1 00:00:00,670 --> 00:00:03,370 So let us have a look at the principle of work and energy. 2 00:00:04,330 --> 00:00:10,150 Now, before we start to look at the definition of work, I want to state that the principle of work 3 00:00:10,150 --> 00:00:17,860 and energy for our case is a tool that we can use to solve some kinetics problems so we can use the 4 00:00:17,860 --> 00:00:21,830 principle of work and energy, for instance, to solve for the velocity of a particle. 5 00:00:22,150 --> 00:00:24,550 And this is very handy. 6 00:00:24,550 --> 00:00:29,070 As you'll see, we'll do an example of applying the principle of work and energy. 7 00:00:29,080 --> 00:00:33,940 So we're going to use it as a tool in our toolbox to solve kinetics problems. 8 00:00:35,440 --> 00:00:42,940 OK, so in physics, work is defined as a measure of energy transfer that occurs to an object that is 9 00:00:42,940 --> 00:00:46,500 moved over a distance while an external force is applied. 10 00:00:47,230 --> 00:00:53,680 So a force will do work on a particle only if the particle is undergoing a displacement in the direction 11 00:00:53,680 --> 00:00:54,340 of the force. 12 00:00:55,900 --> 00:01:03,730 So if if the force is constant, you can calculate the work done on a certain body by multiplying the 13 00:01:03,730 --> 00:01:09,280 length of the path that the body has traveled by the force acting along this path. 14 00:01:09,640 --> 00:01:15,910 Now, as you'll see, sometimes a force will not act in the same direction as the path traveled by the 15 00:01:16,150 --> 00:01:17,440 body or the particle. 16 00:01:17,740 --> 00:01:23,290 But then we need to take the component of the force in the direction of travel and we need to multiply 17 00:01:23,290 --> 00:01:27,310 that component by the distance travelled to get the work done. 18 00:01:29,300 --> 00:01:35,200 So then we also have the concepts of positive and negative work. 19 00:01:35,230 --> 00:01:42,320 So if the force or component of the force and the displacement have the same direction, then the work 20 00:01:42,320 --> 00:01:43,350 done is positive. 21 00:01:44,330 --> 00:01:50,300 Well, if the force or the component of the force and the displacement caused by this force of opposite 22 00:01:50,300 --> 00:01:52,890 directions, then the work done is negative. 23 00:01:53,120 --> 00:01:59,220 So let's look at an example to illustrate when his work positive and when his work negative. 24 00:02:00,140 --> 00:02:03,830 So what I've drawn here is a person doing a bungee jump. 25 00:02:04,070 --> 00:02:12,020 So there's a cliff and the person stands on the edge of the cliff and is attached with a bungee rope 26 00:02:12,440 --> 00:02:14,450 and will jump off the cliff and then. 27 00:02:15,880 --> 00:02:23,890 The bungee rope will slow down the person's velocity and it will pull him back up and so forth, like 28 00:02:23,890 --> 00:02:25,170 someone doing a bungee jump. 29 00:02:25,180 --> 00:02:32,500 So on the left hand side, in the first case, the person has just just jumped over the cliff and the 30 00:02:32,500 --> 00:02:35,530 bungee rope has not yet stretched. 31 00:02:35,950 --> 00:02:38,940 So you can assume that the person is kind of in a freefall. 32 00:02:40,140 --> 00:02:48,270 In this case, gravity does positive work, so the force of gravity pulls the person down and the person 33 00:02:48,270 --> 00:02:54,030 moves in the direction in which the force of gravity acts, that is downwards. 34 00:02:54,900 --> 00:02:58,990 And so the force and the direction of movement is in the same direction. 35 00:02:59,010 --> 00:03:01,590 So the work done is positive. 36 00:03:03,440 --> 00:03:09,320 If we go to the chaos in the middle, this is where the bungee rope has now tightened and started to 37 00:03:09,320 --> 00:03:16,580 stretch, so now the bungee rope applies a force to the person in the upward direction, but the person 38 00:03:16,580 --> 00:03:18,190 is still moving downward. 39 00:03:18,200 --> 00:03:21,030 Remember, this bungee jump is still stretching out. 40 00:03:21,860 --> 00:03:27,860 So gravity still does positive work on the person because the person is moving downward still. 41 00:03:28,610 --> 00:03:33,030 But the spring force, that's the force and the bungee rope works upwards. 42 00:03:33,470 --> 00:03:38,060 So that would be a negative work because it's in the opposite direction of movement. 43 00:03:38,810 --> 00:03:46,850 Now we go to the case of the right person has now moved down up until a point where the velocity became 44 00:03:46,850 --> 00:03:49,340 zero and is now starting to move up again. 45 00:03:49,370 --> 00:03:51,440 So the Bunshiro pulls the person up. 46 00:03:52,370 --> 00:03:57,920 Now, the work done by gravity is negative because the person moves in the opposite direction. 47 00:03:59,940 --> 00:04:06,000 Then the direction in which the force of gravity acts, but now the work of the spring force is positive 48 00:04:06,120 --> 00:04:09,770 because the spring force works up and the person moves up. 49 00:04:10,380 --> 00:04:18,240 So these three cases illustrates to you when a work will be negative and when it will be positive. 50 00:04:18,240 --> 00:04:23,490 It has to do with the direction of the force and the direction of the movement. 51 00:04:24,690 --> 00:04:31,080 Now, it's very important to have the correct sign for the work because you will see when we apply the 52 00:04:31,080 --> 00:04:40,800 principle of work and energy, the work either adds kinetic energy or it subtracts takes away kinetic 53 00:04:40,800 --> 00:04:45,480 energy, and that determines the final velocity of a particle. 54 00:04:45,480 --> 00:04:52,740 And so it's important that we realize where the energy is added to the system or taken away from the 55 00:04:52,750 --> 00:04:53,310 system. 56 00:04:53,700 --> 00:04:58,550 It's added when work is positive and it's taken away when work is negative. 57 00:04:58,890 --> 00:05:00,720 We will look at an example in the end. 58 00:05:03,240 --> 00:05:11,700 So now that we know how work is defined in the in terms of physics and we know when work is positive 59 00:05:11,700 --> 00:05:15,400 and when work is negative, we can state the principle of work on energy. 60 00:05:16,050 --> 00:05:18,120 Now, the principle of work and energy says. 61 00:05:19,180 --> 00:05:21,430 T one plus the some of the work. 62 00:05:22,670 --> 00:05:28,860 Then on the particle is equal to T to now D one and two, those are kinetic energies. 63 00:05:28,880 --> 00:05:34,490 So it says the kinetic energy you start off with of a certain particle. 64 00:05:35,590 --> 00:05:41,620 Plus, the work done on that particle will give you this kinetic energy after the work has been done 65 00:05:41,620 --> 00:05:47,770 on the particle, now that work can be positive or it can be negative if it's positive in the kinetic 66 00:05:47,770 --> 00:05:53,650 energy after the work has been done on, the particle will be larger and the kinetic energy before the 67 00:05:53,650 --> 00:05:54,450 work has been done. 68 00:05:55,030 --> 00:05:58,840 If the work is negative, then the kinetic energy will decrease. 69 00:06:00,320 --> 00:06:08,890 So one, that's kinetic energy, one, we know kinetic energy is given by half in the square, so that 70 00:06:08,930 --> 00:06:15,800 be halftimes, the mass times, the initial velocity squared plus the work done will give you the kinetic 71 00:06:15,800 --> 00:06:20,060 energy in the end, which is one of in the two squared. 72 00:06:20,090 --> 00:06:27,980 So the velocity of the particle has either gone up with positive work or it has decreased with negative 73 00:06:27,980 --> 00:06:28,260 work. 74 00:06:29,030 --> 00:06:31,370 Now the work part can be done. 75 00:06:31,370 --> 00:06:39,830 In our case, we will consider a constant force the force by linear spring and also the force of gravitation. 76 00:06:39,830 --> 00:06:44,960 And so for each one of these forces, there's a slightly different formula that you use to calculate 77 00:06:44,960 --> 00:06:45,470 the work. 78 00:06:46,280 --> 00:06:52,340 And then if we have the velocity at the start and we have the mass and we can calculate the work, then 79 00:06:52,340 --> 00:06:56,480 we can calculate the velocity of the particle after the work has been applied. 80 00:06:56,870 --> 00:07:00,170 And that will be the example that we'll look at in the end as well. 81 00:07:01,370 --> 00:07:02,740 So let us look. 82 00:07:02,750 --> 00:07:07,880 Our work can be calculated for different types of force applied to a body. 83 00:07:08,270 --> 00:07:12,490 So first of all, let's look at a constant force applied to a body. 84 00:07:13,070 --> 00:07:19,310 And what I've drawn here in red is that let's say it's great and it has a certain mass and is a force, 85 00:07:19,310 --> 00:07:26,420 if applied to it at a certain angle to the horizontal, this red great moves only horizontally. 86 00:07:26,690 --> 00:07:29,930 So there's only work done in the horizontal direction. 87 00:07:29,930 --> 00:07:31,640 It's not lifted off the ground. 88 00:07:31,640 --> 00:07:33,800 It's only moved in a horizontal direction. 89 00:07:34,780 --> 00:07:41,080 So we will calculate that work by taking the force component in a horizontal direction and multiplying 90 00:07:41,080 --> 00:07:45,340 it by the distance at this great travels and that distance here is indicated by us. 91 00:07:46,330 --> 00:07:53,270 So how do we get the component of the force in the horizontal direction that would be if cosine of theta? 92 00:07:53,620 --> 00:08:02,080 And if we multiply that by the distance, we get the work done on this that we can also write down as 93 00:08:02,080 --> 00:08:11,380 if DOT is the DOT product indicates that we take cosine of the angle between the force and the direction 94 00:08:11,380 --> 00:08:12,130 of movement. 95 00:08:13,240 --> 00:08:19,300 And if we then write down the principle of work and energy equation, it would be one of M.V. squared, 96 00:08:19,300 --> 00:08:27,430 the kinetic energy plus the work done if cost theta is is equal to half M.V. two squared. 97 00:08:28,150 --> 00:08:31,150 So that's in the case of a constant force applied at an angle. 98 00:08:32,270 --> 00:08:33,500 If we look at a spring. 99 00:08:35,860 --> 00:08:42,820 I have drawn a crate that is pressed against a spring that is attached to a wall, so the spring will 100 00:08:42,820 --> 00:08:51,070 exert a force to the right, and if we push the crate to the left, it means the force of the spring 101 00:08:51,070 --> 00:08:53,260 is in the opposite direction of movement. 102 00:08:53,260 --> 00:08:55,180 If we push the crate to the left. 103 00:08:56,560 --> 00:09:06,280 So are we calculate spring work is to say you caused by this spring is minus of K, which is the spring 104 00:09:06,280 --> 00:09:10,990 constant multiplied by X2 squared minus X one squared. 105 00:09:11,680 --> 00:09:17,690 Now the Xs, those are the positions, the start and the end position of this movement. 106 00:09:18,370 --> 00:09:23,770 Now it is important that we use the correct points for the start. 107 00:09:23,770 --> 00:09:30,520 In the end, you'll notice that in the standard definition of work by spring it is given with a negative 108 00:09:30,520 --> 00:09:30,970 sign. 109 00:09:31,540 --> 00:09:41,380 So that means that X two must be larger than X one if we push the grade to the left in this picture, 110 00:09:41,740 --> 00:09:46,990 because the amount in brackets need to be positive so that we can have a negative work. 111 00:09:48,650 --> 00:09:55,340 OK, I will explain that again in the example, but if we then write down the principle of work and 112 00:09:55,340 --> 00:10:03,650 energy with a spring force, we'll see it's half M.V. one squared, minus one half K X2 squared minus 113 00:10:03,650 --> 00:10:09,800 X, one squared is equal to a half in the two squared. 114 00:10:10,610 --> 00:10:13,740 And then the third force that we consider is that of gravity. 115 00:10:14,060 --> 00:10:23,960 So if a great falls down, gravity does work on this crate and that work is given by minus M g y to 116 00:10:23,960 --> 00:10:31,580 minus Y one y y two is the end position and one is the start position now. 117 00:10:32,950 --> 00:10:35,410 Why two is lower than why one? 118 00:10:36,450 --> 00:10:42,780 Because we want that distance in the brackets to be negative, because we multiply that with a negative 119 00:10:42,780 --> 00:10:46,730 sign in front of the M, we get positive work when it falls down. 120 00:10:47,160 --> 00:10:52,260 In the case of the bungee jumper, when the bungee jump, it moves up, then work. 121 00:10:53,670 --> 00:10:55,050 Done is negative. 122 00:10:55,480 --> 00:11:01,830 OK, so there's a standard way to define the work done by gravity, and that is, what, a minus sign 123 00:11:01,830 --> 00:11:02,910 in front of the IMG? 124 00:11:03,930 --> 00:11:11,550 But you should just be sure that it makes sense if the object moves in the same direction as the force, 125 00:11:11,550 --> 00:11:15,680 then the work that you get out of that equation has to be positive. 126 00:11:15,700 --> 00:11:17,320 So just double check that always. 127 00:11:18,720 --> 00:11:25,140 So if we write down the principle of work and energy with gravity as a force and is one half of one 128 00:11:25,140 --> 00:11:31,710 squared minus N g y two minus one is equal to one half in the squared. 129 00:11:33,010 --> 00:11:41,170 Now we can have different forces doing work on a body at the same time, like we saw in the bungee jumping 130 00:11:41,170 --> 00:11:44,730 example, we've got the spring force of the bungee robe and you have gravity. 131 00:11:45,550 --> 00:11:51,610 So in the next video, we're going to look at an example where a spring falls and the force of gravity 132 00:11:51,610 --> 00:11:52,600 works in on a body. 133 00:11:52,960 --> 00:11:57,130 And we want to calculate the final velocity of the body.