**Concept:** Intro to Connected Wheels

Hey guys! In this video we're going to talk about problems where we have multiple wheels or objects like wheels such as cylinders or discs or gears and when we have multiple of these things chained together, connected to each other by either a chain or a belts much like a bicycle. Let's check out how these work. These problems where we have two, it could be more than two but it's almost always two wheel-like problems. When I say wheels, I mean things like discs, cylinders, etc. They're pretty common in rotational kinematics. Let's check them out. There's two basic cases and we have a case where the wheels are rotating around a fixed axis. In other words you have something like these two wheels here. Imagine that there's a chain around them. But this wheel was bolted let's say to the wall and this wheel was bolted to the wall. If they start spinning they're not going to move sideways. In this case we have w because it's going to spin but the wheel itself, there's a w. Let's call this w1 and then there will therefore be obviously w2 here. But this wheel will have no velocity of the center of mass. This point doesn't move sideways. It does not move sideways. Same thing with this point, Vcm2 doesn't move sideways. There's no velocity at the center of mass. An example of this is if these were pulleys or gears that are like I said attached to the wall. The example I gave you earlier is a toilet paper that's fixed to the wall and it rotates around itself like toilet paper usually works. The other one is a static bicycle. There's two basic ways you can have a static bicycle. One is your lift your bicycle from the floor and then if you spin the wheels, guess what? The rotation of the wheels doesn't cause the bike to move sideways because it's basically now fixed, it's not free to move. The other way is if you flip your bike upside down. Basically if the wheels are not touching the floor, it's a free axis, itÕs a free wheel. Then another cases we're going to have situations where two wheels are connected and they're both free to move. That's a bicycle right. As the wheels spin, they spin together. We're going to look into that later. The big thing to remember, the big thing to know here is that whenever a chain connects to two wheels or connects two wheels, we have that the tangential velocity at the edge of one wheel equals the tangential velocity at the edge of the second wheel. Let's pick a point here and a point here and I can say that this VT equals this VT. They're all the same. VT1, VT2. VT2 is the same as VT1. In fact you could pick a point anywhere and because it's the same chain the tangential velocity has to be the same. Remember also that when you have a fixed disk, let me write this here. For fixed axis, remember that VT = rw. The tangential velocity at a point away from the center, VT is given by rw. In this case we're talking about 1 so itÕs r1w1. This guy here for example would be r2w2. The VTÕs are the same, so that's good, but the rÕs are different. r1 and r2 are different, the wheels have different sizes. Therefore because these guys are different, then the wÕs will be different as well. The rÕs are different so the angular speeds will be different. In fact, the greater my r, the smaller my omega and vice versa. If you got a tiny wheel and a big wheel, the tiny wheel will spin much faster while the big wheel is slowly spinning. There's an inverse relationship there. V1 equals V2 and V equals rw so we can write this. We can write that VT1 = VT2 so r1w1 = r2w2. This is the big equation for this video, the most important one. What I want to do is I want to show how thereÕs three variations of this equation and that's because we not only have w or Omega to talk about, to describe how quickly something spins, we also have frequency, period, and RPM. All four of these are useful in describing how quickly something spins. I'm going to replace w with f, t and RPM and this is going to generate different versions of this equation. w is 2¹f, it is also 2¹ / t. Remember, frequency is RPM / 60. I can also say that w is 2¹ and instead of f, I'm going to say RPM / 60. Can you see that? Yes you can. We got these versions and what I'm going to do is I'm going to replace w here with this, this and this. r1 instead of w I'm going to write 2¹f1 and then r = 2¹f2. What happens is the two ¹'s cancel so you're left with r1f1, r2f2. If I were to do the same thing with these other guys here, I'm going to do this one more time. If you want to do this with t, you would get r1(2¹ / T1) = r2(2¹ / T2). The 2¹Õs cancel again and we're left with, let me put it over here, r1/T1 = r2/T2. Then the last piece is with RPM and I would do the same thing. I'm going to just skip here for the sake of time but I'm going to say that you can have r1 RPM1 = r2 RPM2. The most important of these four is the first, the other three are just sort of derivative equivalent similar versions of them. But I find that if you know all four of them, you're best suited to solve these problems very quickly. These problems are very straightforward. This is the basic idea when you have two connected wheels. These four equations will work for you. Let's do a quick example here. I got two gears of radius 2 and 3. Let's draw this here, r1 = 2, the little one. Then a bigger one, r2 = 3 and they are free to rotate about a fixed axis. Little tricky here. Remember I talked about fix axis and free axis so even though I use the word free here, they are free to rotate about fixed axis. These are fixed, which is what we've been talking about which means the velocity of the center of mass is zero. What it also means is that r1w1 = r2w2 and all the variations. It says here when you give this smaller pulley or gears, let's just rewrite this. These were gears not pulleys. When you give the smaller gears 40 radians per second, it doesn't tell me which way so I'm just going to spin it this way. I'm going to say that w here is 40 radians per second. I want to know what is the angular speed so this follows this way and it spins this way. If this is w1, what is w2 that the larger one will have? Very straight forward. These are two connected cylinders so all I gotta do is write the equation for them. r1w1 = r2w2. We're looking for w2 so w2 is r1w1 / r2. Just solve the equation. This is 2/3 and w1 is 40 so the answer is going to be 26.7 radians per second. That's it for this one. Very straightforward. Let's check out the next example.

**Example:** Speed of pulleys of different radii

Here we have two pulleys with radii 0.3 and 0.4, so notice that this one's a little bit smaller so r1 is 0.3 and r2 is 0.4, attached as shown. A light cable runs through the edge of both pulleys. Light means the cable has no mass, runs through the edge of both pulleys. The equation for connected pulleys when they're fixed in place is that r1w1 = r2w2. We're supposed to use r which is the distance to center. When it says that the cable runs through the edge of both pulleys, the word edge here tells us that the distance to the center in this case happens to be big R, the radius, that's what's going to be most of the time. If you're not sure, you can pretty safely guess that that's what it is but the problem should tell you. That means I'm going to have big R1w1 = big R2w2. It says you pull down the other end causing the pulleys to spin. If you're going to pull down this way, this guy is going to spin with w1 and this guy is going to spin with w2. Then it says when the cable has a speed of 5, what is the angular speed of each. When this cable has a V = 5, what is w1 and what is w2? What I want to remind you is that the velocity here is the same as the velocity here, which is the same as the velocity here which is the same velocity at any point here. We can write that V cable is VT1 = VT2. V cable which is 5 is what equals R1w1 and equals R2w2. That's what we're going to use to solve this question. If I want to know what is w1, I can look into this part of the equation right here. To solve for w1, IÕm going to say 5 = R1w1, so w1 is 5/0.3 and 5 divided by 0.3 is 16.7 rad/s. To find w2, same thing. 5 = R2w2, so w2 is 5/0.4 which is 12.5 rad/s. That's it for w1 w2. The key point that I want to highlight here is that not only are these two velocities the same at the edge, which allows us to write that R1 = R2, but also that they equal the velocity of the cable that pulls them. That's what's special about this problem. It's this blue piece right here that equals the velocity of the cable as well. Please remember that just in case you see something like it. That's it. Let me know if you have any questions.

In cars, a gear train (shown in the figure) is used in the transmission to allow the wheels of the car to rotate at high angular speeds while at low engine rpm's, allowing for less gas to be used. If the drive gear in a gear train is connected directly the engine, and the driven gear directly to the wheels (meaning both rotate with their respective gears), what gear ratio would allow the car to move at 60 mph while the engine rotates at 1500 rpm? Note that the definition of gear ratio is the driven gear radius : drive gear radius. Consider this for wheels with a 45 cm diameter.

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On a byclice, the pedals rotate a gear, which I will call G1. A second gear, G2, spins the back wheels, and it moves via a chain that connects the chain as shown in the figure. If G1 is rotating at 100rpm, how fast will G2 be rotating? Note: R1=10cm and R2=5cm

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