Practice: A disc of radius 10 m rotates around itself with a constant 180 RPM. Calculate the linear speed at a point 7 m from the center of the disc.
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Concept #1: Converting Between Linear & Rotational
Hey guys! You may remember that one of the very first things I showed you in rotation is how it can connect linear displacement _x and angular or rotational displacement __ using a tiny equation. There's two more equations that we can use to connect velocity and acceleration between linear and rotational. Let's check it out. We have these tiny equations that are going to link, that are going to connect, that are going to allow us to convert from one to the other between linear and rotational. Linear, we're also going to refer to linear as tangential. Linear or tangential, both of these are going on a straight line. Then I connect a linear or tangential to rotational which is also referred to as angular. It's important you know that these words mean the same thing. The linear variable is x and the rotational equivalent is __. From that we get that _x is the change in position, the change in x and __ is a change in angular or rotational position _. The way that _x and __ connect is by this equation right here, we've used this. Similarly, V connects to its angular equivalent, w, using a very similar equation. _x is r__ and V tangential. This T here means tangential velocity is rw. I want to point out that there's a pattern here. This is the linear variable r and the rotational variable. Same thing here, linear variable, r rotational variable. I'm going to remove all these little circles so it's not messy. Simply with a and _. a is going to be r, and if you see the pattern, the equivalent of V is w. The equivalent of a is _. This is also the tangential acceleration. These are the two new equations that we're going to be able to use. When do they come up? Usually it's on a problem like this. You have a disc and this disc spins with angular speed w. If you pick a point in this disc, a point here, and I want to know what is the velocity, the linear velocity of this point. This point moves with linear velocity or tangential velocity that looks like this, VT. You might remember that when I have a point going around the circle, the point has tangential velocity and it also has centripetal acceleration. It turns out that VT is connected to w by this equation. VT = rw which is a very useful relationship equation. Let's keep going. I want to quickly mention that there are four types of acceleration. I already mentioned two here. We have ac and actually I already mentioned 3. We have ac, we have at, and we have _. There's a fourth one but we're going to talk about that later. I want to just be very clear here that this equation right here at = r_ refers to the tangential acceleration. It doesn't refer to the centripetal acceleration. It doesn't refer to the angular or rotational acceleration. There's four types of acceleration. Most of them have two names so it's going to be a mess but I'll show you pretty soon. A few more points here. Whenever you have a rigid body or a shape, let's say this is a cylinder that spins around itself, all rotational quantities __, w and _ are the same at every point. Let me show you this real quick. Let's imagine a line here and imagine this is a huge disk and there's people on top of it or whatever. You have a guy A over here on that point and guy B is over here. Imagine that this disc spins from here to here, to that point right there. Guy A is going to be here and guy B is going to be here. Notice how they all spin on the same line. If I'm here and you're here and this spins, we're still in the same place. We're moving together. Our __, our change in angle will be the same. By the way, this happens even if we're not in the same line. It's just easier to see it if it's the same line. __ is the same. Because w is defined in terms of __ / _t, w is also going to be the same. Since w is the same, _ depends on w. All these three things are the same. Long story short, if you're in a circle, all the objects on top of a circle have the same __ as they move. They're going to experience the same _ and the same w. All the rotational quantities will be the same. However, the linear speeds might be different since they depend on r which is radial distance or distance to the center. That's another way to think about it. They might be different. The best way to illustrate this is by doing an example. We can do a very straightforward. I have a wheel of radius 8, so let's draw this here, put a little radius here. Radius of this wheel is 8 meters. It spins around its central axis. What that means is that imagine a circle and imagine a sort of an invisible line through the circle and it's free to spin around that invisible line. I'm going to draw this here. You don't have to draw it. I'm going to delete it. Imagine the imaginary line that goes through this thing almost as if you stuck a thing through it and then it's free to spin around that. That's what that means. Let's get this out of here. Basically it spins around its center which is how these things always work at 10 radians per second. That's our w is going to be 10 radians per second. We wanna know the angular and linear speeds at different points. I want to know at a point in the middle of the wheel on the central axis, so we're going to call this point one at a distance 4 meters from the center. If the radius is 8 meters, 4 meters is halfway in. I'm going to draw this here. This is point two and at the edge of the wheel, point three. What we want to know is we want to know v1, v2 and v3 and I want to know w1, w2, and w3. That's what it says. I want the angular which is w and linear v speeds at these three points. First thing is to realize that all these points have the same because they're on the same disc, they have the same w and that w is the same w as the disc. That's the first part, w1 = w2 = w3 = w disc. This is more of a conceptual to know if you know that. All of these would be 10 radians per second so please remember all of them are the same and they are the same as the disc. If you're on top of a disc, you're spinning with the disc. You have the same w. What about v1, v2, v3? This is going to be a little bit different. The point the tangential or linear velocity of an object on a disc is given by V tangential. V1 tangential which is rw, in this case r1w1. All these objects have V2T. IÕm gonna write this for all of them, r2w2, and V3T is r3w3. Now all these objects have the same w, but they have different rÕs which means they're going to have different VÕs. Let's calculate this real quick. The first r here is how far from the center is that point. Remember r is the distance to the center. The first point is at the center, at the middle of the wheel. What's the distance from the center to the center? Zero. r1 is actually 0, so it doesn't really matter what this is, the answer will be 0. We'll talk about that in a second. Let's go to the next one. r2 is at a distance of 4 meters from the center, so this is 4 and w is 10, so the answer is 40 meters per second and this is at a distance 8, it's at the edge, w is 10 so this is 80 meters per second. These are the answers. Let me talk about it real quick. If you're at the edge, you move faster. Think about you and a friend inside of a carousel that's spinning. If you're at the edge, you are going to feel faster, you are in fact moving faster on the linear direction. Anyway you have a harder force, a stronger force pulling into the middle. If you're at the edge of a spin, you are faster. If you are dead at the center, if you could be in a center of a carousel and it's spinning, you're basically doing this. You're rotating in place and you have no V. Velocity is when you're moving sideways, when you're spinning in place that's w. If you're spinning in place as opposed to spinning sort of youÕre just doing this, spinning in place, as opposed to spinning like this. You spin around yourself so you have no velocity only w. If you're at the edge, you are faster. That's it. Let's try the next example.
Example #1: Length of string rotating point mass
Here we have a small object that rotates at the end of a light string. Here's a string and you got a small object, so IÕm gonna do this and it's spinning like this. Imagine that if you spin a string, it forms a circular path. The object reaches 120 rpm from rest in just 4 seconds. I'm giving you a ton of information here. First you started at rest so w initial is zero. You reach the final rpm of 120 and you did this in just 4 seconds. It also says here that tangential acceleration, tangential acceleration remember is at, after the 4 seconds is 15 m/s^2 and we want to know what is the length of the string. We haven't talked about the length of string yet but I hope you can figure out that the length of the string is this distance here. It's the radius of the circle that forms the circular path you get or it is the radial distance from the center of rotation which is your hand and the edge of rotation which is where the object is. Little r, the distance to the middle is your length. Essentially what we're looking for is little r. Think of it as little r or not as L because there's no LÕs in any of these equations so you're not going to find out. This is a little bit of a mess because we're going to have to use the combination of equations here. If you look through all the equations we've used so far, you might first think about one of the three or four motion equations. You might think of that because I gave you w initial is 0. I gave you _t. I gave you rpm which we can convert to w final. If you do that, you're going to have three out of five variables once you convert. However notice that if you look through all those four equations, there are no rÕs in them. You're not going to be able to solve for r by doing this. If you look a little further, I do have an equation that I gave you recently that links at with r. That equation is at = r_. I know at so all I have to do is find _. r = at which is 15 right there divided by _. What we're going to have to do is find _ and plug in here. How do I find _? _ is one of my five variables of motion so I'm going to be able to use these three guys to find _, and that's what we're going to do now. First I'm going to convert from rpm into w final. Remember that w final is 2¹f and F is rpm/60. 120/60 is 2 therefore w final is 2¹ and instead of f I'm going to put a 2 which is 4¹. I'm going to rewrite this here just to clean it up. w initial = 0, w final is 4¹, _t is 4. We're looking for _ and the ignored variable is the __. Notice how I know three things. The ignored variable is __. The only equation that doesn't have __ is the first one out of the four. w final = w initial + at. If we're looking for _, we just got to move everything out of the way. Initial is zero so _ is w final / t. w final is we found it here, itÕs 4¹ divided by time, time is 4. 4 cancels with 4 and alpha is 3.1415 rad/s^2. All you got to do is plug in this number here and we're good. 15 divided by 3.14 and if you divide the two you get that r is 4.77 meters and that is the final answer. ThatÕs it for this one. It's an interesting question that combines these two equations. The basic idea here is just that old school physics hassle of you got stuck in one and you're going to have to go find the other and just kind of work your way through it. There's not a very clear path. There's a few different ways you could have done this. But the most important thing is try to figure out what's the equation that has my variable and then look for all different ways to find all the letters, all the variables, you have to solve for. That's it for this one. Let me know if you guys have any questions.
Practice: A disc of radius 10 m rotates around itself with a constant 180 RPM. Calculate the linear speed at a point 7 m from the center of the disc.
Practice: A rock rotates around a light, 4-m long string. The rock is initially at rest, but reaches 150 RPM in 3 seconds. Calculate its tangential acceleration after 3 s.
BONUS: Calculate its tangential speed after 3 s.
Practice: A 4 m long blade initially at rest begins to spin with 3 rad/s2 around its axis, which is located at the middle of the blade. It accelerates for 10 s. Find the tangential speed of a point at the tip of the blade 10 s after it starts rotating.
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