1 00:00:02,000 --> 00:00:07,520 Let us look at how we would apply Newton's second law in cylindrical coordinates now cylindrical coordinates. 2 00:00:08,760 --> 00:00:15,320 Is a system that works very handy when you have a particle that moves in three dimensions and where 3 00:00:15,330 --> 00:00:27,090 rotation is involved, so let's say someone sliding down a water slide that spirals down around the 4 00:00:27,090 --> 00:00:28,440 pole, let's say. 5 00:00:28,680 --> 00:00:30,570 So let me explain it this way. 6 00:00:30,570 --> 00:00:38,220 By using this figure, if you have a point and a particle or let's call it a let's call it a body, 7 00:00:38,220 --> 00:00:41,250 because it has mass indicated over here. 8 00:00:41,820 --> 00:00:46,200 Then this particle can move along a curved path in three dimensions. 9 00:00:46,200 --> 00:00:53,550 And we can describe the exact position of this particular making use of three variables, namely Theta, 10 00:00:53,670 --> 00:00:55,020 R and Z. 11 00:00:56,040 --> 00:00:58,170 Now, Theta will give us the angle. 12 00:00:59,190 --> 00:01:10,020 Measured from some fixed line or will give us you can see it as the radius of a cylinder, so the radial 13 00:01:10,020 --> 00:01:17,310 direction, how far out from the center of this particle is situated and then Z gives us the height. 14 00:01:18,480 --> 00:01:25,320 Of this particle, so by specifying a value for Theta, a value for R and a value for Z, we know exactly 15 00:01:25,320 --> 00:01:27,510 where this particle is now. 16 00:01:27,510 --> 00:01:33,320 Theta is given in Radians and then R and Z, usually given in meters. 17 00:01:33,870 --> 00:01:37,940 Now, for each one of these components, we apply Newton's second law. 18 00:01:38,070 --> 00:01:43,830 As we know, Newton's second law states F is equal to M multiplied by eight. 19 00:01:44,670 --> 00:01:48,030 So we are going to have a sum of the forces in this direction. 20 00:01:48,570 --> 00:01:51,270 We're going to have a sum of the forces in the theater direction. 21 00:01:51,290 --> 00:01:54,600 We're going to have a sum of the forces in the radial direction. 22 00:01:54,600 --> 00:01:59,090 So we're going to break down any applied force onto this body. 23 00:01:59,460 --> 00:02:01,950 We're going to break it down in its three components. 24 00:02:02,130 --> 00:02:07,500 And it might be that it only is acting in one of these three or maybe all three at the same time. 25 00:02:08,130 --> 00:02:10,320 And we're going to multiply it by the mass. 26 00:02:10,650 --> 00:02:12,180 And the mass is constant. 27 00:02:12,180 --> 00:02:13,690 We assume the mass to be constant. 28 00:02:14,070 --> 00:02:20,760 So the math doesn't change for the different directions, but the force changes for the different directions. 29 00:02:20,760 --> 00:02:26,940 And therefore, the particle has different accelerations in each one of these three directions. 30 00:02:27,450 --> 00:02:34,170 And it is the formula for the acceleration in each one of these different directions that I'm going 31 00:02:34,170 --> 00:02:35,300 to give you in this lecture. 32 00:02:35,310 --> 00:02:41,040 And if you have that, you can just simply multiply it by the mass to get the force in that specific 33 00:02:41,040 --> 00:02:41,680 direction. 34 00:02:42,630 --> 00:02:49,500 So let's take Newton's second law and cylindrical coordinates, says the sum of the forces in the right 35 00:02:49,650 --> 00:02:55,260 direction, plus the sum of the forces in the theater direction, plus the sum of the forces in this 36 00:02:55,260 --> 00:03:01,290 in the Z direction is equal to the mass multiplied by the acceleration and other acceleration is the 37 00:03:01,290 --> 00:03:09,240 sum of the acceleration in the right direction, plus the acceleration of the theater direction and 38 00:03:09,240 --> 00:03:11,370 the acceleration in the Z direction. 39 00:03:11,550 --> 00:03:14,980 And for each one of these accelerations, we have a different formula. 40 00:03:15,480 --> 00:03:17,370 So four in the radial direction. 41 00:03:19,350 --> 00:03:24,060 We have multiplied by eight where A is given by our double dots. 42 00:03:24,070 --> 00:03:27,420 So the second time derivative of the radius. 43 00:03:28,620 --> 00:03:37,680 Miners are multiplied by Theatergoer Square now DataDot, that is the first time derivative of the theater 44 00:03:37,680 --> 00:03:40,920 angle, so that means it's the angular velocity. 45 00:03:40,930 --> 00:03:42,900 We also call it the angular velocity. 46 00:03:44,250 --> 00:03:53,370 Then in the theater direction we have a multiplied by a C, D as F, so that is multiplied by the acceleration 47 00:03:53,370 --> 00:03:57,090 in the theater direction is given by our theater double dot. 48 00:03:57,840 --> 00:04:04,200 So that is the second derivative of the angle plus two or dot theta dot. 49 00:04:04,950 --> 00:04:10,410 And in the direction we have, the sum of the forces in the in the same direction is mass multiplied 50 00:04:10,410 --> 00:04:17,310 by the acceleration in the same direction that is mass multiplied by Z double dot. 51 00:04:17,340 --> 00:04:21,510 So the second time derivative of the Z coordinate. 52 00:04:21,960 --> 00:04:29,520 So in a question you'll typically be given a function of time for, let's say, the Z coordinates. 53 00:04:29,520 --> 00:04:35,340 You might say that Z increases by the function two T squared. 54 00:04:35,890 --> 00:04:36,300 Right. 55 00:04:36,690 --> 00:04:44,130 And then you can take the second derivative of that and the second derivative of to the squared while 56 00:04:44,130 --> 00:04:48,160 that is four and they have a constant acceleration of four. 57 00:04:48,450 --> 00:04:54,450 So we will do an example where we calculate the different accelerations in the radio, the Theta and 58 00:04:54,450 --> 00:05:02,490 the Z direction, and then multiply it by the mass and then get the forces in each one of these directions.