It's one week later, and I still find myself playing Breath of the Wild, so I'm going to continue with my theme of Zelda and lift physics from last week. However, as there really isn't much else to discuss about lift in Breath of the Wild, we're going to jump backwards in the series to my favorite 3D Zelda game: Majora's Mask. For the uninitiated, in Majora's Mask, there are two main mechanics to the game: looping time and transformation masks. We're going to focus on the second one, because one such transformation mask turns you into a creature which appears to be made of wood called a deku scrub, and, as a deku scrub, you can launch yourself into the air (with the assistance of flowers—if you haven't played it, just don't ask) and then hover around via two propeller-like flowers held above your head. I think you all know where this is going now: we're extending the idea of lift form last week to propellers, or rotors, and I'm going to ruin the physics of my favorite Zelda game. Let's do this.
To start off again, I had to get some data, which involves research. This, of course, meant playing the game. So I dusted off my Nintendo 64 and messed around a bit in Majora's Mask. The controls took some getting used to again after so much Breath of the Wild, and also the N64 controller, but I got the data I needed: as a deku scrub, Link is about half as tall as an adult human, so we'll say three feet. The flowers he holds above his head seem to have a total diameter of deku scrub link's height, and look like this:
He can shoot bubbles out of his mouth to attack. I promise this is a real game. And that it's really good.
Based upon my observations, I'd say that the middle of the flower is about a quarter of its total diameter, and the petals are each about that wide. We'll say three-quarters of a foot. That makes each petal roughly rectangular with dimensions of one and one-eighth feet by three-quarters of a foot. We're doing imperial measurements here because that's what I started with in the Breath of the Wild post, so now I'm stuck there. Next week will be back to metric, I promise.
So, now that we have these measurements, we need to determine the physics of lift for rotors. This is actually pretty simple. Remember the equation from last week? Well, that still applies. The rotors are still moving through the air to generate lift, only they're moving in a circle rather than a straight line. So we can still use this equation:
However, there is one tiny snag, which you may have already caught. Since the rotors, which are our airfoils, are moving in a circle, each "slice" or the rotor (or petal, or airfoil) moving outwards from the base (or stem, or axel) will be moving at a different speed. Much like a record spinning, where the grooves near the edge move so much faster than those near the middle, even though the record itself is spinning at the same rate of rotations per second. This rate of rotations per second is called angular velocity, and it's measured in radians per second. Radians are just a measurement of a portion of a circle, where 0 radians is no progress along a circle, and 2π radians is one full 360 degree rotation. In fact, radians can be converted to degrees very simply. Angular velocity, too, can be converted into linear velocity, which is the "normal" or "straight-line" velocity that we're used to, without too much fuss, and the conversion equation looks like this:
Where v is the linear velocity, ω is the angular velocity, and r is the distance from the center of rotation (the radius). That means we can re-write our equation like this:
But now we have a dependence on r in our lift equation, which is not a fixed number like all the rest of those values. This is going to be trouble, because for each different value of r, we need to calculate L for that particular radial slice of our rotor (petal, airfoil, etc.). Now, we can make increasingly better estimates of the "true" value of L for the whole rotor by doing the equation for more and more granular values of r between rmin (the base of our rotor, where it joins the axel), and rmax (the outermost edge of our rotor), but the best way to do this is using my favorite kind of math: calculus.
What we can do here is define an infinitesimally thin slice of our rotor, which we'll call dr. This is going to change A in our equation, which was a measurement of area, but our infinitesimally thin slice of rotor is essentially one-dimensional—it has no depth—so it's going to just be the width of our rotor. This does not change with any variance in r, because the petals for the deku flowers are roughly rectangular, and I want to make the math easier for me, and easier to follow for you. Now that we have an infinitesimally slice of our rotor, we need to add up all infinitesimally thin slices between rmin and rmax. This is where integration comes in.
To add up all these infinitesimal slices, we need to take the integral with respect to dr of the whole equation, which will give us the result for the sum of all infinitely thin slices of rotor. Thus is the power of calculus, and it's pretty cool. Our equation now looks like this:
I've changed A to W because it's no longer an area, it's just the width of the rotor, and I've multiplied the square through to ω and r. Now, it's just a matter of simplifying and solving the equation. Even if you don't know calculus, this shouldn't be too difficult to follow along with. To simplify the whole thing, we can remove any values from the integral that are constant regardless of the value of r. That's because the integral only deals with summing all infinitely small pieces of r (that's what the dr signifies), so anything that doesn't depend on r won't change as we sum, and we don't need to include it when solving the integral. Our equation now looks much simpler:
This just means that we multiple this whole mess of constants by the result of the summation of all infinitesimally thin slices of the value of r from rmin to rmax. Solving this integral (if you haven't taken calculus, just trust me on this. If you have, feel free to check my work) yields the equation
Which we can just plug all our values the we determined from the research up above and get the lift from a single rotor. Since there are eight rotors in total (four per flower, two flowers), we multiply the whole equation by eight to get the total lift force.
So, if we plug in our values, what do we get? Well, I don't know. I couldn't really determine a good value for ω from my playing—err, research, as the petals just move too fast. So I'm going to go about this in a more unorthodox way: we're going to determine how much lift we need for deku scrub Link to hover, and then determine ω from that, and see if it's a reasonable angular velocity. And to do that, we're going to need to figure out the weight of deku scrub Link.
I mentioned above that Link, as a deku scrub, is about three feet tall, or as tall as the children in the game. He also appears to be composed entirely of wood. It looks like a hardwood with a slightly distinct grain, so I used my extensive knowledge of wood to determine that he appears to be made of cherry. Now, I know that cherry has a density of about 50 pounds per cubic foot, and we'll say that deku Link is about 2 cubic feet, which makes him weight a whipping 100 pounds. Which just doesn't seem right to me, because he can hop on water, and I always got the impression that he was supposed to be super light. So let's instead choose a much lighter wood, like cedar, which clocks in at about 20 pounds per cubic foot, and puts deku Link at forty pounds. That means, to hover, we need his propellers/flowers to generate forty pounds of lift. Now, let's re-arrange our equation to be solving for ω, since we now know L:
Plugging in all our values yields 29.28 radians per second, which is about four and a half revolutions per second. Which is... surprisingly reasonable. I'd be perfectly willing to believe that those flowers are rotating that fast. Deku scrub Link can totally hover using those flowers, provided that he's made of the right materials to be light enough. Once again, Majora's Mask proves its worth as my favorite 3D Zelda by being spot-on with the physics. Now, if you'll excuse me, I need to get back to playing Breath of the Wild.