1

concept

## Bicycle Problems (Static)

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Hey, guys. We're now gonna talk about bicycle problems, which are special type of connected wheel problems that we've seen, however, bike problems a little bit more complicated. Let's check it out. So it says your bike problems a bit more complicated. That's because they have more parts. They have five in total. In fact, um, in the simplest example we've covered so far, you have two cylinders or two wheels connected to each other. So that's two parts, by case, as many as five. Now, the reason I say as many as five is because most bike problems I'm going to talk about all the five parts. Uh, they may talk about two or three, but we don't know which ones. Someone to cover all of them. Okay, so let's check it out. What we're gonna do is we're gonna see some relationships between the pieces. So the pedals number one, um, the pedals. We're gonna draw them over here. Um, number one. The pedals caused the middle sprocket to spin middle sprocket. Is this green? Over here. This'll is the This is the front of the bike over here. Just to be clear, um, this is the middle sprocket, which is the gear like thing that your pedals air connected to. Okay, um, I'm gonna go in disorder. I'm gonna go pedals first called one, and then I'm gonna go middle sprockets second, because you interact directly with the pedal. That's the first thing you touch. The first thing that starts spinning and it causes the other one to spin. Now, these two things are spinning on the same axis of rotation. They're both spinning around the same central points on day one causes the other to spend. So we're gonna say that w of one has to equal w of the other. Okay, so I'm gonna say w one equals W two. Okay, Um, that's the first one I'm gonna put it here is Well, w one equals W two. So it's more visual. The chain there's a chain that connects both sprockets. I'm gonna draw a little chain here with green check and export sprockets. What that does that means that any point here is going to have the same linear velocity. A Xeni point here. Okay, so if I write this, I get that, um, let's call this number three over here again. Going sequential order number one, cause number two Just spin, which caused number three to spin. Um, points two and three over here have the same tangential velocity at the edge of the sprocket. So I can write that VT one. I'm sorry. VT two equals VT three. But remember, the tangential velocity at the edge of a circle is a potential lawsuit. The edge of a circle is VT. But over here, bt is our omega. If you are a distance, are okay. In this case, you're always going to be a distance. Are will be the radius because in bicycles, the chain is always on the very outside on the very edge off the circle. So our little are will always be the radius. So I'm gonna write Big are to Omega two equals big are three omega three. That's the second relationship you need to know. The third thing that happens is the back sprocket Number three over here is connected to the back wheel back wheels. The blue one. It's called it a four. And again it follows the sequence Number one cause number two to spin, which caused number three to spin, which causes number four to spin. Okay. Now, these two guys here have a similar relationship as these two guys. Okay, one and two are on top of each other. One causes to to spin. They spend together on the same central access. It's the same thing that happens here between three. And four. Okay, so I'm gonna say that Omega of three, which is the back sprocket, equals omega four, which is the back wheel. Okay, let me also add here that the these two points here are related. And Aiken, right? So what I wrote over here are to Omega two equals R three, omega three. I wanna have them in the diagram here, so it's nice to see, but I also wanna have him here. It's a little bit more organized. So here, we're gonna write that Omega three equals Omega form. Cool. So these are the three relationships you have, um, and bikes. Um, Now, the last thing we want to talk about is if the bike is not free to move on, but it wouldn't be free to move if the wheels are not touching the ground. Right. So if you lift your bike, for example, that we don't touch the ground. So if you spend the pedal, the back, the back tire, the back wheel will spin. But the front wheels not gonna spend because there's nothing connected to it. Okay, so in this case, the front wheel doesn't spin. So this is a special situation, which is when you have bicycles that are static. Okay, that's what we're looking into right here. Static bikes. You could lift the bike or you could flip it upside down. There's nothing connecting to the front wheel. The front wheel really only spins. If you are touching the floor, um, you rotate this pedals which make the metal sprocket rotate, which make the bottom of the back sprocket rotate, which in terms makes the back we rotate. And because, um, that causes the bike to go forward, which means that this will spin as well. So this is actually the last thing that happens. The front well, basically just spends because the back is already moving and you're touching the floor over here. All right, so that's it for that. Let's quickly do an example and see what we get here. So you turn your bike upside down for maintenance. This means that the bike won't move. So I'm already thinking. Okay, Bike doesn't really move. So v front in omega front, which, if you remember, we number that number five will make a five are both going to be zero. The middle and back sprockets have diameter 16 and 10. You spend the pedals at eight. You wanna know the angular velocity, which is w for all of these guys. So here it's talking about the middle sprocket, the back sprocket, the back wheel, the front wheel and the pedal. So this problems actual, including all five elements again, usually won't see that. But we're doing this here for just for completion. The back sprocket is always smaller. The middle sprocket is bigger. But then obviously the tire that we on the back is even bigger. So I got the little petals here. Okay, Pedals, one sprockets too. Three and wheel four in sequence. Okay, these guys are connected. You don't necessarily gonna do the whole thing, But I wanna do it just to get in the habit. The front wheel doesn't matter. It's not part of this thing. It's not gonna do anything. Okay, So I know that the the pedals spin a eight radios per second. Radiance per second is Omega. So this means that omega one is eight. That's given. Okay. Part eggs. Asking for the middle sprocket metal sprocket is Omega too, Remember, one and two are they go together. In fact, it's the same number for Omega. Okay, so I'm just gonna put here that omega two is eight as well because they rotate together very easy for part B. What about the back sprocket? Well, I wanna know Omega three. And what I know about Omega three is there's a way to connect, uh, back sprocket to the middle Sprocket by using this equation right here. Okay, so that's what we're gonna do. Um, are to w two equals R three W three. Okay, so w three becomes are two w two. So are too. W two divided by r three. Before we plugging numbers notice that we were given diameters instead of radio, So we're just gonna convert the to remember, in physics, you're always gonna use radius and not diameters. So we were given that the middle has a diameter so d to of 16 which means are too is 0.8 and then the back sprocket over here D three is 10. So are three is 0.5 now, Technically, if you plugged in diameters, right, if you did this D and D, it would have worked because they would have canceled. Um, but just to develop the habit of always switching to radius, just in case you can't really switch, you can't really use diameter. So we're gonna do that. All right. So even though we would have worked So this is gonna be 0. w two we just found here is eight. And then this divides by 0. All right. And if you do this, you get 12. 12.8 regions per second for part C. We want to know the back wheel. So we want to know what is w four eso. What do you think w four would be what's special about four. And how does four connect to three? They're on top of the same thing. They spin together. So w four is the same as W three. So it's 12.8, and lastly, the front wheel doesn't spin it all. So I hope you thought Do Omega is zero. Okay, that's it for this one. Let's see the next one

2

example

## RPM of pedals of static bicyle

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Alright, So here you lift your bike slightly and you begin to spend the back wheel. Since the bikes lifted, the spinning the back wheel will not move, will not cause the front wheel to move or to spin the middle and back Sprockets have diameters d and two D. So let's draw that real quick. I got the back sprocket, which is a little one, and I got the metal sprocket here. I'm gonna also draw the wheel and I'm going to draw the pedal just in case pedals one which causes metal sprocket to to spin which caused the back spark sprocket three to spin which caused the back wheel four to spend. I'm giving the diameters here and the diameters of the middle sprocket d two is two d and of the back sprocket d three is deep now I don't know the value of tea, but I do know that the the middle one is twice the radius or twice the diameter of the back one. Now, we don't really use diameters in physics, so I'm gonna change this into radius radios. Tombs gonna write this to our in the radius three. Our number radius is just half the diameter. You'd basically be dividing both of these guys by two. Um, I could do our and to our instead, as long as this number is double this number. We're good. Okay. So I want to know if you spin the back wheel right here. Um, with an rpm at x r p m. In other words, if rpm of the back wheel, which is four is X what will be the rpm in terms of X for the pedals, which is one. Okay, so we're going all the way from 4 to 1. Um, typically, you spin one which causes to to spin, which causes three to spin, which causes for to spin. But this whole thing is connected, so there's not necessarily a sequence you could spend four, and then it goes all the way and causing one to spin. Okay, so we have to be able to trace a connection between these. Well, remember these two guys were connected. These two guys were connected and these two guys were connected. Let's write those connections. So between one and two, the connections that they have the same omega Omega one equals omega too. But in this problem. We don't have Omega's. We have RPMs. So let's change that. And I want to remind you that we can write the relationship between them like this. Omega is two pi f. But F is our PM over 60 So let's do that here. F is our PM over 60. Now. If I plug this on both sides, look what I get. I get W two pine. Our PM 1/60 equals two pi our PM to over 60. What that means is that I can just cancel everything and I'm left with our PM equals R P M. That's the first relationship. Okay, that rpm one equals R P m two because they spin together the relationship between two and three. Let's put this over here. The relationship between two and three is that you have. They have the same V's. They're connected. So the two equals V three. The tensions velocity, which means you can write this as our to Omega two equals R three Omega three. Here you can do a similar thing where you replace Omega with two pi r p m. Over 60. The two Pi MD 60 will cancel on both sides. So this becomes just are too. Our PM two equals r three r p m three. Okay, so that's the second relationship and the third relationship here, it's kind of squeezes here. Sorry about that. Is the relation between three and 43 and four spin on the same axis of rotation. So Omega three equals omega four. And as I've done here, we can just rewrite this as our PM three equals R p m four. Okay, What we're looking for is our PM one which is right here. And what I have is our PM four, which is right here. OK, Our PM four is X. So we're gonna try to connect them using, um, these three equations in green rpm forests X therefore, rpm three is X as well. So this guy here is X. What I'm going to do is solve for r p m two because our PM two is the same as our PM one. So it comes down to this equation here. I'm gonna rewrite this as our two instead of rpm to I'm gonna write rpm one because they're the same. And this is what I'm looking for. Equals R. Three and our PM three is what I know, which is X, Okay? And I want the answer to be in terms of X. So rpm one equals are three X over our to let me disappear here and or three is our are two is to our times x the arse cancel and you end up with X over to X over to. And what that means is that basically the the pedals will spend on behalf the rpm off the back wheel. Okay, now we solve this sort of well, not sort of would solve this mathematically, but it might have been easier to actually just kind of think about this stuff. Okay, Now interesting. Here is this equation. This is a linear relationship, and what that means is that if two wheels, if a wheel has double the radius or double the diameter, it's goingto have half the speed. The bigger you are, the slower you are. OK, the smaller you are, the faster you go. But that that that relationship only applies between the two cylinders. So you could have thought If this guy's ex, then this guy is X. This guy here's bigger double the size that could be X over to and therefore the pedals must be X over to as well. Okay, so the back wheels X, which means the back sprocket has to be X. When I cross it over to the other side, it's doubled the radius. So it's gonna be half the speed, half the rpm andan. These two guys have the same. That might have been a little bit easier to dio, um, so that you don't run the risk of getting confused with the math and all the equations. Whatever you prefer. Alright, that's it for this one. Tricky question. Hope makes sense. Let me know if you guys have any questions.

3

concept

## Bicycle Problems (Moving)

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Hey, guys, we're not gonna look into bicycle problems where the bike is actually free to move. So the wheels air touching the floor. So as they spin it also caused the bike to move sideways. Check it out. Um, so movie bikes. Yea. But first, I want to remind you what happens if the bike doesn't move. If it's not free to move sideways. Okay, if the bike doesn't move when the wheel spin, you have what's what we call a fixed access or fixed wheel. Andi, this means that the velocity of the center of mass off the wheel will be zero. Okay, neither wills or will spend. Additionally, the both of a loss in the front wheel and the omega in the front wheel will be zero. Remember the omega in the back wheel? The back wheel could be spinning because you could lift the bike and move the pedals, and then that caused the back to spend. But the front wouldn't spend unless you spend the front yourself. Okay. All right. Now, if the bike is moving, we have a free access, which is a situation where you have both Omega and the Omega Envy. So It's sort of the toilet paper that's rolling around the floor. Okay, in this case, because the bike is one unit, the back wheel in the front wheel are moving sideways together. They don't become farther apart, they move together. What that means is that this velocity here, the loss of center mass here when I clean it up. So it's not a mass in the velocity center mass here are actually the same. Okay. And that's what typically ah, problem would call the velocity the bike If the problem says the bike moves the 10 m per second. This means that this moves with 10. And this moves with 10. This way. Okay, let's clean it up so we don't make a huge mess. All right, Now, remember, for free access, which is this situation here we have that This velocity right here we have the velocity center mass is, um, our omega. Where are the race of the wheel? So this relationship here can be rewritten if V C. M is our omega, then v c m equals V. C. M is gonna be our omega equals are omega. Now, in this case, I'm gonna write front front back back. Okay. So we can write those two now for most bikes. Um, the front wheel on the back wheel are supposed to have the same radios, same diameter. Now, the reason I say most is because you could get a physics problem that doesn't have it that way. That's not really supposed to be like that, but they could give you one of those. Um, And if that's the case, we can say that's Omega. So basically, what happens is if these two are there the same, these two guys with cancel, Right? Okay, So if our front equals are back, the ours would cancel. Then you have that Omega front equals Omega back. So not only do they have the same V, but they have the same w. So let's sort of recap Here. You have pedal one sprocket too back, sprocket Three back. Well, four front wheel five. And the relationships are that these two guys spinning the same access. So the omega is the same. Omega equals omega, too. Um, these two guys spend the same access, so omega three equals omega four. The chain that connects Thies too. Make it so that's their V's are the same. So I can say that V two equals V three. And what this means that I can write that are to Omega two equals R three omega three. Okay. And this is really, um, the important one. Is this one here? That's the useful one. Okay, the first this is just to get to that. All right. Boom. And then the last relationship here, which this is old stuff, by the way, Um, the new thing here is that there's also relationship between front wheel and back wheel, which is this right here. Okay, so I'm going to write that are four omega four equals R five omega five. And obviously, if the art of the same they canceled. So Omega four is Omega five becomes the same. Cool. So this is how a moving wheel works theme on Lee New thing. If the wheels moving is this, I'm gonna put a little plus here to indicate that this is what's new. Okay, back and put a little new here. Um, and then obviously that this guy would actually move. Okay, This is now actually touching the floor. Let's do an example. So it says here the wheels on your bike have radius 62 both 66. Both of them. Okay, so let's draw both wheels. Um And then it says if you ride with 15 so that's the bike equals 15. Calculate the linear speeds, the center of mass of both wheels and the angular speed of both wheels. So we're not talking about pedals or sprockets or anything. Just these two wheels. I'm gonna call this just for the sake of simplicity. I only have two things, So I'm gonna do R one and R two. Now I'm giving the radius years. That's good. 0.66 0.66. And we want to know what is the linear speed of the center of mass. So I wanna know what is V. C. M. One and what is V c m. Two v c m of any wheel that moves while rolling is our omega. So V C M one is our one omega one. Um, and V C m two is our to Omega too. But the key thing to remember here, there's two things to remember. These two wheels move together, so these numbers are actually the same. Okay, also, they're also both 15. Okay, remember if the bike moves of 15 to the right, both wheels move with 15 to the right. So what I'm gonna do is I'm gonna do this. I'm gonna say this equals 15 and this equals 15. Okay? And that's the answer to part A Is that both of these guys equal 15 now for part B, I want to know what is Omega one and what is Omega Shoe? Well, if you look at this equation, I can use this here to solve, okay? And so it's just basically plugging into the equations. Let's do that. So Omega one will be 15 divided by R one or divided by 150.66. And the answer to that is 20 to radiance per second. Second, we will have the same omega because it's the same numbers. So I've Omega two equals 15 divided by r. Two arches, the same 20.66. So the answer is also 22.7 radiance per second. Okay, so that's the answer for parts. Be now just to recap again. What happened here? I told you the velocity bike was 15. So automatically you would know that the boss of the wheels. The linear velocity. The wheels at the center of mass in the middle of them is 15 as well. Once you know that this is 15 and you have the radius of both wheels. You could just plug it into that equation, um, in solve for w Very straightforward. Cool. That's it for this one. Lets the next example.

4

example

## Angular speeds of moving bicyle

4m

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Alright. So here you have the wheels on your bike with radius 70. Both of them. So let's draw that real quick. Yeah, and e got the middle BRAC sprockets. I'm giving the radio, I hear so little guy, middle guy this and then you got the pedals here. Okay, if you read the question, doesn't actually mention the paddles that I put it here just so that we get in the habit of doing this one. The middle sprocket to back sprocket three back we or four. And this is five. Okay, the wheels have radius. So r four equals R five, which is 50.70 the middle sprocket in the back sprocket middle and back or 15 and eight. So middle is to r two equals 20.15 and r three equals points 08 Okay, if you ride with 20 this means that the bike equals 20. Let's say you're going that way, which means that V center of mass five equals 20. Let's get this out of the way on. I'm gonna draw this, um e just drive here. I'm gonna write it up here. That the center of mass of four, which is the back wheel. He's going to equal 20 as well. Remember, if you move in 20 the center mass of the wheels Are you gonna move with 20 as well? Okay, so we want to calculate the angular speed omega of the front wheel. Front wheel is five. Okay. How do we get this? Well, I know the radius, and I know the V c m. Okay. Remember when you have a wheel that's free. Um, you have that. The c m of that wheel is our omega here. We're talking about five. So I'm gonna put five year, five year, five year, and I wanna find Omega five. So, omega five, I have these two numbers, so it's just a matter of plugging it in V C. M is 20 and the radius is 0.7. Okay. And if you do this, the answer is 28 0. radiance per second. All right, B, what about the back wheel? Well, back. Well, what's gonna be the same exact thing? Because the numbers are the same. So what is Omega four? Well, omega V C m four equals R four omega four. The radius and the V C. M are the same, right? It's moving 20 and the radius 200.7. Which means Omega four will be the same 28.6. If you calculate you get the same number. Okay. For part C, can you get it out of the way for part C? We want to know what is, um the English speed of the back sprocket. Remember, the back sprocket has the same angular speed is the back wheel. So we've already calculated this. Basically, Omega three is the same as Omega Four. So it's also 28.6 radiance per second. So so far, these first three things all have the same Omega, then for party. What about the middle sprocket? Let's let's give ourselves a little bit more room here. Sorry about that. I'm gonna sort of go backwards here. I wanna know what is Omega to? Well, I just found 32 is connected to three. Using this equation are to Omega two equals R three, omega three. So if I want to find this, I just have to move things around. So our three omega three divided by r. Two are three is 0.8 right here. Omega three is 28. and R two is 0 15. And if you calculate everything here, multiply this whole thing, you get 15.3 radiance per second. All right, so that's what this one hopefully makes sense very similar to the static bike. Uh, you just have this additional thing where the wheels now both have center of mass velocity, center of mass. And there's this new equation. Um, you know, we have to take care of All right, that's it for this. Well, I mean, if you guys have any questions