BENG 100: Frontiers of Biomedical Engineering
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Frontiers of Biomedical Engineering
BENG 100 - Lecture 19 - Biomechanics and Orthopedics (cont.)
Chapter 1. Introduction to Locomotion [00:00:00]
Professor Mark Saltzman: So, today we’re going to continue talking about the subject of biomechanics. On Tuesday we talked about properties of materials and particularly elasticity and viscosity and viscoelasticity. So, you know something about how biological materials behave, at least in terms of how they respond in terms of deformations to forces. Certainly there’s a lot more to learn about that but we covered the basics on Tuesday.
Today I want to talk about how organisms move, and how they move through a physical world where they have to obey the laws of physics. This is one important role of biomechanics, to sort of understand, if you’re thinking about human biomechanics, how humans perform as sort of mechanical machines. That’s one important role of biomechanics, how organisms, not just humans but all kinds of organisms function in an engineering sense. A big part of that is how they survive, and not only survive, but prosper in a world where they have to obey the laws of physics. It’s interesting to try to understand this because if we understood sort of how the human machine works, for example, under normal conditions you can think about, ‘How can I enhance performance in certain conditions?’ That of course is a big part of exercise physiology is understanding how humans work as biomechanical machines.
What we’re going to talk about today with the example I want to use as just an example of this is motion in biological systems. In particular, motion that humans can’t do on their own, flying. Motions that we do all the time walking, and a kind of motion that we do occasionally, some of us, swimming. Think about how you would apply what you know about physics to understanding those processes. Why, in general, do organisms have an ability move? Think about organisms like us or organisms like bacteria, it’s one of the essential ingredients of life. One of the ways that we recognize that things are alive is because they’re able to move from one spot to another, so why is motion so important? Why is motion important in the life of a microorganism, a protozoa, or a bacteria that’s swimming around in the ocean, why should it move?
Professor Mark Saltzman: To find nutrients; to eat and to find the nutrients or sub-straits that are essential to its growth. If you think about the movements that you make during the course of the day, how many of them are motivated by your need for food? You move in order to find things that you can eat and that’s one reason why living organisms move. Why else do they move? Justin?
Student: To avoid danger.
Professor Mark Saltzman: To avoid danger, to get out of harm’s way. If you’re a microorganism what would be dangerous to you is often chemicals. There might be chemicals around you that are toxic to you, and so you’ll swim to get away from those chemicals. If you’re a rabbit in the woods, you might be running to get away from a fox or a wolf or something that’s going to eat you. If you’re a human you might run out of Prospect Street to avoid getting hit by a bus, that’s a good use of motion for self preservation; one more key reason why animals move; essential for propagation of the species, not just survival of the organism. Justin?
Professor Mark Saltzman: To reproduce, they move to reproduce, to get together and that might be some of the reasons why you move everyday. Certainly in the animal world, in general, movement for finding mates, for reproduction is an essential part of it. So, easy to see why movement is critical, for not just the life of a microorganism, not just the life of a multicellular organism, but for survival of the species as well.
Movement requires force and we’re going to think about forces. In particular, the forces that an organism needs to develop somehow or generate in order to overcome forces that resist motion, so just some elementary physics here which most of you probably remember from high school physics, is that if an object, the object here is this green oval. If it’s going to be moving with a constant speed what does that mean? Moving with a constant speed means not accelerating. There has to be some force applied in order to create the motion but the forces must balance, so the overall forces acting are zero. For an object that’s not accelerating the sum of the forces is zero because the acceleration is zero.
If we think about this green object moving to the right, to the left, to the–to your left then there must be a force that’s creating the motion to the left and that force is just big enough to overcome the resistance to motion. That force is equally balanced with whatever force resists motion. Of course the force that resists motion is going to be different if you’re an organism that’s trying to fly, than if you’re an organism that’s trying to move along the ground, than if you’re an organism that’s trying to swim. So, what we’re going to think about today are the forces that resist that motion and how organisms overcome that resistance. If we think about flight first then the situation is a little bit more complicated.
What’s the force that creates motion? Well, there’s got to be some propulsive force that’s moving us–that’s moving the organism forward or to the left here. There has to be some propulsive force that opposes the force that resists that motion, and that force is drag. It’s drag on the object that’s flying the air and that drag comes from the interaction of the surface of that object with the air. It has to move air out of the way in order to move ahead. In order to move with a constant speed, there has to be a force that’s pushing it to the right and that force has to be exactly opposite to the drag force which is resisting that movement. That drag force comes from displacement of air, friction of air.
You have experienced this drag force if you are driving in a car and you stick your hand out the open window, you can feel the force that’s pushing your arm backward. In order for you to hold your arm straight out you have to apply a force in the forward direction that’s directly opposed to the force that’s from the air that’s pushing your arm back. As the car goes faster, it takes more force to move it ahead, it takes more force to hold it steady. So, the amount of force that you need to apply varies with the speed that you go. The faster the car is going the more force you have to apply to hold your arm out steady, the faster a bird flies the more force it has to apply in order to overcome this drag force that’s opposing its forward motion. That’s the dynamics, that’s the force balance in the horizontal direction; propulsive force overcoming the drag force.
You know it also depends on shape. If you hold your arm out this way, you hold your arm out this way, you feel a different force from the air. The shape or the profile of the object that’s in this field also matters, keep that in mind as well. If you’re going to fly there’s another force you have to overcome and that’s the force of gravity. That’s the force of gravity that is pulling you down towards the Earth. To overcome that force, that constant force, there has to be a force that’s moving up. If you’re going to maintain a constant height, a zero velocity up and down was what you would need to maintain a constant height, then you’d have to have a force that’s lifting you up that’s exactly opposite to the force of gravity. If you’re going to fly you have to have that upward force as well.
Chapter 2. The Mechanics of Flight [00:09:11]
What kinds of objects can fly? Planes can fly, this is a picture of an old fashioned plane. Maybe you’ve ridden on a plane that has a propeller on it, maybe not, but it’s a little bit easier to describe this way; that the forward force that overcomes drag is provided by a propeller that spins and creates a force in the forward direction. In order to move at a constant velocity, depending on what that velocity is, you have to overcome a certain amount of drag and the propeller spins at a speed that will provide that force to move forward. That seems, if not a complete description, that seems like a pretty adequate description of how you move ahead.
How do you get off the ground and maintain off the ground, maintain your height off the ground? In order to do that, you have to overcome this constant force of gravity which is pulling you back. That’s the main problem with flight, overcoming the force of gravity and in airplanes that’s done through design of wings. A typical, sort of airfoil or wing design is shown here. What happens is it’s using this flow of air, this constant flow of air over the body, over the wings for example, by changing the shape so that there’s more surface area on top of the wing than there is on the bottom. You change the relative velocities that air can move over the wing, over the top of the wing versus over the bottom of the wing, you create a flow separation or a disturbance here where the pressure drops relative to the pressure in the ambient air.
The pressure drops because of this turbulent air flow pattern. If there’s less pressure here than there is down here, you get a force acting up, pressure is force per area. If there’s a difference in pressures from top to bottom, more pressure on the bottom than there is on the top, you’ll have a lift force going up. You can fly if that lift force is large enough to overcome, or at least balance the force of gravity. Does that make sense? You can imagine that that lift force varies with speed that you’re moving through the air. For example, there’s a critical speed you have to reach before you get enough lift to take off and that lift force depends on the design of the exact shape of the wings or the airfoil, certainly depends on their size. If I’ve got bigger wings and pressure’s what’s moving me, force is pressure per area, bigger wings more force up.
The basic physics here is fairly straight forward, although implementing this took some time, people have been trying to fly as long as there were people watching birds. It took awhile for us to figure out how to implement this and to implement it as reliably as we can now. How do birds do this? How do birds fly? Well, birds don’t have stationary wings, typically. Although their wings can be stationary; if you watch a hawk or an eagle or a large bird, they glide for a lot of their flight once they’re airborne. If you watch, my office overlooks the Grove Street cemetery, there’s some hawks that live in the cemetery, must be a good place for hawks. T hey take off and then they glide for long periods, they can circle and there’s very little motion of their wings. They’re basically doing–they’re taking advantage of the forward thrust that they have created and they’re coasting but their wings are doing this. They’re holding their wings in a position that creates an airfoil, creates the pressure drip to move them up.
To take off or to create this forward thrust they move their wings, they don’t have propellers. So, they use their wings to create thrust by sort of pushing them against the air backwards, by creating a forward thrust which overcomes the drag. Yet their wings are shaped like an airfoil so they create thrust, they bring them back into the airfoil position, and they get lift. So, it’s more complicated than what I’m describing but you can see beating your wings like this, creating thrust in one direction, holding them in a position where you create lift. Then, varying between those two things you can both move forward and supply the lift you need to overcome the force of gravity. If you watch different kinds of birds you’ll see that they execute different kinds of patterns, and they do that because they’re different. Some birds are smaller than others, some are bigger, some have different shaped wings, different shaped bodies. So, their particular motion is adapted to the physics of their body and their need to move it through the air.
I talked a bit about wing size. Wing size is one thing that you could use in an airplane, for example, bigger wings, more upward lift. Bigger wings, of course, heavier. So, if it’s heavier it’s going to require a bigger engine so that you can get the thrust needed to move at a certain velocity. If you look at birds of different sizes, or flying animals of any kind of different sizes, you’ll notice that there’s a correlation between the size of the wing and the size of the animal. The bigger the bird, typically, the bigger the wings for just that reason; bigger bird heavier, needs more lift in order to get up into the air.
You also notice other kinds of adaptations; that birds that have big wings, big bodies, also tend to have strong legs. Little birds, isn’t it amazing, little birds you see perched on a tree limb or something, very small legs. Big birds, hawks, falcons, eagles, strong legs; why would a big bird need strong legs? Well, they actually pick up prey and things and so they use them for acquiring food, that’s a good reason. Why do they need them for flight? They need to be able to lift off and so to get that initial–maybe they get some boost from their legs in the lift off phase, some forward thrust and some upward lift, just from the muscular structure of their legs. They’re not always in flight, they need to land sometimes and they land on their legs typically. If you’re going to bring a big object down to the ground there’s going to be some force involved with that. You need large legs in order to survive that force. Just giving you some examples of how the physical structure of animals is adapted to their function and a lot of it is their function as mechanical objects. It’s a familiar concept but one can sort of break it apart.
An interesting kind of bird is shown here, a hummingbird; have you ever watched a hummingbird? Hummingbirds do something that’s really amazing, they hover, which means they can stay up in the air without having any forward thrust. When I was talking about the airplane example or even the bird example, I was talking about using the forward motion supplied by thrust in order to create the pressure drop that brings you up. But if you stop moving forward that pressure drop disappears, you go down. Forward motion is required for an airplane to stay up in the air, for a bird, most birds that stay up in the air.
Hummingbirds can hover which is a remarkable physical achievement, and they do that because they can actually turn their wings in both directions. They can turn them upside down and that’s shown in this diagram here, that they can make the wing go backward and forward, and turn it right side up and upside down, so they can create a motion where they’re creating no net forward thrust or backward thrust, but always creating lift because they’re changing the position or the orientation of their wings. More complicated than that, if you’re interested I’m sure you can find information on it, but again they can do this because they have specially designed wings that can flip over during their motion. They have the muscles that allow them to make this kind of a motion, and they’re fairly small so they don’t need much lift force in order to keep them aloft to overcome the problem of gravity. Hummingbirds can hover because they generate lift in both the upstroke and the down stroke. As long as their wings are in motion they’re constantly creating lift, and so they stay off the ground. That’s something about flight, which is interesting, and I think maybe the simplest one to think about because one can break down the forces that are involved fairly easily.
Chapter 3. The Physics of Walking [00:18:29]
Let’s think about walking and running, or a mode–nobody flew here this morning, you got here to class somehow, how did you get here? Most of you probably walked, maybe some of you ran, but most of you probably walked, why? Of all the choices that you had why did you walk instead of slithering on the ground or crawling, or running? Why did you walk? Why did you make that choice? You might didn’t have even thought about it, you probably didn’t get to the room–to the door of your room and say should I crawl, slither, run, or walk you probably just started walking. Why was that your first choice? Justin?
Student: It seems natural.
Professor Mark Saltzman: It comes naturally. Why do you think that’s your natural motion for moving along the surface of the Earth? Why is that your natural choice? Why have you decided, even without thinking about it, that that’s your normal mode of motion? Yeah?
Professor Mark Saltzman: It’s the easiest, it’s the least resistant, it’s the most efficient way to move from one place to another, and its efficiency that’s important. Now, how would you define efficiency here? How much energy did you need to expend to get from your room to here and how can you get from your room to here minimizing the amount of energy that you spend? That’s the subconscious decision that you make and the answer is that you walk.
What force are you overcoming when you’re walking here? What’s the force that opposes your motion? The force that creates your motion is the movement of the muscles or your leg and your skeleton. So, like rocking back and forth like this I’m using my muscles to create motion side to side, and you use a similar strategy when you walk. What force am I overcoming when I do this? What force did you have to overcome in order to get here? Well, you’re not–there is a drag force because you’re moving through air. So just like a flying bird, you’re experiencing drag from the atmosphere but it turns out that that’s not as large as the frictional force of your body touching the ground. When you walked you probably didn’t walk like with your feet on the ground because that would take a lot of energy. Try it, try walking back home with your feet constantly on the ground, that takes a lot more energy than picking them up.
You pay a price for picking them up. Now, you’re introducing another problem, actually two problems, in that when you pick up a foot you have to overcome gravity, so you’re using some of your motion to go in the wrong direction. Instead of moving towards class you’re moving up by picking your leg up, which is not getting you any closer to where you want to be. You pick it up in order to release from the frictional forces that are holding it back, the frictional forces that resist motion. How do you walk then? You don’t walk with your feet constantly in contact with the ground, although that would be easier in a sense, because if your feet are both on the ground balance is not so hard. Once you lift a foot off the ground the other problem it introduces is balance. It’s much harder for me to stay stationary on one leg than it is on two legs because gravity is acting on my body and trying to pull me back down again. An additional problem arises with this lifting up of one foot and that is the problem of balance. How do we solve that? We solve that by controlling our muscles, the muscles that move us from side to side maybe, but that takes energy.
I want to move forward and the easiest way is to walk and you probably didn’t think about it when you were walking here today but you’re doing a fairly complicated set of operations. You’re lifting one foot off the ground to avoid gravity, and you’re letting it swing forward. The heel of your forward foot hits the ground, your foot then moves, the toe of your back foot leaves the ground, swings forward, heel strike, toe lift, swing, heel strike, toe lift, swing, one foot’s always on the ground when you walk. You don’t think about it but it’s a fairly complicated motion that you can do automatically now, but if you do think about it when you’re walking to your next class, you’ll notice that one foots always on the ground. That helps you solve the problem of balance.
What else happens when you’re doing this motion? You might not notice it at all but you’re pelvis is actually undergoing a fairly complicated set of movements. I’m going to exaggerate it so you can’t laugh. If you watch somebody else walk or you put your hands on your pelvic bone, the pelvic bone is this sort of girdle that both of your hips are connected to, you’ve seen it on a skeleton. When I lift up my leg then I’m tilting my pelvis up, I’m tilting this side up in order to get the leg off the ground. I’m also bending my knee a little bit to make my leg shorter, and I’m moving my foot in order to allow, when the toe lifts off, allow it to swing without hitting the ground. You don’t want to hit the ground because of friction, that would be a waste of energy, you want to lift it just enough so that you clear the ground and you do that by tilting your pelvis up and by bending your knee a little and manipulating your foot.
You let that leg swing forward, the toe hits, and now you’re doing something else with your pelvis, and that you’re going like this. You’re not only lifting up to get it off the ground, and then up on the other side, you’re also twisting it like this, so your pelvis twists, your pelvis tilts, and it actually undergoes–if you could put a sort of–I’m exaggerating it–not as much as others could but I’m trying to exaggerate that your hips sort of go through a figure-eight kind of motion as you’re moving forward. It’s a fairly complicated motion that you do without even thinking about it because you’ve done a lot of walking and you’ve learned how to do it very efficiently. Does this make sense?
If you look at the efficiency of walking you are creating forward motion, you’re creating it in opposition to friction. You’re doing it by lifting a foot off the ground, freeing it from friction, swinging it forward. The cost is you had to spend some energy going up, and then you recover that energy when you come back down. What’s energy called that you store as you move up? Potential energy; so one of the other things that happens, if you notice somebody walking, and it’s hard to notice if you just see me walking like this, maybe you can, but my head is actually moving up and down when I walk because my center of gravity is rising and falling as I pick my leg off the ground and let it swing.
You can’t see that so easily when you’re watching a whole–have you ever seen somebody walk with a fence in front of them and you see their head and it moves up and down a little bit? There’s an inter-conversion there between potential energy when I’m up and kinetic energy when I’m down. I can exaggerate that way too if I walk funny. You’re putting some of your energy into potential energy, creating potential energy with your muscles, and then recovering that when you come back down. The more of that you have to do the more energy it takes. These movements of the pelvis I mentioned before sort of smooth out this up and down movement, so you’re not wasting so much energy going up and down, you’re mainly going forward.
Chapter 4. Efficiencies of Walking, Running, Cycling [00:26:53]
If you walked–did anybody walk here with somebody? Did you walk by yourself or did you walk in–anybody walk in a pair? Nobody wants to admit that they walked some–well have you ever walked with somebody else? Did you ever notice that sometimes when you’re walking with somebody else they want to go faster than you or they want to go slower than you? Why is that? Why do some people walk more slowly than others? Why do some people walk more quickly than others? Their legs are longer so their bodies are shaped differently. So, they’ve figured out how to make their walking efficient for them and that means they’ve figured out how to do this motion with their body in the most efficient way.
Now, why would longer legs make it more efficient to move faster? Well, because when you swing your leg you get a bigger bang for the buck, you get farther with one stride, that’s part of it. The other part is that your leg, when it’s in this swing motion, behaves like another object you probably studied in physics, the pendulum. You’ve seen pendulum’s that swing, you notice that the timing of pendulum’s–anybody have a metronome? Metronomes are pendulums like that, too, that have an adjustable weight. If you move the weight one way or the other, then the pendulum starts swinging at a different frequency. Depending on the length and weight of your leg, as a pendulum, it has a natural frequency at which it wants to swing.
If I want to walk with the least energy possible I want to just let my leg swing. I don’t want to put any energy into it, I’ll put some energy into lifting it off the ground, but once I lift it off it’s back here, what if I can just let it go and it swings forward? That’s the most efficient motion. That motion would have a timing that depends on the size of your body and the length of your leg, and the weight of your leg, and all the other characteristics that are particular to you. If I want to walk slower than my natural state, I have to slow something down. My leg might swing forward fast with a certain distance. In order for me to move slower I have to actually slow down my leg. I have to slow down the swing forward, that takes energy, it takes muscles to slow down, and so sometimes you feel like, if you’re walking more slowly than you want to, that it’s actually taking more energy than it should. Likewise, if you have to walk faster than your normal pace, it will seem like and will be that you’re using more energy than you would like to.
Walking has an efficiency; you can change your speed of course, you’ve learned how to do that. If you plot on a graph, like this one, this is a graph for a typical person who’s walking on a flat outdoor surface. What it shows–the blue line there shows the efficiency, or how much energy you use as a function of speed of walking, how much energy you use as a function of speed. So, there’s a flat part at the beginning, not too much different. If I’m below two miles an hour walking it takes about the same amount of energy. But as speed goes up, the energy consumption goes up. You have to walk faster, faster than your natural stride, and so you’re using more energy maybe to swing your leg forward faster so that you can walk. You might even, if you start walking fast, try this on your way to the next class or try to walk with somebody and watch them, as you walk faster your head’s going up and down more. You’re taking advantage of this potential kinetic transition to get more speed, but the cost of that is that you’re moving body up and down more and so that is less efficient because movement up and down is wasted.
At a certain point, you see the blue line goes up and up and up, and it’s exponential. If you–it starts to look exponential. If you continue to try to walk faster and faster it’s going to eventually start to become really uncomfortable. You’re naturally going to want to switch to a different form of motion and that’s running. Now, you can avoid that and you could try this yourself, try walking faster and faster, and faster and don’t let yourself run, just walk faster and faster, and pretty soon you’re doing the crazy kind of walk that Olympic fast walkers do, that feels very unnatural. You might get a lot of speed but it feels very unnatural. I’m not–you’re kind of–can anybody do it? You’ve seen it. You’re moving up and down a lot, your pelvis is doing a very unusual pattern, not the subtle movements that it does during normal walking, but it’s moving a lot more. That’s what you notice, the fast walkers they’re moving from side to side, they’re pelvis is really moving a lot.
It becomes quite inefficient because you’re wasting energy. In fact, your body is telling you to make the transition from walking into running. If I want to move at this speed it’s more efficient for me to run, and that’s what this graph shows you as well. If you extended the walking line up a little bit, at a speed of about 5 mph, running becomes more efficient than walking. You can feel that transition if you try to walk fast. There will become a point, at about 5 mph, when you’re going to want to start to run because that will feel like a more efficient way for your body to move. Now, what is it that causes that, mechanically, that transition to take place? What’s different about running than walking?
Well, one of the things that’s different is that you don’t always have one foot on the ground when you’re running. In fact, if you watch somebody run or you run yourself, you’ll know that for a lot of the time you’re only–you’re not–no foot’s on the ground. There’s not time that both feet are on the ground, one foot hits, you sort of bounce off of it, you bounce off the other one, you bounce off the other one, you’re sort of hopping off one foot. Try to do that slow it’s really hard. Why is that hard? Because it’s hard to balance on one foot, it’s impossible to balance on no feet when you are standing still.
As your speed increases you create inertia. You create forward motion, and that forward motion or inertia stabilizes your body. It makes balance less of an issue. If you’re moving forward at 5 mph, you’re not worried about sideways motion so much because you have this forward motion which is going to continue to carry you forward, and balance isn’t so much of a problem anymore. You can sacrifice having both feet on the ground. You can even sacrifice having any foot on the ground for some period and you’ll just fly and you’ll fly forward no balance is required, so you’re not spending any energy on balance.
Running is sort of like flight in a way, only it’s very short flight. It’s like flying and hopping, hopping on one foot. Also at that speed, when you’re moving at 5 mph, if you see somebody that’s running, particularly an experienced runner, and what makes experienced runners better than inexperienced runners? They’ve become more efficient at doing this running motion and so they can move at fast speeds very efficiently. One of the things you’ll notice in very efficient fast runners or good runners is that there’s very little up and down motion of their head even though they’re hopping and flying, they’re not moving up and down very much. They’ve learned how to do this without moving very much in the gravitational field, keeping their center of gravity about the same which allows them to convert all of their energy into forward motion. Does this make sense?
There’s another peculiar thing on this slide and it’s the black line, cycling. If I told you that you had to move this afternoon at 12 mph to get from here to somewhere else in New Haven in a couple of minutes, most of you if you had the choice would choose to use a bicycle to do that rather than walking and running. Particularly if you knew you had to move that fast you’d pick a bicycle. It won’t be surprising for you to see that cycling is much more efficient at higher speeds than either walking or running, so if I want to move 12 mph, cycling is much more efficient than running; walking is not even shown here because it would be so much less efficient.
On the other hand it doesn’t make obvious sense because you’re going to agree to move fast, not with your–not only with your body but with another thing as well, which adds weight. Even a very lightweight bicycle weighs 30 lb. So, you’re adding 30 lb to your weight but you would gladly do that if you knew you had to move fast because you know it’s going to be efficient. Why is cycling so much more efficient than running? Well, one reason is when you’re cycling you’re sort of a in a constant profile. You’re moving your legs but your center of gravity stays relatively uniform, very little up and down. All of your motion, all of your energy that’s created by your legs is converted into forward motion.
The other thing you know is that now you have a mechanical–you have a machine that’s going to convert the energy of your legs efficiently into motion of a wheel and that wheel overcomes friction very efficiently by actually using friction. It’s friction between the wheel and the ground that keeps the bicycle stable and allows it to generate high speeds. This machine allows you to overcome the problems of friction and gravity very smoothly, and so cycling is a much more efficient activity. A lot has been written about the biomechanics of cycling if you’re interested in learning more about that.
Chapter 5. Mechanics and Efficiency of Swimming [00:37:57]
Well, I want to talk about swimming, another mode that we use, not everyday. Some of you might swim everyday, most people don’t. Microorganisms that live in the ocean, fish that live in water, of course swim all the time. Swimming is their primary mode of motion and they swim to get food, get away from danger, reproduce, all of those things. What’s the major problem that you have to overcome when you swim? Well, it’s friction or drag, it’s the same problem really that all movement has to overcome, but we’re not thinking so much about friction on the Earth now we’re thinking more about the drag force that’s created from moving through a fluid.
It’s a bigger problem now because the fluid is viscous. Air has a relatively low viscosity; you know what viscosity is from Tuesday’s lecture. Air has a relatively low viscosity, water has a higher viscosity, molasses or honey has an even higher viscosity, but it’s harder to move through water than it is through air because it’s thicker, it’s harder to slip the plains of packets of water passed each other to go back to the description I gave you of viscosity last time. In fact, if you’re going to swim at a constant velocity, then you have to create a propulsive force moving forward that is exactly opposite to the drag force that is created by moving at that velocity.
Just like the example of putting your arms out of a moving vehicle, which I don’t recommend. I’m sorry I’m using it as an example because we’ve all done it but I don’t recommend experiments of that kind. Imagine that you’re putting your hand out in water, the same kind of thing would happen, only the force would be even greater because of the greater viscosity of the water. You have to create a bigger propulsive force in order to overcome this drag. That’s what fish have to do, that’s what microorganisms that swim have to do.
Most microorganisms that swim for a living or most fish that swim for a living have evolved mechanisms to move quickly through water by reducing their profile, so fish often have streamlined shapes. They might be flat. The look like flat plates sometimes, some kinds of fish. That’s an easier shape, a more streamlined shape creates less drag as it moves through water than a bulky shape like a cube or a sphere. Streamlining or shape is an adaptation that animals can use in order to move faster. You know this if you swim, if you want to swim and just glide through the water, push off from the side and just glide, if you go like this you’re not going to glide very far. If you put your hands in front of yourself or down at your side and make your body more streamlined you’re going to go a long ways, so streamlining helps.
It turns out that there’s a very simple relationship between the drag force and the velocity that you move forward for objects that are spherical or roughly spherical. I show that equation here. The drag force is equal to 6 pi times the speed or the velocity that you’re moving times the viscosity, times the size–and here the size would be the radius of the sphere (Fd=6πvµr). This equation applies to spheres. It applies very well to objects like microorganisms, like bacteria that aren’t entirely spherical but they’re not too different from spherical either. If I am this bacterium and I want to move with a constant speed, I have to create a force that exactly overcomes this drag force that’s created and that drag force depends on speed. To move faster I have to overcome a bigger drag force, I have to create more propulsive force just like the airplane. A fish the same thing, in order for it to move forward at a certain speed, it has to overcome the drag force and it does that by moving its muscles, by moving its flippers and fins to create a propulsive force.
What I did here is a simple calculation between different kinds of swimmers. Swimmers that are basically different in their size. For a bacterium it’s very small, its size is .0001 cm, that’s about 1 micron, so a very small object. I asked the question how much–if I ask the question, ‘How much drag force is created for it to move at a certain speed? ’ What was a reasonable speed? I picked a speed of 10 body lengths /s; 10 body lengths /s would be a speed of .0001 cm/s, so just multiply this by 10 /s. The viscosity of water is .01 centipoise or .01 g/(cm*s). If you knew these factors, you knew how big the bacterium was, you knew the viscosity of the medium it’s going through, you knew how fast it was going to move, you could plug into this equation and calculation the drag force. If you calculated the drag force, that’s how much force that object has to be generating in order to move.
You could do the same thing for a fish. Now, a fish is bigger, let’s just say lots of different sizes of fish. Let’s say a 10 cm fish moving through the same medium, moves 10 body lengths /s that would be 100 cm/s. A Yale student who wants to swim might have a size of about 170 cm, larger than the fish or the bacterium. To move 10 body lengths /s, I converted that to mph, that’s about 0.5 mph. I’m not a swimmer, is that fast? Sounds fast; it’s slower than you normally walk, but you can’t swim as fast as you can walk. Let’s just use that as an estimate and it’s moving through the same viscosity. One of the things that I asked you to do in the homework for next week is to take this data and sort of consider what life is like for a swimming bacterium, for a swimming fish, for a swimming Yale student, just with these factors in mind, their sizes, viscosity being the same, and their speeds of motion somewhat different. You’ll have a chance to think about this quantitatively for next time and we’ll come back to it in section.
Chapter 6. Design in Biomechanics and Conclusion [00:45:04]
I want to finish by talking about another element of biomechanics that I hope I’m able to talk a little bit more about a couple of weeks from now. That’s the issue of design in biomechanics, and one of the main issues of design in biomechanics is the design of artificial replacements for joints like the hip. Artificial hip replacements have now been around for hundreds of years. W e are quite skilled at making artificial hip replacements, surgeons are quite skilled at doing them. A surgeon who does a lot of hip replacements can do an operation like this in about an hour. That includes taking your diseased hip out, putting an artificial hip back in, and getting you to the post-op all in about an hour. It’s a pretty remarkable achievement from the surgical side.
I want to show you a little bit about how these work from the biomedical engineering side. To do that I first wanted to show you the hip joint, which you might have seen before. This is skeleton, this is half of the pelvis I was talking about before, it’s motion during walking. You see half of it here, so this is the person’s midline, this is the right hip, the pelvis, the crest there. If you put your own fingers on the sides of your hips, you can feel the bone that sort of sticks out at the top about where my belt lies here, that’s the crest of the ileum or one of the bones that makes up the pelvis. For the experiment, you’re going to do while you’re going to your next class where you think about how your pelvis moves, keep your fingers there that will be convenient and silly if you walk that way.
The hip is connected right below that. You can feel–if you feel down a little bit from the crest of your ileum, you can feel this part of a long bone called the femur that makes up the thigh. It’s the bone of your–the upper part of your leg. That bone extends down to your knee this way and it connects to your pelvis this way, through a structure that’s a continuous structure of the bone called the femoral neck and the femoral head. What it looks like is a neck and a head. The head being sort of a ball and the ball of the femur fits into a cup in the pelvis, and it is this ball moving in this cup which gives your hip its range of motion. Your hip has remarkable range of motion, you can move it all different sorts of ways. Not just forward and backward, but you can move it to the side. It’s the ball moving in a cup that creates all of this versatility of motion; you can swing it around different ways.
Well, when this joint gets diseased, movement becomes much more difficult. In general, it’s a degenerative disease that the bones and the cartilage start to break down, and so now this cup and ball don’t move so smoothly over one another. When they don’t move smoothly over another, they generate friction, they generate pain. So, degenerating hips can cause tremendous amounts of pain for people. What’s in between these two–this is just a skeleton so there’s nothing in between these two, but in our hips what is it that lubricates the space between the hip of the cup in the pelvis and the ball of the femur, what lubricates that space? Well, there’s some fluid there, but the main thing that lubricates it is protective layer of cartilage or soft skeletal material. Not bone but a soft spongy material that lubricates between bones, and that can break down in diseases like arthritis.
Well, this is what an artificial hip looks like. The modern hip is composed of two parts, there’s a cup which goes into your pelvis that replaces the normal structures into which the femur goes. So, it replaces the cartilage and the bony structure around the pelvis, and the end of the femur, the neck and the head are replaced by a piece of metal. The metal sets in the metal ball of the femur, the artificial femur sets in the cup of the pelvis. Now, what lubricates it now is not cartilage but a polymer. Because if you had a metal cup and a metal femur going in, metal on metal, even if it was very smooth metal, that’s not–there’s friction associated with that motion. If you put a polymer, in particular, what are used most often now is high density polyethylene or something like Teflon, a metal ball slides very easily over a Teflon cup, particularly if they’re machined to fit together precisely. So that’s what is used currently and what allows normal motion to be restored. Now, the problem, the main problem here is how to get this ball and this neck attached to the femur. Because, if you think about it, why is it metal? It’s metal because if you’re standing on one leg all of the weight of your body is on that one leg. The whole weight of your body is on the femur, the whole weight of the body is actually right here. The whole weight of your body is on this very small–depends on the integrity of that very small piece of material.
What we learned last time is that materials have elasticity. You don’t want something that bends a lot there. You want something that’s strong, that doesn’t bend, even when there’s a high stress applied to it like the stress of the weight of your body. Metals are really the materials that best can supply that kind of mechanical strength, that kind of resistance to bending, resistance to fracture. The problem is how to get this on here. The way that that’s solved, currently, is to have the neck and the ball of the femur attached to a much longer piece of metal. When you get an artificial hip, then this part of the new replacement femur goes down into the bone of your leg, inserted into the bone so that it’s more stable. There’s all of this metal into your bone to sort of stabilize it because the last thing you want, or what you want to achieve, is for this part of the femur to move without any slippage with the rest of your femur.
I’m going to come back to this concept later and talk more when we talk about artificial organs about how artificial hips are made, but I wanted you to at least have a picture of how this particular piece of biomedical engineering works. We’ll see you at section this afternoon.
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