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 rfront = rback, the r's would cancel and you have that ωfront = ωback. 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 vbike = 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.