Ch 10: Rotational KinematicsSee all chapters

Sections | |||
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Rotational Position & Displacement | 29 mins | 0 completed | Learn |

More Connect Wheels (Bicycles) | 30 mins | 0 completed | Learn |

Rotational Velocity & Acceleration | 22 mins | 0 completed | Learn |

Equations of Rotational Motion | 21 mins | 0 completed | Learn |

Converting Between Linear & Rotational | 27 mins | 0 completed | Learn |

Rolling Motion (Free Wheels) | 17 mins | 0 completed | Learn |

Intro to Connected Wheels | 13 mins | 0 completed | Learn |

Types of Acceleration in Rotation | 27 mins | 0 completed | Learn |

Concept #1: Bicycle Problems (Static)

**Transcript**

Hey guys! We're now going to talk about bicycle problems, which are special type of connected wheel problems that we've seen. However, bike problem is a little bit more complicated. Let's check it out. It says your bike problem is a bit more complicated, and that's because they have more parts, they have five in total in fact. In the simplest example we've covered so far, you have two cylinders or two wheels connected to each other. That's two parts. Bike has as many as five now. The reason I said as many as five is because most bike problems aren't going to talk about all the five parts. They may talk about two or three, but we don't know which ones so I'm going to cover all of them. Let's check it out. What we're going to do is we're going to see some relationships between the pieces. 1, the pedals cause the middle sprocket to spin. 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. I'm going to go in this order. I'm going to go pedals first called 1 and then IÕm gonna go middle sprocket 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. These two things are spinning on the same axis of rotation. They're both spinning around the same central points and the one causes the other to spin. We're going to say that w of 1 has to equal to w of the other. I'm going to say w1 = w2. That's the first one. I'm going to put it here as, w1 = w2, so it's more visual. The chain, there's a chain that connects both sprockets. IÕm gonna draw a little chain here with green. The chain connects both sprockets. What that does, that means that any point here is going to have the same linear velocity as any point here. If I write this, I get that let's call this number 3 over here. Again, going sequential order. Number 1 causes number 2 to spin which cause number 3 to spin. Points two and three over here have the same tangential velocity at the edge of the sprocket so I can write that VT2 = VT3. But remember, the tangential velocity at the edge of a circle is VT = rw if you are at distance r. In this case, you're always going to be at distance r will be the radius because in bicycles, the chain is always on the very outside of the very edge of the circle. Little r will always be the radius. I'm going to write R2w2 = R3w3. That's the second relationship you need to know. The third thing that happens at the back sprocket, number 3 over here, is connected to the back wheel. Back wheel is the blue one. LetÕs call that a 4. Again it follows the sequence number 1 causes number 2 to spin which causes number 3 to spin which causes number 4 to spin. These two guys here have a similar relationship as these two guys. One and two are on top of each other, one causes two to spin. They spin together on the same central axis. ItÕs the same thing that happens here between three and four. I'm going to say that w3 which is the back sprocket = w4 which is the back wheel. Let me also add here that these two points here are related and itÕs what I wrote over here, R2w2 = R3w3. I want to have them in the diagram here so it's nice to see, but I also want to have them here. It's a little bit more organized. Here we're going to write that w3 = w4. These are the three relationships you have in bikes. 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. If you lift your bike for example, the wheels don't touch the ground. If you spin the pedal, the back tire, the back wheel will spin but the front wheel is not going to spend because there's nothing connected to it. In this case the front wheel doesn't spin, so this is a special situation which is when you have bicycles that are static. ThatÕs what weÕre 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 these pedals which make the middle sprocket rotate which make the back sprocket rotate which in turns makes the back wheel rotate and because that causes the bike to go forward, which means that this will spin as well. 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. That's it for that. Let's quickly do an example and see what we get here. You turn your bike upside down for maintenance. This means that the bike won't move. I'm already thinking, the bike doesn't really move so V front and w front, which if you remember we numbered at number 5, w5 are both going to be 0. The middle and back sprockets have diameter 16 and 10. You spin the pedals at 8, and you want to know the angular velocity which is w for all of these guys. Here it's talking about the middle sprocket, the back sprocket, the back wheel, the front wheel and the pedals. This problem is actually putting all five elements. Again, you usually you won't see that but we're doing this here just for completion. The back sprocket is always smaller. The middle sprocket is bigger but then obviously the tire, the wheel in the back is even bigger. We got that little pedals here. Pedal is one, sprocket is two, three and wheel 4 in sequence. These guys are connected. You don't necessarily have to draw 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. ItÕs not going to do anything. I know that the pedals spin at 8 rad/s. Radians per second is w, so this means that w1 is 8. That's given. Part A is asking for the middle sprocket. Middle sprocket is w2. Remember, 1 & 2, they go together. In fact it's the same number for w. I'm just going to put here that w2 is 8 as well because they rotate together. Very easy. For Part B, what about the back sprocket? I want to know w3, and what I know about w3 is there's a way to connect the back sprocket to the middle sprocket by using this equation right here. That's what we're going to do. R2w2 = R3w3. w3 becomes R2w2 / R3. Before we plug in numbers, notice that we're given diameters instead of radii so we're just going to convert it. Remember in physics you're always going to use radius and not diameters. We were given that the middle has a diameter, so D2 of 0.16 which means R2 is 0.08. Then the back sprocket over here, D3 is 0.10, so R3 is 0.05. Technically if you plug in diameters, if you did this, D and D, it would've worked because they would have cancelled. But just to develop the habit of always switching to radius just in case you can't really use diameter. We're going to do that even though it would have worked. This is going to be 0.08, w2 we just found here was 8, and then this divides by 0.05. 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 w4. What do you think w4 would be? What's special about 4 and how does 4 connect to 3? TheyÕre on top of the same thing, they spin together, so w4 is the same as w3, so it's 12.8. Lastly, the front wheel doesn't spin at all so I hope you thought w is zero. That's it for this one. LetÕs do the next one.

Example #1: RPM of pedals of static bicyle

**Transcript**

Here you lift your bikes slightly and you begin to spin the back wheel. Since the bike is lifted, the spin in 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 2D. Let's draw that real quick. I got the back sprocket which is the little 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. Pedal is 1 which causes middle sprocket 2 to spin which caused the back sprocket 3 to spin which caused 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. I donÕt 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. We don't really use diameters in Physics so I'm going to change this into radius. I'm just going to write this 2R in the radius 3 = R. The radius is just 1/2 of the diameter. YouÕd 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. 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 would be the RPM in terms of X for the pedals which is 1? 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. We have to be able to trace the connection between these. Remember, these two guys are connected, these two guys are connected, and these two guys are connected. Let's write those connections. Between 1 and 2, the connection is that they have the same w, w1 = w2. But in this problem, we don't wÕs, we have rpms. Let's change that. I want to remind you that we can write the relationship between them like this. w is 2¹f but f is rpm/60, so let's do that here. f is rpm/60. If I plug this on both sides, look what I get. I get 2¹ rpm1/60 = 2¹ rpm2 / 60. What that means is that I can just cancel everything and I'm left with rpm = rpm. That's the first relationship, that rpm1 = rpm2 because they spin together. The relationship between 2 and 3 is that they have the same vÕs, they're connected so v2 equals v3, the tangential velocity, which means you can write this as R2w2 = R3w3. Here you can do a similar thing where you replace w with 2¹ rpm / 60. The 2¹ and the 60 will cancel on both sides, so this becomes just R2rpm2 = R3rpm3, that's the second relationship. The third relationship here is the relationship between 3 & 4. 3 and 4 spin on the same axis of rotation, so w3 = w4, and as I've done here, we can just rewrite this as rpm3 = rpm4. What we're looking for is rpm1 which is right here and what I have is rpm4, which is right here. 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. This guy here is X. What I'm going to do is solve for rpm2 because rpm2 is the same as rpm1. 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. Rpm3 is what I know, which is X. I want the answer to be in terms of X. rpm1 = R3x / R2. R3 is R. R2 is 2R times x. The RÕs cancel and you're left with x/2. What that means is that basically the pedals will spin at half the RPM of the back wheel. We solved this mathematically, but it might have been easier to actually just kind of think about this stuff. 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 going to have half the speed. The bigger you are, the slower you are. The smaller you are, the faster you go. But that relationship only applies between the two cylinders. You could have thought if this guy is X, then this guy is X. This guy here is bigger, double the size. ThatÕs gonna be x/2 and therefore the pedals must be x/2 as well. The back wheel is X, which means the back sprocket has to be X. When I cross it over to the other side, itÕs double the radius so it's going to be half the speed, half the RPM. 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. That's it for this one. Tricky question. I hope it makes sense. Let me know if you guys have any questions.

Concept #2: Bicycle Problems (Moving)

**Transcript**

Hey guys! We're now gonna look into bicycle problems where the bike is actually free to move. The wheels are touching the floor so as they spin, it also causes the bike to move sideways. Check it out. Moving bikes. But first I want to remind you what happens if the bike doesn't move, if it's not free to move sideways. If the bike doesn't move when the wheels spin, you have what we call a fixed axis or a fixed wheel. This means that the velocity at the center of mass of the wheel will be zero. Neither wheels will spin. Additionally, both the velocity in the front wheel and the w in the front wheel will be zero. Remember, the w 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. If the bike is moving, we have a free axis which is a situation where you have both w and V. ItÕs sort of a toilet paper that's rolling around the floor. 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. What that means is that this velocity here, velocity of center mass and the velocity of center of mass here are actually the same. 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 with 10, and this moves with 10 this way. Remember, for a free axis which is this situation here, we have that this velocity right here, we have that the velocity of the center mass is Rw, where R is the radius of the wheel. This relationship here can be rewritten. If Vcm is rw, then Vcm = Vcm is going to be our Rw = Rw. In this case, IÕm gonna write front, front, back, back, so we can write those two. For most bikes, the front wheel and the back wheel are supposed to have the same radius, the same diameter. The reason I say most is because you could get a physics problem that doesn't have it that way, thatÕs not really supposed to be like that but they could give you one of those. If that's the case, we can say that wÉ Basically what happens is if these two rÕs are the same, these two guys would cancel. If r front = r back, the rÕs would cancel, and then you have that w front = w back. Not only do they have the same V, but they have the same w. Let's sort of recap here. You have pedal 1, sprocket 2, back sprocket 3, back wheel 4, front wheel 5. The relationships are that these two guides spin on the same axis, so their w is the same, w1 = w2. These two guys spin on the same axis, so w3 = w4. The chain that connects these two make it so that these are the same so I can say that V2 = V3, and what this means is that I can write that R2w2 = R3w3. This is really the important one is this one here. That's the useful one. The first, this is just to get to that. Then the last relationship here which this is old stuff by the way, the new thing here is that there is also relationship between front wheel and back wheel which is this right her. I'm going to write that R4w4 = R5w5. Obviously, if the rÕs are the same, they cancel so w4 is w5 becomes the same. 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, and then obviously that this guy would actually move. This is now actually touching the floor. Let's do an example. It says here the wheels on your bike have radius 66, both of them. Let's draw both wheels. 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. 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 I only have two things, so I'm going to do R1 and R2. I'm given the radius here, so that's good, 0.66 and 0.66, and we want to know what is the linear speed of the center of mass. I want to know what is Vcm1 and what is Vcm2. Vcm of any wheel that moves while rolling is Rw, so Vcm1 is R1w1 and Vcm2 is R2w2. But the key thing to remember here, there's two things to remember Ð these two wheels move together so these numbers are actually the same. Also, they're also both 15. Remember, if the bike moves with 15 to the right, both wheels move with 15 to the right. What I'm going to do is I'm going to do this. I'm going to say this equals 15 and this equals 15. That's the answer to Part A, is that both of these guys equal 15. For Part B, I want to know what is w1 and what is w2. If you look at this equation, I can use this here to solve. It's just basically plugging into the equations. Let's do that. W1 will be 15 divided by R1 or 15/.66 and the answer to that is 22.7 radians per second. Second wheel will have the same w because it's the same numbers. I have w2 -= 15/R2, R2 is the same, 0.66. The answer is also 22.7 radians per second. That's the answer for Part B. 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 w. Very straightforward. That's it for this one. Let's do the next example.

Example #2: Angular speeds of moving bicyle

**Transcript**

Here you have the wheels on your bike with radius 70, both of them. Let's draw that real quick. I got the middle and back sprockets. I'm given the radii here, so little guy, middle guy, this and then you got the pedals here. If you read, the question doesnÕt actually mention the pedal, but I put it here just so that we get in the habit of doing this. One, the middle sprocket 2, back sprocket 3, back wheel 4, and this is 5. The wheels have radius so R4 = R5, which is 0.70. The middle sprocket and the back sprocket are 15 and 8. Middle is 2,R2 = 0.15 and R3 = 0.08. If you ride with 20, this means that V bike equals 20. Let's say you're going that way, which means that V center of mass 5 equals 20. IÕm going to write up here that V center of mass of 4, which is the back wheel, is going to equal 20 as well. Remember if you move with 20, the center of mass of the wheels are going to move with 20 as well. We want to calculate the angular speed, w, of the front wheel. The front wheel is 5. How do we get this? I know the radius and I know the Vcm. Remember, when you have a wheel that's free, you have that Vcm of that wheel is Rw. Here we're talking about 5, so I'm going to put 5 here, 5 here, 5 here. I want to find w5. W5, I have these two numbers, so itÕs just a matter of plugging it in. Vcm is 20, and the radius is 0.7. If you do this, the answer is 28.6 radians per second. B, what about the back wheel? Back wheel, it's going to be the same exact thing because the numbers are the same. What is w4? W4 = Vcm4 = R4w4. The radius and the Vcm are the same. It's moving with 20 and the radius is 0.7, which means w4 will be the same, 28.6 if you calculate it you get the same number. For Part C, we want to know what is the angular speed of the back sprocket. Remember, the back sprocket has the same angular speed as the back wheel. We've already calculated this basically. W3 is the same as w4, so it's also 28.6 radians per second. So far, these first three things all have the same w. Then for Part D, what about the middle sprocket? Let's give ourselves a little bit more room here. I want to know what is w2. I just found 3. 2 is connected to 3 using this equation: R2w2 = R3w3. If I want to find this, I just have to move things around. R3w3 / R2. R3 is 0.08 right here. W3 is 28.6 and R2 is 0.15. If you calculate everything here, multiply this whole thing, you get 15.3 radians per second. That's it for this one. Hopefully it makes sense. Very similar to the static bike. You just have this additional thing where the wheels now both have the velocity of center of mass, and there's this new equation that we have to take care of. That's it for this one. Let me know if you have any questions.

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Concept #1: Bicycle Problems (Static)

Example #1: RPM of pedals of static bicyle

Concept #2: Bicycle Problems (Moving)

Example #2: Angular speeds of moving bicyle

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