Rotational Velocity & Acceleration

Hey guys! Now that we've seen how rotational position and displacement relate to linear position displacement, we're going to look into velocity and acceleration. Let's check it out. The rotational equivalence of linear velocity and linear acceleration are rotational velocity and rotational acceleration or angular velocity and angular acceleration. Similar to how x becomes _ and delta _x becomes __ in rotation, V and a will take different letters as well. Average velocity if you remember is _x / _t and the units are meters per second. Average acceleration or I should say acceleration is _V / _t, change in velocity over change in time, and it's measured in meters per second squared. That's if you are dealing with linear problems. If you have rotational problems, angular motion problems, instead of V we're going to use w. w (omega) is a Greek w, so it's like a curly w. ItÕs essentially w. Instead of _x / _t, remember instead of _x we now __. W is __ / _t. V is how quickly I can get from here to here. It's a measurement of how quickly I move between two points. w is a measurement of how quickly you spin in a circle. Remember also that this was in meters and this is in radians. Instead of meters, you have radians. Instead of meters per second, you're going to have radians per second. The acceleration is very similar. Instead of a we're going to have _ which is a Greek a, and same thing here, acceleration is velocity over time. Acceleration would be angular or rotational velocity over time, so itÕs __ / _t, and it's radians per second squared. You might start to see a pattern. The variables are x or _x, and then V and a. They become _, w and _ and the pattern is that English letters are representing linear motion and Greek letters are represent the angular or rotational motion. These are all Greek letters Ð _, _ and _. _ is a way to indicate how quickly something spins. There are actually three additional variables that will help us describe how something moves and they're related to w, in fact theyÕre all related to each other. You might remember I showed you right here we just did w is __ / _t. You might remember that we talked about if you have a complete revolution, then your __ is 2¹. The entire angle for a complete revolution is 2¹. The time that it takes for a full cycle is called period t. One way that you can rewrite w is not just __ / _t, but also 2¹/t. Remember also that period and frequency are inverse of each other. Instead of 2¹ / t, I could also write this as 2¹f. You have w is a measurement of how quickly something spins, period is a measurement of how quickly something spins and frequency is also a measure of how quickly you spin and they're all related by this equation. Last one we're going to talk about is rpm. rpm stands for revolutions per minute so 1 rpm is 1 revolution per minute. A Hertz which is a unit of frequency, frequency is measured in Hertz is one revolution per second. You can see how these two are related. For example if I tell you something spins with 120 rpm or at a rate of 120 rpm, rpm is simply 120 revolutions per minute and what I can do is I can say I can put a minute up here and convert this into seconds by divided by 60. One minute equals 60 seconds, I can do this. Look what I end up with. 120 divided by 60 is 2. I put 2 revolutions per second. Revolutions per second is frequency. Revolutions per second right here is frequency. If you have rpm, you can convert to frequency by dividing by 60. I'm going to quickly write another equation here which is that frequency is rpm over 60. We have a way to connect all of these guys. w, T, f and rpm are all connected. Typically what you're going to do is we're going to convert from any of these 3 Ð T, f or rpm Ð back into w using these equations. The idea you see more of this later is most of the equations I give you will have w but not any of the other letters. I have a little diagram to connect all of these things. If you have rpm, you will want to convert it to frequency. The way you do this is by using f = rpm / 60. That's how you go from rpm to frequency. Then from frequency, you can convert into either period using the fact that period is the inverse of frequency or you can convert into w using the fact that w is 2¹f. Obviously you can convert it backwards in any direction. Generally you want to end up here, but you might have to go from let's say w to rpm. We'll do some of this stuff. These are the 4 units to tell you how quickly something spins and you may have to convert among them. IÕm trying to highlight this. There you go. One last point before I do an example and I've mentioned this briefly before. Rotational equations which is what I'm showing you will feel already by now. They work for two different situations Ð one is when you have a point mass. A point mass is a tiny object of negligible size that spins around the circular path. We call it a point mass because we just represented it by a point. It has no volume. I'm going to say that the radius of this object is 0. Imagine a sphere with radius. It has no radius. It has no volume. Radius is zero. You can also write if you want volume = 0. IÕm actually gonna write out volume so you don't think it's velocity. It's a point. It could be a small object that we simplified down to a point, so that's point mass. The other one is when you have a rigid body which is something where the radius is not zero. The radius is greater than zero so it has volume. I refer to this as either a rigid body, that's sort of the classic textbook name, or a shape. The reason why I like to think of it as a shape is because in these problems usually we'll be told what the shape is. If I tell you have a small object, that's a point mass, and if I tell you have a cylinder usually I'll tell you that it's what the radius is and then you treat that a little bit separately. You can either have a point mass, IÕm gonna draw a tiny little M going around a circle. The circle has a radius r, in this case little r is the distance from the to the edge of the circle and you are going around at the edge. R is the radius of the circle and then little r is how far you are from the center. Those are the same thing or you can have like a cylinder for example spinning around itself. Let me draw a little cylinder here, it looks like a cake, and a cylinder that rotates around itself and that cylinder has a radius of R. You can have either one of these two situations. We'll look at both a lot. That's a quick intro. I gave you some equations how to connect things together between these four different variables and weÕre now gonna do a problem. I have a 30-kilogram disc, so mass = 330, radius = 2. Let's draw a disc, a circle, the radius is 2. This is a disc. It's a rigid body or a shape. The radius of this thing is 2 and it rotates at a constant 120 rpm. I'm going to write here that rpm equals 120. We want to know its period and angular speed. For Part A, I want to know what is big T and then for Part B I want to know what is w, which is w, angular speed same thing angular velocity same thing as rotational velocity or rotational speed. How do we tie it together? If you look at this little diagram here, we can convert from one to the other. I'm going to convert rpm into frequency and then I'm going to convert from frequency into period and into w. First weÕre gonna go frequency = rpm/60, so it's 120 divided by 60 is 2. The units are Hertz but I don't actually want frequency, I want period. Period will be 1/frequency, _ = 0.5. Period is measured in seconds. To find angular speed w, I just used that w is 2¹ / T or I can use that w 2¹f. I'm going to use 2¹f because it's more straightforward. 2¹ frequency is 2, so I get 4¹ and if you multiply all of this, the answer is 12.6 radians per second. That's it for this one. Let's do the next one.

Here we have an example. WeÕre calculating the rotational velocity of the earth as, in two different situations, as it rotates around itself and rotates around the Sun. LetÕs look at the first one, Earth's around itself. Actually I want w of Earth around self. w is either __ / _T or 2_ / t or 2_f. As the Earth rotates around itself, what's the __, what's the time? I'm not telling how many degrees it's going through. I'm not telling you this either. One of the things you can do is you can then use some information you know about the earth. You know how long the period of the earth is, how long here it takes to make a complete revolution around itself. If itÕs a complete revolution around itself, that takes 24 hours. It takes 24 hours, so all I can do is I can just write w = 2_ / 24 hours. Obviously I can't just leave it as 24 hours, I have to convert that into seconds. It's going to be 2_ / 24*60*60 and you multiply all of this, you get that the w for the earth is 7.27 x 10^-5 radians per second. It's a very small number, and that's because the earth takes a very long time to spin around itself relative to like a DVD or something. B, what is the rotational velocity of the earth rotating around the Sun? Very similar problem. w equals, I can use any of these equations. In this case what I'm going to use is again 2_ / T, and the period of the earth going around the Sun is 365 days. You don't have to worry about leap years or any of that stuff. Just simplify it by doing this. It's 2_ / 365. It's basically all of this, this is one day, 24 hours. It's all of this times 365. I'm going to repeat the bottom there. This is going to be an even smaller number. I get 1.99 x 10^-7 radians per second. The last thing I want to point out to you guys is that if the earth is rotating around itself, in this case we're talking about a rigid body that's spinning around itself because it's sort of a sphere not exactly but you could approximate it as a sphere that spins around itself. The Earth rotating around the Sun actually acts as a tiny point object. Why? Because relative to the size of the Sun and relative to the distance between the two, the earth is so tiny that you would consider it a point mass going around the circle. Here you have a rigid body spinning around itself. That's an important distinction in these two examples. An object can depending on the situation be thought of as a rigid body that spins around itself or a tiny point mass that spins around a circular path. That's it for this one. Hopefully it makes sense. Let me know if you guys have any questions.