Practice: While you drive, your tires, all of radius 0.40 m, rotate 10,000 times. How far did you drive, in meters?
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|Rotational Position & Displacement||29 mins||0 completed|
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Concept #1: Rotational Position & Displacement
Hey guys! In this video we're going to start talking about rotational motion also known as rotational kinematics. I'm going to take you back all the way to the beginning of physics and we're going to look at basic motion, one-dimensional motion, and we're going to contrast and compare the two. YouÕre gonna see that a lot of things are similar but there's some differences. Let's check it out. Rotational motion is when you have motion around a central point. Imagine you have a tiny object that spins like this around a central point. It forms a circular path. That's one type. The other type is when you have a cylinder that spins around itself. You may remember that we described your location using position which is variable X. We used to call this simply position but now that we have linear and rotational motion, we may want to specify that this is the linear position. If you don't see the word linear, you will assume it's linear. The other one is going to be rotational position, which is describing where you are in a circle. For example if you are moving in a circle, think of this as a track and you can only move around that track either this way or this way. You're trapped. There's two ways you could describe your position. You could do it by saying this is coordinates x,y and then if I move over here, I have a new x and a new y. It's a two-dimensional grid. It's a surface so you could do that. The problem is that's more complicated than it needs to be because now I have x's and y's changing. What's actually easier is to use a single variable, a single number to describe where you are. We do this using angles. For example you may remember that this is 0 degrees and this would be 90 right there. We're going to say that this is, just make it up something, 80 degrees. That's easier because I'm using a single number to represent where you are around the circle. That's what rotational position is and it uses the variable _. You might remember that _ is what you use to represent angles or degrees. Now notice here that I have the words rotational and angular. I need you to know that these words basically mean the same thing. They're used interchangeably. You see a lot of words like angular velocity. It just means rotational velocity. These words are used interchangeably. Whereas in linear motion, you used X. In rotational motion, we used _. What I'm going to say here is that X becomes _. The X equivalent in rotation is _. Later on youÕre gonna get some equations where old equations but instead of using X, we're going to use _. Let's look a little deeper into the differences between the two. Position is defined and how far you are from the origin. You may remember this. It's your distance from the origin. Rotational position is the same thing. ItÕs how far you are from the origin. The difference is that the first one we measure using meters and the second one we measure using angles. You could do either radians or degrees, but we're going to use radians most of the time. We're going to use radians, which is abbreviated rad. Origin, if you remember, origin is simply where x = 0. For example, let's say we got a line here and you are here and then there's two points. Let's draw three points here. Just two points, whatever, that's fine. Let's say that these two points are 10 meters apart. This would be 0 and then this would be 10 and maybe you are at 7. If this is x = 0, this is where the origin is. But we could have done this a little bit different and then we say that your position Xyou is +7. But we could have put the origin right here. We could have arbitrarily said I want this to be x = 0, and then this distance here is 93, so your X, Xyou, would have been negative 3. The point that I'm trying to make here is that origin in linear position is arbitrary. Arbitrary meaning up to you. You can change it and sometimes the problem will tell you, but it could change. It doesn't have to be a fixed thing. In rotation, itÕs a little bit different. In rotation origin is still where position, in this case theta, equals zero Ð zero degrees or zero radians. Let me put a little meters here. 0 degrees or 0 radians. The difference is that whereas here it's arbitrary, it's up to you unless the problem tells you, in rotational position the origin is always fixed. It's always fixed. ItÕs fixed at the positive x axis. Zero is always here. Remember the unit circle. This is always the origin. That's non-negotiable. Whereas here you could put it whatever you want if you're given that kind of liberty in a problem. The last thing is direction is also arbitrary. You could say that this is positive or you could say that this is the direction of positive. Either-or works and then you adjust accordingly. If you're in rotation, direction is fixed. Clockwise which follows the clock duh, goes this way is negative, and counterclockwise which goes this way is positive. Direction here is also fixed. It's not up to you. One quick note here which is it might seem backwards and I like to think of this as backwards. Why couldn't they have made the direction of the clock positive? Why is it the clock is backwards? It's because this stuff actually follows the unit circle. You might remember that the unit circle grows like this, the angles grow like this. The unit circle and the clock are backwards from each other and we use the unit circle and that's it. Those are the key differences between linear position and rotational position. We're going to quickly talk about the displacement now. The rotation equivalent of linear displacement, position is x, displacement is changing position which is _X. Rotational position was _, so rotational displacement is simply __. Instead of _x, the equivalent is __. If you're moving this way, we measure the _x. If youÕre moving this way, you can measure your __. These two quantities here Ð _x and __ Ð are linked. They're connected. They can be converted from one to the other using the following equation: _x = r__. This r here, you can loosely refer to it as radius, I'll talk about this a little bit more. But what it really is is radial distance which is distance to the center. Radius would be the radius of a cylinder, but if it's a distance of points spinning around the circle, then you're talking about distance to the center. That's a technicality. Don't worry about that too much. You might have seen this. You might remember this equation. You've seen this before. In math this looks like this: s = r_. In fact most textbooks, I think every textbook actually, talks about this equation like this. But I like to use _x instead of s because that's what you used to and __ because we're looking at the displacement. This is the arc length equation and that's where this stuff comes from. I'm going to use this version right here and it should be fine. Quick points about this equation, really important equation. This equation speaks radians. What do I mean by speaks radians? If you're plugging in a __ into this equation, we'll do an example just now. But if you're plugging in a __, that number that you're plugging into the equation has to be in radians otherwise the equation doesn't work. Also instead of plugging in __, you're solving for __, the answer will be in radians. Either you're giving the equation radians or if the equation is giving you an angle, it's giving you that in radians that's why I say here that the input must be in radians. You have to plug in in radians for the equation to work, and the output will be radians. If you get an answer out of that equation, if you get a delta theta out of the equation, you'll be in radians. What the hell is a radian? One radian is approximately 57 degrees. 57 degrees is somewhere around here, somewhere in the first quadrant so that's what roughly one radian is. It's just a different way of measuring angles. To convert between radians and degrees, you just have to remember that 360 degrees equals 2¹. Most people remember that. What a lot of people don't realize is that the unit for ¹ is radians that's why this conversion works. ¹ is 3.1415 radians. Another way you can do this is just by saying ¹ radians = 180 degrees. IÕm gonna do a quick example. We have an object that moves along a circle of radius 10. Let's draw this. I got a tiny little object. It spins around a circle here and it has around a circle of a radius of 10 meters which means that the radial distance if you go around a circle of radius 10, it means a radial distance to the middle is 10. It says here that you start at 30 above the x-axis and then you go all the way to 120 above the x-axis. Let me draw another circle here just so I can put the angles. 30 is somewhere here, you start here. Then remember this is 0, 30, this is 90, so 120 will be somewhere here. You're going from something like this from here to here. We want to know what is your angular displacement. Angular means rotational. I'm asking what is your __. Very straightforward definition of __ is __ is _final Ð _initial. Then the angles are 120 minus 30, so this is just 90 degrees. I have to be very careful. If I had a negative here, like 45 down here, I'd have to plug it in as a negative okay and that makes it a little bit different. I just have to be careful. That's the answer for Part A. It just asked for angular displacement. It didn't say if I wanted it radians or degrees, so degrees is fine. Then for Part B, it wants linear displacement. Linear displacement is _x. I just showed you how I can connect _x and __ so we're basically converting from one to the other. _x is r__. I have r. r is 10 meters. __ is 90 degrees. Here I hope you're saying no, it's not. This is wrong. You're supposed to use radians. I want you to actually write this out and then cross it out so you remember not to do this. It has to be in rad. What we're going to do is weÕre gonna quickly convert the two. 90 degrees I converted it using this ratio here, which means I'm going to put them in a fraction. I'm going to say over here I have degrees at the top so I want degrees at the bottom, 180 degrees. Then up top I have _ radians. Then what happens is the degree symbol cancels and I'm left with just radians. Then you just multiply this in the calculator. YouÕre going to put 90*_ / 180. If you do that, I have it here. Actually I have it here. It's _/2. That's a little cleaner way of doing and then the other version is 1.7. They're both radian. Now I can plug this in here. 10*1.57 and the answer will be 15.7 meters. Why is it meters? Because meters is the unit of _x. That's it for this one. Hopefully it makes sense. Let me know if you guys have any questions.
Concept #2: Displacement in Multiple Revolutions
Hey guys! Now we're going to talk about rotational displacement when you go around the circle multiple times. Let's check it out. If you make one full revolution Ð revolution is a complete circle, a complete cycle, a complete rotation, a complete spin there's all these words Ð if you make a full one around a circle, you've gone a total change in angle of 360 degrees. Everyone knows that a full one is 360 degrees or 2¹ radians. Therefore _X, remember the _X equation is _X = r__. That's how you link these two variables. It becomes 2¹r. What I've done here is I replaced __ with 2¹ because that's what __ is if you make a revolution. You might recognize this equation that _X = 2¹r. This is the circumference equation. The definition of circumference is the size, the length of the border of a circle, and that's linear distance. If you're driving your car around a circle, your odometer which tells you how far you've traveled, would give you a quantity equal to 2¹r. That's the linear distance as you go in a rotational motion around a circle that's if you spin once. If you spin once, you get 360. What happens if you spin twice? Then you get 360*2. If you spin any n times, then your __ is 360*n or 2¹*n, obviously in radians. Instead of having a _X of 2¹r, you get a _X of 2¹r * N, where N is the number of rotations. Two more things you may need to know. If you want to know how many revolutions you go through and we'll do example of this just now, all you got to do is divide your number of angles by either 360 degrees or by 2¹. For example if I tell you I spun 720 degrees and I want to know how many revolutions that is, you divide that by 360 and you get 2. That means that I spun twice. Same thing with if it's in radians, if it's in ¹, you can just divide it by 2¹. The last thing is let's say you go around the circle many times and you end up over here. If I want to know how far you end up, all you got to do is you keep subtracting by 360 interior angles less than 360. For example if you spun 410 degrees and I want to know how far from zero you end up, 410 is more than 360 so you make multiple revolutions. All I got to do is subtract by 360 and you see that the answer is 50. You keep doing this until your final answer is less than 360, which it is so we're good to go. If it wasn't you would subtract by 360 again. Same thing with radians. If it's in radians, you just keep subtracting by 2¹ until the answer is less than 2¹. Let's do a quick example here. Starting from zero degrees, itÕs rollerball, a circle, starting from here. 0 degrees is always the positive x-axis. You make 2.2 revolutions. We're using the letter n to represent the number of revolutions, 2.2 around a circular path of radius 20. If you have a circular path of radius 20, that means that your radial distance from the middle, little r, is 20. You could use little r or big R interchangeably. Little r is technically more correct because big R is reserved for the radius of like a disc. Little r is the distance from the center, but the words are used kind of interchangeably. What is your rotational displacement in degrees? Rotational displacement is __ and we want that in degrees. I'm going to put a little deg here to indicate that we want to do this in degrees and not in radians. If you spin once, you spin 360 degrees but if you spin 2.2 times, you just multiply them. This is going to give you 792 degrees. 792 is how much you spun. That's it. For part B, how many degrees from zero are you? Again, we're just going to subtract 792 until we get to a number that's less than 360. I want to know how far from zero. 792 Ð 360, that gives you 432. We're going to have to keep going because we're not below 360, minus 360 and then finally that's the answer, 72 degrees. That's the final answer. For part C, what is a linear displacement? Linear displacement remember is _X. If I want to know _X, _X is r__, r is the distance here 20, and __ has to be in radians. I cannot use 792. I cannot use that. I'm going to have to use in radians. In radians this is going to be 2¹*2.2. 2¹ is a full rotation times 2.2 because we rotated 2.2 times. If you multiply all of this, put it in the calculator, ¹ is 3.1415 or your calculator has a button for that. If you do all this, you get the distance is 276 meters. That's it for this. Hopefully it made sense. Let me know if you guys have any questions.
Practice: While you drive, your tires, all of radius 0.40 m, rotate 10,000 times. How far did you drive, in meters?
Practice: An object moves a total distance of 1,000 m around a circle of radius 30 m. How many degrees does the object go through?
BONUS: How many complete revolutions does it make?
Practice: A car travels a total of 2,000 m and 1140° around a circular path, starting from 0° . What is the radius of the circular path?
BONUS: How far (in degrees) from 0° does the car end up?
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