Hey guys, we're now going to talk about bicycle problems, which are a special type of connected wheel problems that we've seen. However, bike problems are a bit more complicated. Let's check it out. It says here bike problems are a bit more complicated and that's because they have more parts. They have 5 in total, in fact. In the simplest example we've covered so far, you have 2 cylinders, or 2 wheels connected to each other. So that's 2 parts. Bike has as many as 5. Now the reason I say as many as 5 is because most bike problems aren't going to talk about all the 5 parts. They may talk about 2 or 3, but we don't know which ones. So I'm going to cover all of them. Okay? So let's check it out. What we're going to do is we're going to see some relationships between the pieces. So the pedals, number 1, the pedals, I'm going to draw them over here. Number 1, the pedals cause the middle sprocket to spin. The middle sprocket is this green over here. This is the front of the bike over here, just to be clear. This is the middle sprocket which is the gear-like thing that your pedals are connected to. Okay? I'm going to go in this order. I'm going to go pedals first, call it 1, and then I'm going to go middle sprocket second because you interact directly with the pedal. They're the first thing you touch, and the first thing that starts spinning, and it causes the other to spin. Now, these two things are spinning on the same axis of rotation. They're both spinning around the same central points, and one causes the other to spin. So we're going to say that ω 1 ≡ ω 2. Okay? That's the first one. I'm going to put it here as well, ω 1 = ω 2, so it's more visual. The chain, there's a chain that connects both sprockets. I'm going to draw a little chain here with green. The chain connects both sprockets. What that does, that means any point here is going to have the same linear velocity as any point here. Okay? So if I write this, let's call this number 3 over here, again going in a sequential order, as number 1 caused number 2 to spin, which caused number 3 to spin. Points 2 and 3 over here have the same tangential velocity at the edge of the sprocket, so I can write that vt 2=vt 3. But remember, the tangential velocity at the edge of a circle is vt is rω, if you are a distance r. Okay? In this case, you're always going to be a distance r, which will be the radius because, in bicycles, the chain is always on the very outside, on the very edge of the circle. So r will always be the radius. So I'm going to write R 2ω 2 = R 3ω 3. That's the second relationship you need to know. The third thing that happens is the back sprocket, number 3 over here, is connected to the back wheel. The back wheel is the blue one. Let's call that number 4. And again, it follows a sequence, number 1 caused number 2 to spin, which caused number 3 to spin, which causes number 4 to spin. Okay. Now these two here have a similar relationship as these two. Okay. Number 1 and 2 are on top of each other. Number 1 causes number 2 to spin. They spin together around the same central axis. It's the same thing that happens here between 3 and 4. Okay. So I'm going to say that ω 3, which is the back sprocket, equals ω 4, which is the back wheel. Okay. Let me also add here that these two points here are related and I can write, this is what I wrote over here, R 2ω 2 = R 3ω 3. I want to have them in diagram here, so it's nice to see, but I also want to have them here so it's a little bit more organized. So here we're going to write that ω 3 equals ω 4. Cool. So these are the three relationships you have, in bikes. Now the last thing we want to talk about is if the bike is not free to move, and it wouldn't be free to move if the wheels are not touching the ground. Right? So if you lift your bike, for example, the wheels don't touch the ground. So if you spin the pedal, the back, the back tire, the back wheel will spin, but the front wheel is not going to spin 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. You rotate the pedals, which make the middle sprocket rotate, which makes the back sprocket rotate, which in turn makes the back wheel rotate, and because 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 wheel basically just spins because the bike is already moving and you're touching the floor over here. Alright, 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 vfront, and ωfront, which, if you remember, we numbered at number 5, ω 5 are both going to be 0. The middle and back sprockets have a diameter of 16 and 10. You spin the pedals at 8. You want to know the angular velocity, which is ω 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 pedals. So this problem is actually including all 5 elements. Again, usually, you won't see that but we're doing this here for completion. The back sprocket is always smaller. The middle sprocket is bigger, but then obviously the tire, the wheel on the back is even bigger. So you got the little pedals here. Okay. Pedals 1, sprocket 2, 3, and wheel 4 in sequence. Okay. These guys are connected. You don't necessarily need to do the whole thing, but I want to do it, just so we get in the habit. The front wheel doesn't matter. It's not part of this thing. It's not going to do anything. Okay. So I know that the pedals spin at 8 radians per second. Radians per second is ω. So this means that ω 1 is 8. That's given. Okay. Part A is asking for the middle sprocket. The middle sprocket is ω 2. Remember, 1 and 2, they go together. In fact, it's the same number for omega. K. So I'm just going to put here that ω 2 is 8 as well, because they rotate together. Very easy. For part B, what about the back sprocket? Well, I want to know ω 3. And what I know about ω 3 is there's a way to connect the back sprocket to the middle sprocket by using this equation right here. Okay. So that's what we're going to do. R 2ω 2 ≡ R 3ω 3. Okay. So ω 3 becomes, R 2ω 2 divided by R 3. Now before we plug in numbers, notice that we were given diameters instead of radii. So we're just going to convert to 2. Remember, in physics, you're always going to use radius and not diameters. So we were given that the middle has a diameter, so d 2 of 0.16, which means r 2 is 0.08. And then the back sprocket over here, d 3 is 0.10. So r 3 is 0.05. Now, technically, if you plug in diameters, right, if you did this, d and d, it would have worked because they would have canceled. 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 going to do that. Alright? So even though it would have worked. So this is going to be 0.8. ω 2, we just found here is 8, and then this divides by, 0.05. All right. And if you do this, you get 12.8 radians per second. For part C, we want to know the back wheel. So we want to know what is ω 4. So what do you think ω 4 would be? What's special about 4? And how does 4 connect to 3? They're on top of the same thing. They spin together. So ω 4 is the same as ω 3, so it's 12.8. And lastly, the front wheel doesn't spin at all, so I hope you thought ω 5 is 0. Okay. That's it for this one. Let's do the next one.

# More Connect Wheels (Bicycles) - Online Tutor, Practice Problems & Exam Prep

### Bicycle Problems (Static)

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### RPM of pedals of static bicyle

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Alright. So here, you lift your bike slightly and you begin to spin the back wheel. Since the bike is lifted, spinning the back wheel will not cause the front wheel to move or to spin. The middle and back sprockets have diameters D and 2d. So let's draw that real quick. I got the back sprocket, which is the smaller one, and I got the middle sprocket here. I'm going to also draw the wheel, and I'm going to draw the pedal just in case. The pedal is 1, which causes middle sprocket 2 to spin, which causes the back sprocket 3 to spin, which causes the back wheel 4 to spin. I'm given the diameters here, and the diameters of the middle sprocket, D₂ is 2d, and of the back sprocket, d₃ is d. Now I don't know the value of d, but I do know that 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 going to change this into radius. Radius 2, I'm just going to write this as 2r and radius 3 as r. Now radius is just half the diameter. You basically be dividing both of these guys by 2. I can do r and 2r 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, with an RPM, at x RPM, in other words, if RPM of the back wheel, which is 4, is x, what will be the RPM in terms of x for the pedals, which is 1. Okay. So we're going all the way from 4 to 1. Typically, you spin 1, which causes 2 to spin, which causes 3 to spin, which causes 4 to spin. But this whole thing is connected, so there's not necessarily a sequence. You could spin 4, and then it goes all the way and causing 1 to spin. Okay. So we have to be able to trace a connection between these. Well, remember, these two guys are connected, these two guys are connected, and these two guys are connected. Let's write those, connections. So between 1 and 2, the connections that they have the same omega, ω₁ = ω₂. But in this problem, we don't have omegas. We have RPMs. So let's change that. And I want to remind you that we can write the relationship between them like this. So ω is 2πf, but f is RPM over 60. So let's do that here. f is RPM over 60. Now if I plug this on both sides, look what I get. I get ω₂π RPM₁ over 60 equals 2π, RPM₂ over 60. What that means is that I can just cancel everything and I'm left with RPM equals RPM. That's the first relationship. Okay? That RPM₁ equals RPM₂ because they spin together. The relationship between 2 and 3, let's put this over here, the relationship between 2 and 3, is that they have the same v's. They're connected. So v₂ = v₃, the tangential velocity, which means you can write this as r₂ω₂ = r₃ω₃. Here, you can do a similar thing where you replace ω with 2πRPM over 60. The 2π and the 60 will cancel on both sides. So this becomes just r₂RPM₂ = r₃RPM₃. Okay. So that's the second relationship. And the third relationship here, it's kind of squeezed this here. Sorry about that. It's the relationship between 3 and 4. 3 and 4 spin on the same axis of rotation, so ω₃ = ω₄. And as I've done here, we can just rewrite this as RPM₃ = RPM₄. Okay. What we're looking for is RPM₁, which is right here. And what I have is RPM₄, which is right here. Okay. RPM₄ is x. So we're going to try to connect them using these three equations in green. RPM₄ is x. Therefore, RPM₃ is x as well. So this guy here is x. What I'm going to do is solve for RPM₂ because RPM₂ is the same as RPM₁. So it comes down to this equation here. I'm going to rewrite this as r₂. Instead of RPM₂, I'm going to write RPM₁ because they're the same, and this is what I'm looking for, equals r₃. And RPM₃ is what I know, which is x. Okay. And I want the answer to be in terms of x. So RPM₁ = r₃x/r₂. Let me disappear here. And r₃ is r. r₂ is 2r times x. The r's cancel and you end up with x/2. x/2. And what that means is that basically the pedals will spin at half the RPM of the back wheel. Okay? Now we solve this sort of, well, not sort of. We solved 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 a wheel has double the radius or double the diameter, it's going to have half the speed. The bigger you are, the slower you are. Okay. The smaller you are, the faster you go. But that relationship only applies between the two cylinders. So you could have thought, if this guy is x, then this guy is x. This guy here is bigger, double the size, so it's going to be x/2. And therefore, the pedals must be x/2 as well. Okay? So the back wheel is 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 going to be half the speed, half the RPM, and then these two guys have the same. That might have been a little bit easier to do 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 it makes sense. Let me know if you guys have any questions.

### Bicycle Problems (Moving)

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Hey guys, we're now going to look into bicycle problems where the bike is actually free to move. So the wheels are touching the floor. So as they spin, it also causes the bike to move sideways. Check it out. So moving bikes, yay. But first, I want to remind you what happens if the bike doesn't move. It's not free to move sideways. Okay. If the bike doesn't move when the wheels spin, you have what's what we call a fixed axis or a fixed wheel. And this means that the velocity at the center of mass of the wheel will be 0. Okay? Neither wheels will spin. Additionally, the both the velocity in the front wheel and the ω in the front wheel will be 0. Remember the ω 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 spin. But the front wouldn't spin unless you spin the front yourself. Okay? Alright.

Now, if the bike is moving, we have a free axis, which is a situation where you have both ω and v, ω and v. So it's sort of like a toilet paper that's rolling around the floor. Okay. In this case, because the bike is one unit, the back wheel and the front wheel are moving sideways together. They don't become farther apart; they move together. But that means is that this velocity here, velocity of the center of mass here, I'm going to clean it up so it's not a math problem, and the velocity of center of mass here are actually the same. Okay? And that's what typically a problem would call the velocity of the bike. If the problem says the bike moves at 10 meters per second, this means that this moves at 10 and this moves with 10 this way. Okay? Let's clean that up so we don't make a huge mess. Alright.

Now remember, for a free axis, which is this situation here, we have that this velocity right here, we have the velocity of the center of mass is rω, where r is the radius of the wheel. So this relationship here can be rewritten. If VCM = rω, then VCM = rω. Now in this case, I'm going to write front, front, back, back. Okay. So we can write those two. Now for most bikes, the front wheel and the back wheel are supposed to have the same radius, 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. And if that's the case, we can say that ω so basically what happens is if these two r's are the same, these two guys would cancel. Right? Okay. So if r_{front} = r_{back}, the r's would cancel and you have that ω_{front} = ω_{back}. So not only do they have the same v, but they have the same ω. Okay.

So let's sort of recap here. You have pedal, 1, sprocket 2, back sprocket 3, back wheel 4, front wheel 5. And the relationships are that these two guys spin on the same axis, so the ω1 = ω2. These two guys spin on the same axis. So ω3 = ω4. The chain that connects these two makes it so that their v's are the same. So I can say that v2 = v3. And what this means is that I can write that r2ω2 = r3ω3. Okay. And this is really the important one here. That's the useful one. Okay. The first thing is just to get to that. Alright. Boom. And then the last relationship here, which this is old stuff by the way, the new thing here is that there's also a relationship between the front wheel and back wheel, which is this right here. Okay. So I'm going to write that r4ω4 = r5ω5. And obviously, if the r's are the same, they cancel so ω4 = ω5 becomes the same. Cool? So this is how a moving wheel works. The only new thing if the wheel is moving is this. I'm going to put a little plus here to indicate that this is what's new. Okay. Maybe I can put a little new here, 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 a radius of 0.66, both of them. Okay. So let's draw both wheels. And then it says, if you ride with 15, so that's v_{bike} = 15, calculate the linear speeds of 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 going to call this just for the sake of simplicity, radius here, so that's good, 0.66, 0.66. And we want to know what is the linear speed of the center of mass. So I want to know what is VCM1 and what is VCM2. VCM of any wheel that moves while rolling is rω. So VCM1 = r1ω1 And VCM2 = r2ω2. But the key thing to remember here, there are 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 with 15 to the right, both wheels move with 15 to the right. So what I'm going to do is I'm going to 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 ω1 and what is ω2. Well, if you look at this equation, I can use this here to solve. Okay? So it's just basically plugging into the equations. Let's do that. So ω1 will be 15 divided by r1 or 15 divided by 0.66. And the answer to that is 22.7 radians per second. Secondly, we will have the same ω because it's the same numbers. So I have ω2 = 15 divided by r2. r2 is the same, 0.66. So the answer is also 22.7 radians per second. Okay. So that's the answer for parts b. Now just to recap again, what happened here? I told you the velocity of the bike was 15. So automatically, you would know that the velocity of the wheels, the linear velocity of the wheels at the center of mass, the middle of them is, 15 as well. Once you know that this is 15 and you have the radius of both wheels, you can just plug it into that equation, and solve for ω. Very straightforward. Cool. That's it for this one. Let's do the next example.

### Angular speeds of moving bicyle

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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. And I got the middle and back sprockets. I'm giving the radii here. So little guy, middle guy. This, and then you got the pedals here. Okay. If you read, the question doesn't actually mention the pedals, but I'll put it here just so that we get in the habit of doing this. 1, the middle sprocket 2, back sprocket 3, back wheel 4, and this is 5. Okay. The wheels have radius, so r 4 = r 5 = 0.70. The middle sprocket and the back sprocket, middle and back are 15 and 8. So middle is 2, r 2 = 0.15 and r 3 = 0.08. Okay. If you ride with 20, this means that v b i k e = 20. Let's say you're going that way, which means that v c e n t e r o f m a s s 5 = 20. Let's get this out of the way. And I'm going to draw this, now I'll just draw it here. I'm going to write it up here, that v c e n t e r o f m a s s 4 = 20 as well. Remember, if you move at 20, the center of mass of the wheels is going to move at 20 as well. Okay? So we want to calculate the angular speed, omega, of the front wheel. The front wheel is 5. Okay. How do we get this? Well, I know the radius and I know the VCM. Okay. Remember, when you have a wheel that's free, you have that VCM of that wheel is r Ω. Here we're talking about 5. So I'm gonna put 5 here, 5 here, 5 here, and I want to find Ω5. So Ω5, I have these two numbers, so it's just a matter of plugging it in. VCM is 20, and the radius is 0.7. K? And if you do this, the answer is 28.6 radians per second.

B, what about the back wheel? Well, the back wheel, it's going to be the same exact thing because the numbers are the same. So what is Ω4? Well, Ω v c m 4 = r 4 Ω4. The radius and the VCM are the same. Right? It's moving with 20 and the radius is point 7, which means Ω4 will be the same, 28.6. If you calculate, you get the same number. Okay. For part C let me get it out of the way. For part C, we want to know what is, the angular speed of the back sprocket. Now remember, the back sprocket has the same angular speed as the back wheel. So we've already calculated this basically. Ω3 is the same as Ω4, So it's also 28.6 radians per second. So, so far, these first three things all have the same omega. And then for part D, what about the middle sprocket? Let's give ourselves a little bit more room here. Sorry about that. I'm going to sort of go backwards here. I want to know what is Ω2. Well, I just found 3. 2 is connected to 3 using this equation, r 2 Ω2 = r 3 Ω3. So if I want to find this, I just have to move things around. So r 3 Ω3 ÷ r 2 . r 3 = 0.08 right here. Ω3 = 28.6, and r 2 = 0.15. And if you calculate everything here, multiply this whole thing, you get 15.3 radians per second. Alright. So that's it for this one. Hopefully, it makes sense. Very similar to the static bike, but you just have this additional thing where the wheels now both have the center of mass, a velocity of the center of mass, and there's this new equation, that we have to take care of. Alright? That's it for this one. Let me know if you guys have any questions.