Hey, guys. So in this video, we're going to talk about problems where we have multiple wheels or objects like wheels such as cylinders, discs, or gears. And when we have multiple of these things chained together connected to each other by either a chain or a belt much like a bicycle. Let's check out how these work. So these problems where we have two, it could be more than two, but it's almost always two, wheel-like problems. So when I say wheels, I mean things like discs, cylinders, etc. They're pretty common in rotational kinematics. So, let's check them out. There are 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. So if they start spinning, they're not going to move sideways. Okay? So in this case, we have
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Intro to Connected Wheels: Study with Video Lessons, Practice Problems & Examples
In problems involving connected wheels, such as gears or pulleys, the tangential velocity at the edge of one wheel equals that of another. This relationship is expressed as
Intro to Connected Wheels
Video transcript
Speed of pulleys of different radii
Video transcript
Alright. So here we have 2 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, 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 not, when they're fixed in place is that r1ω1 equals r2ω2. So we're supposed to use r, which is the distance to the center. Now 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, which that's what's going to be most of the time. Okay? So if you're not sure, you can, pretty safely guess that that's what it is, but the problem should tell you. Okay. So that means I'm going to have big R1ω1, big R2ω2. It says you pull down the other end causing the pulleys to spin. So if you're going to pull down this way, this guy is going to spin with ω1 and this guy is going to spin with ω2. And then it says, when the cable has a speed of 5, what is the angular speed of each? So when this cable has a v equals 5, what is ω1 and what is ω2? Okay. And what I wanna 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 as the velocity at any point here. So we can write that vcable2 is vcable1. So vcable, which is 5, is what equals r1ω1 and equals r2ω2. Okay? And that's what we're going to use to solve this question. So if I want to know what is ω1, I can look into this part of the equation right here. Okay. So to solve for ω1, I'm going to say 5 equals r1ω1. So ω1 is 5 divided by 0.3, and 5 divided by 0.3 is 16.7 radians per second. And to find ω2, same thing, 5 equals r2ω2. So ω2 is 5 divided by 0.4, which is 12.5 radians per second. Okay? So that's it for ω2, ω1. The key point that I want to highlight here is that not only are these velocities the same at the edge, which allows us to write that r1 equals 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. Okay? So please remember that just in case you see something like it. Alright? So that's it. Let me know if you have any questions.
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More setsHere’s what students ask on this topic:
What is the relationship between the tangential velocities of two connected wheels?
The tangential velocities at the edges of two connected wheels are equal. This relationship is crucial in problems involving gears, pulleys, or any connected rotational systems. Mathematically, this is expressed as:
Since the tangential velocity
How do you calculate the angular speed of a larger wheel connected to a smaller wheel?
To calculate the angular speed of a larger wheel connected to a smaller wheel, you use the relationship between their radii and angular speeds. The formula is:
Solving for the angular speed of the larger wheel
For example, if the smaller wheel has a radius of 2 units and an angular speed of 40 radians per second, and the larger wheel has a radius of 3 units, the angular speed of the larger wheel is:
What are the different ways to describe how quickly something spins?
There are four main ways to describe how quickly something spins:
- Angular Velocity (
): Measured in radians per second (rad/s). - Frequency (
): Measured in Hertz (Hz), it represents the number of rotations per second. - Period (
): Measured in seconds (s), it is the time taken for one complete rotation. It is the inverse of frequency: .Math input error Math input error Math input error Math input error - Revolutions Per Minute (RPM): It represents the number of rotations per minute. It is related to frequency by:
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How do you convert angular velocity to frequency and period?
To convert angular velocity (
Frequency:
Solving for
Period:
The period is the inverse of frequency:
Combining the two equations, we get:
What is the significance of the equation r1ω1=r2ω2 in connected wheels problems?
The equation