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.