:08:06,858 So, when you see F versus x, 222 00:08:06,858 --> 00:08:09,648 you should think area equals work. 223 00:08:09,648 --> 00:08:12,702 But, be careful, area above the x-axis is gonna count 224 00:08:12,702 --> 00:08:16,161 as positive work done, and area underneath the x-axis 225 00:08:16,161 --> 00:08:18,251 is gonna count as negative work done, 226 00:08:18,251 --> 00:08:20,573 and make sure the x-axis really is position. 227 00:08:20,573 --> 00:08:22,651 If you get a force versus time graph, 228 00:08:22,651 --> 00:08:24,741 the area's impulse, not work. 229 00:08:24,741 --> 00:08:27,764 So, what would an example of work as area look like? 230 00:08:27,764 --> 00:08:30,012 Let's say a box started at x equals zero 231 00:08:30,012 --> 00:08:33,158 with a velocity of five meters per second to the right 232 00:08:33,158 --> 00:08:35,736 and a net horizontal force on the box is given 233 00:08:35,736 --> 00:08:37,116 by the graph below. 234 00:08:37,116 --> 00:08:40,244 We want to know, at what position other than x equals zero, 235 00:08:40,245 --> 00:08:42,236 will the box, again, have a velocity 236 00:08:42,236 --> 00:08:45,175 of five meters per second to the right? 237 00:08:45,175 --> 00:08:47,218 Well, since the box will end with the same speed 238 00:08:47,218 --> 00:08:50,457 that it began with, the change in kinetic energy is gonna 239 00:08:50,457 --> 00:08:51,688 equal zero. 240 00:08:51,688 --> 00:08:54,556 But, that means the net work would also equal zero, 241 00:08:54,556 --> 00:08:56,773 since the net work is equal to the change 242 00:08:56,773 --> 00:08:57,957 in kinetic energy. 243 00:08:57,957 --> 00:09:00,256 So, if the box starts at x equals zero, 244 00:09:00,256 --> 00:09:02,683 how far do we have to go in order for us 245 00:09:02,683 --> 00:09:04,772 to have no net work done. 246 00:09:04,772 --> 00:09:06,166 Between zero and three meters, 247 00:09:06,166 --> 00:09:08,244 the work done is gonna be negative, 248 00:09:08,244 --> 00:09:10,217 and the area of this triangle is gonna be 249 00:09:10,217 --> 00:09:12,470 one-half the base times the height 250 00:09:12,470 --> 00:09:15,128 which is one-half times three meters 251 00:09:15,128 --> 00:09:16,951 time negative six Newtons 252 00:09:16,951 --> 00:09:19,413 which is negative nine Joules of work done, 253 00:09:19,413 --> 00:09:20,841 and the area under this triangle, 254 00:09:20,841 --> 00:09:22,721 between three and five seconds, 255 00:09:22,721 --> 00:09:25,055 would again be one-half base times height, 256 00:09:25,055 --> 00:09:27,911 which is one-half times two meters 257 00:09:27,911 --> 00:09:29,954 times height of four Newtons 258 00:09:29,954 --> 00:09:32,729 which is positive four Joules of work done. 259 00:09:32,729 --> 00:09:35,841 So, by the time that the box has made it to five meters, 260 00:09:35,841 --> 00:09:37,478 there's been a total amount of work done 261 00:09:37,478 --> 00:09:39,253 of negative nine plus four, 262 00:09:39,253 --> 00:09:41,773 which is negative five Joules of work. 263 00:09:41,773 --> 00:09:43,454 But, we want no net work done. 264 00:09:43,454 --> 00:09:45,117 So, we're gonna have to keep going 265 00:09:45,117 --> 00:09:47,683 until this positive area contribution is gonna 266 00:09:47,683 --> 00:09:49,227 equal the negative area. 267 00:09:49,227 --> 00:09:50,376 In other words, if I can make it 268 00:09:50,376 --> 00:09:52,385 so that all of this negative area is equal 269 00:09:52,385 --> 00:09:54,266 to all of the positive area, 270 00:09:54,266 --> 00:09:56,355 my net work's gonna equal zero. 271 00:09:56,355 --> 00:09:58,109 My negative area is negative nine. 272 00:09:58,109 --> 00:10:01,011 My positive area, so far, is positive four. 273 00:10:01,011 --> 00:10:03,087 If I continue on to the six meter mark, 274 00:10:03,087 --> 00:10:06,036 I've pick up another positive four Joules of work 275 00:10:06,036 --> 00:10:08,057 since the height of this rectangle is four 276 00:10:08,057 --> 00:10:09,775 and the width is one meter, 277 00:10:09,775 --> 00:10:10,913 which means we're almost there. 278 00:10:10,913 --> 00:10:12,178 Four plus four is eight. 279 00:10:12,178 --> 00:10:13,836 I'd only need to pick up one more Joule, 280 00:10:13,836 --> 00:10:15,675 so I can't go all the way to seven meters. 281 00:10:15,675 --> 00:10:18,320 I'd only need to go one more fourth of a meter 282 00:10:18,320 --> 00:10:20,317 to pick up one more Joule 283 00:10:20,317 --> 00:10:23,300 so that one plus four plus four is equal 284 00:10:23,300 --> 00:10:24,612 to negative nine. 285 00:10:24,612 --> 00:10:26,278 So, the net work would equal zero 286 00:10:26,278 --> 00:10:29,361 somewhere between x equals six and x equals seven 287 00:10:29,361 --> 00:10:30,626 which would ensure that the change 288 00:10:30,626 --> 00:10:33,169 in kinetic energy is zero and we would end 289 00:10:33,169 --> 00:10:35,952 with the same speed that we began with. 290 00:10:35,952 --> 00:10:37,546 What does power mean? 291 00:10:37,546 --> 00:10:41,273 In physics power is the amount of work done per time, 292 00:10:41,273 --> 00:10:42,329 which can also be thought of 293 00:10:42,329 --> 00:10:45,034 as the amount of energy transferred per time. 294 00:10:45,034 --> 00:10:47,937 In other words, the amount of Joules per second 295 00:10:47,937 --> 00:10:49,214 that are transferred, 296 00:10:49,214 --> 00:10:52,279 and the name given to a Joule per second is a Watt. 297 00:10:52,279 --> 00:10:53,470 So, you can solve for the power 298 00:10:53,470 --> 00:10:55,634 by finding the work divided by the time 299 00:10:55,634 --> 00:10:58,525 or the change in energy divided by the time. 300 00:10:58,525 --> 00:10:59,907 And you can increase the amount of power 301 00:10:59,907 --> 00:11:01,741 by increasing the work done 302 00:11:01,741 --> 00:11:03,866 or decreasing the amount of time it takes 303 00:11:03,866 --> 00:11:05,456 for that work to be done. 304 00:11:05,456 --> 00:11:08,556 And just like energy and work, power is not a vector. 305 00:11:08,556 --> 00:11:11,296 So, what's an example problem involving power look like? 306 00:11:11,296 --> 00:11:13,862 Let's say a box of mass M slid all the way 307 00:11:13,862 --> 00:11:15,708 down a frictionless ramp of height H 308 00:11:15,708 --> 00:11:18,889 and angle two-theta as seen in this diagram, 309 00:11:18,889 --> 00:11:21,257 and a separate mass M slides all the way 310 00:11:21,257 --> 00:11:22,836 down a frictionless ramp of height H 311 00:11:22,836 --> 00:11:25,820 and angle theta as seen in that diagram, 312 00:11:25,820 --> 00:11:27,864 and we want to know how the average power developed 313 00:11:27,864 --> 00:11:30,511 by the force of gravity on the boxes compares 314 00:11:30,511 --> 00:11:32,066 for each incline? 315 00:11:32,066 --> 00:11:33,460 So, we use the formula for power, 316 00:11:33,460 --> 00:11:35,549 power's the work done per time. 317 00:11:35,549 --> 00:11:38,254 The work done on these boxes is gonna equal the change 318 00:11:38,254 --> 00:11:40,193 in kinetic energy of these boxes 319 00:11:40,193 --> 00:11:42,643 which would equal the change in potential energy 320 00:11:42,643 --> 00:11:45,383 of the boxes, but the mass of the boxes are the same, 321 00:11:45,383 --> 00:11:47,775 the gravitational acceleration is the same, 322 00:11:47,775 --> 00:11:49,818 and the height they fall from is the same. 323 00:11:49,818 --> 00:11:51,827 So, the work done on the boxes are equal, 324 00:11:51,827 --> 00:11:53,429 but the time it takes for these boxes 325 00:11:53,429 --> 00:11:55,913 to slide down the ramp is not equal. 326 00:11:55,913 --> 00:11:59,002 The mass on the steeper ramp will reach the bottom faster 327 00:11:59,002 --> 00:12:01,440 which means it has a higher rate of power being done 328 00:12:01,440 --> 00:12:04,017 compared to the mass on the less steep ramp. 329 00:12:04,017 --> 00:12:06,606 So, even though the same amount of work is being done, 330 00:12:06,606 --> 00:12:09,520 the rate at which that work is being done is greater 331 00:12:09,520 --> 00:00:00,000 for the steeper ramp compared to the more shallow ramp.