1 00:00:03,580 --> 00:00:06,760 When solving physics problems, we have fundamental principles. 2 00:00:06,760 --> 00:00:10,030 For instance, we have Newton's second law. 3 00:00:10,030 --> 00:00:12,070 And we also have an energy principle, 4 00:00:12,070 --> 00:00:16,090 work non-conservative equals the change in kinetic energy 5 00:00:16,090 --> 00:00:18,460 plus the change in potential energy. 6 00:00:18,460 --> 00:00:20,440 And many times students are challenged 7 00:00:20,440 --> 00:00:22,170 by-- do I use this law? 8 00:00:22,170 --> 00:00:23,420 Do I use that law? 9 00:00:23,420 --> 00:00:25,220 Why do I use one or the other? 10 00:00:25,220 --> 00:00:26,710 Do I use both at the same time? 11 00:00:26,710 --> 00:00:28,930 Can I get the same answers, et cetera. 12 00:00:28,930 --> 00:00:32,280 Now I want to illustrate a particular category of problems 13 00:00:32,280 --> 00:00:35,650 where we have some type of circular motion, 14 00:00:35,650 --> 00:00:38,830 and why we need to use both of these laws. 15 00:00:38,830 --> 00:00:45,250 So let's consider, let's go back to our dome example at MIT. 16 00:00:45,250 --> 00:00:49,090 And let's say we have an object that's on the dome. 17 00:00:49,090 --> 00:00:54,610 And now let's write out a free body diagram for this object. 18 00:00:54,610 --> 00:00:58,420 So we have our gravitational force. 19 00:00:58,420 --> 00:01:00,520 We have a normal force. 20 00:01:00,520 --> 00:01:04,390 And here, let's just assume that our dome is frictionless, 21 00:01:04,390 --> 00:01:07,510 for simplicity. 22 00:01:07,510 --> 00:01:14,860 So what you see here is that the object is displacing 23 00:01:14,860 --> 00:01:21,310 in a direction-- now here if I choose r hat and theta hat, 24 00:01:21,310 --> 00:01:24,070 and if I choose a coordinate system where 25 00:01:24,070 --> 00:01:26,650 I take my positive angle theta this way, 26 00:01:26,650 --> 00:01:32,830 that the normal force and the displacement are perpendicular. 27 00:01:32,830 --> 00:01:41,860 Now what that means is that the work done by the normal force, 28 00:01:41,860 --> 00:01:47,380 F normal dot ds is 0, because well, we're 29 00:01:47,380 --> 00:01:51,670 calling F normal equal to the normal force 30 00:01:51,670 --> 00:01:58,430 and that the normal force is perpendicular to displacement. 31 00:01:58,430 --> 00:02:00,850 So what that means is all information 32 00:02:00,850 --> 00:02:05,980 about the normal force is not included in the energy 33 00:02:05,980 --> 00:02:06,730 principle. 34 00:02:06,730 --> 00:02:10,000 In fact, the energy principle is-- the work 35 00:02:10,000 --> 00:02:14,690 that's done is just the amount-- this is angle theta, 36 00:02:14,690 --> 00:02:18,520 so this is angle theta-- so the actual work done 37 00:02:18,520 --> 00:02:25,510 by the gravitational force g is the component of g mg sine 38 00:02:25,510 --> 00:02:30,520 theta times the displacement ds. 39 00:02:30,520 --> 00:02:35,240 So the only work that's appearing here, 40 00:02:35,240 --> 00:02:36,530 and this is conservative. 41 00:02:39,230 --> 00:02:42,010 And so when we integrated this work 42 00:02:42,010 --> 00:02:45,280 and got that the work done by the gravitational force 43 00:02:45,280 --> 00:02:50,780 is minus the change in potential energy, that's showing up here. 44 00:02:50,780 --> 00:02:56,530 So the part of the force that's in the direction of the motion 45 00:02:56,530 --> 00:03:00,280 is giving us the energy condition. 46 00:03:00,280 --> 00:03:03,340 But we're losing all information about the forces 47 00:03:03,340 --> 00:03:05,210 in the radial direction. 48 00:03:05,210 --> 00:03:18,160 So because also what we need is Newton's second law 49 00:03:18,160 --> 00:03:25,600 in the direction perpendicular to the displacement. 50 00:03:25,600 --> 00:03:28,400 Displacement is in the theta hat direction, 51 00:03:28,400 --> 00:03:32,290 so we need Newton's second law in the radial direction. 52 00:03:32,290 --> 00:03:37,300 And that we have r hat, is we have a normal force 53 00:03:37,300 --> 00:03:40,420 minus mg co-sine theta. 54 00:03:40,420 --> 00:03:43,930 And the object is undergoing circular motion, 55 00:03:43,930 --> 00:03:51,760 so it's equal to minus mv squared over R, 56 00:03:51,760 --> 00:03:54,850 where R is the radius of the dome. 57 00:03:54,850 --> 00:03:58,329 This equation is not at all included 58 00:03:58,329 --> 00:03:59,829 in the energy condition. 59 00:03:59,829 --> 00:04:05,330 You may say, well what about the tangential Newton's second law, 60 00:04:05,330 --> 00:04:15,910 which is mg sine theta equals mR d squared theta dt squared. 61 00:04:15,910 --> 00:04:19,390 But it's precisely this equation that's 62 00:04:19,390 --> 00:04:29,450 integrated with respect to the displacement 63 00:04:29,450 --> 00:04:32,534 and that gives us our energy principle. 64 00:04:35,200 --> 00:04:38,250 So in summary, the energy principle 65 00:04:38,250 --> 00:04:40,650 is the integration of Newton's second law 66 00:04:40,650 --> 00:04:43,140 in the direction of motion. 67 00:04:43,140 --> 00:04:47,610 And we're completely missing the application 68 00:04:47,610 --> 00:04:50,355 of Newton's second law in the direction perpendicular 69 00:04:50,355 --> 00:04:51,210 to the motion. 70 00:04:51,210 --> 00:04:53,350 Energy does not account for that. 71 00:04:53,350 --> 00:04:56,010 And that's why we needed separately 72 00:04:56,010 --> 00:05:00,660 to apply both of the principles of energy 73 00:05:00,660 --> 00:05:03,960 and Newton's second law in the radial direction 74 00:05:03,960 --> 00:05:07,380 in order to figure out how to solve this problem. 75 00:05:07,380 --> 00:05:11,410 Now this idea is true in general when you have circular motion, 76 00:05:11,410 --> 00:05:12,870 you need the energy equation, which 77 00:05:12,870 --> 00:05:15,040 is the tangentially integrated Newton's second 78 00:05:15,040 --> 00:05:17,110 on the tangential direction. 79 00:05:17,110 --> 00:05:21,060 And you need Newton's second law in the direction perpendicular 80 00:05:21,060 --> 00:05:22,892 to that circular motion.