A Lever Long Enough
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All of this could, at least, seriously mess up the calculations for how long our lever would need to be or, at worst, make the whole affair pointless.
First will be some "ground" and Earthlike gravity. Our fulcrum will rest on the "ground" and will push up against the Earth attempting to lift it as if it were just an extremely large rock. So, how long would a lever need to be to move the world? The formula for this is quite simple:. In other words, the force applied down F 1 times the length distance of the "pushed down" portion of the lever D 1 is equal to the force upward on the other side of the lever F 2 times the length of that side of the lever D 2.
Initially, you might think that this means that the Earth side of the lever is the radius of the Earth. I did too. I figured out all of my calculations before realizing the truth. The good news is that I had to do more math. This is always good news to a math geek.
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The radius of the Earth at the equator is 6, km. The radius from center to pole is 6, km. We need to figure out what side C is. Easy enough. The weight of the Earth is easily Googled: 5. This only leaves us the force pushing down on the other side. No cheating and gathering an army or pushing on it with rocket thrusters. This means that the forces we need for each side of our lever equation are really the weight of our objects times the acceleration — our faked Earth gravity down.
Simplified, this gives us a lever distance of 3. That seems pretty long, but how long is it really? Light is the fastest substance known. One light year is 9. This means our side of the lever would need to be about 4. How long is this? Then c lies halfway between a1 and a2. If a1, a2 were different masses, it would be the weighted average. OK, I have to go now, but work through the calculations. It should take you 10 mintues. Apply to a sphere. Thus, gravity attraction of uniform sphere is same as point mass. Similarly, for any homogenous body, or apply to a cube, and approximate the body as a limit of cube unions.
Give me a lever long enough and a fulcrum on which to place it, and I – Soulay
Probably the biggest and most widely effective lever in physics is that the simplest possible answer is the right one. For instance, many natural combinatorial problems have bijections between them. But it is a different matter to deal with something that is not a point source and is close. There are reasonable symmetry arguments to make the idea that the composite attraction of the many particles is along the line of the center, but it is not easy to make a simple argument that the composite also acts as though all the mass is at the center. By the way note that light radiation near a large glowing body does not act as though it comes from a point source e.
Sun and Earth. That is, replace a single shell of mass with concentric shells having alternating mass. Obtain the same net force by telescoping the series from the left and from the right, thus proving that the force exerted by a point mass telescoping from the left is equal to the force exerted by a shell having the same mass telescoping from the right.
Needless to say, these are two fabulous levers. For example, a very simple vector sum argument establishes that the line of the force will be through the center of the sphere.
Give Me a Lever Long Enough and I Shall Move the World: Leadership, Influence and Power
What we want is the distance of the force through this line. But just as you can get a nice estimate for pi by sampling a square with a circle in it — 8th graders find this interesting and fun — you can also get a pretty good estimate of the resultant force by sampling a sphere and just adding in each force vector on the located particle. The skin of a living organism cuts off an outside from an inside. So does the circumference of a circle in a plane. By tracing the way we represent such a severance, we can begin to reconstruct, ….. The history of many branches of mathematics can be read as the sequential discovery of increasingly powerful levers.
Consider for example the following nine levers of dynamics:. These levers are knit together by an evolving mathematical toolset that increasingly emphasizes considerations of universality and naturality. What dynamical levers are missing from this list? What new dynamical levers may be coming in the 21st century? A reliable way to produce good approaches to research problems and levers?
The problem is in knowing what assumptions one is making to begin with. Here, the assumption people often make without realizing it is that the lines should stay inside the grid. Of course they are not required to, and being aware of this is one key of one approach to solving to the problem. Once one becomes aware of the assumptions one is making, and questions them, and finds some of them to be erroneous, one can then use this as a basis for where to go next.
I like your notion that the prefaces and forwards of advanced math books can be inspiring to younger kids — in fact were for me in the 40s and 50s and very likely turned me towards a degree in pure math. My recollection from 40 plus years ago is that he referenced Sheffer early, and Peirce a little later. I just dug the book out from the stacks in my library, and this is the case.
I first encountered this book because it was one of many that were listed in the Whole Earth Catalog. It was slim which was nice , and obscure which was annoying — especially when most of the obscurity was in his choice of terms and phrasing. I think that the obscure way it is written has led to its attraction in some quarters — it seems more philosophically interesting than it is. The first published account of this kind of logic was done by Sheffer in , and later it was discovered that Peirce had a section on these universal operators in the early s in his unpublished writings.
If Brown were reinventing the wheel, I am guessing Bertrand Russell would have noticed, having worked with both Sheffer and Brown. Spencer-Brown has suceeded in doing what is very rare indeed. He has revealed a new calculus of great power and simplicity. I congratulate him.
We would be dealing with a theory of two-sided forms that can be used only in a one-sided way.
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Notify me of new posts via email. Create a free website or blog at WordPress. The role of levers in finding mathematical proofs Archimedes of Syracuse c. Archimedes is famous for the quote: Give me a place to stand on, and I will move the Earth. Replace the function by Often this change does not affect the theorem being proved, but now , which could dramatically reduce the number of cases that are needed later in the proof.
Two Other Simple Examples Maximum-likelihood estimation often involves setting parameters to maximize the probability of a series of independent events. Since the events are independent, this is just the product of the events. If one wishes to keep all quantities non-negative, one can instead minimize This preserves numerical accuracy and is easier to differentiate. A Nontrivial Example I would like to give a beautiful example of a lever from one of the talks at the workshop.
One can take the input vector and use random sampling to divide the signal up into pieces. This Lever Extends David, in his talk, also outlined several lower bound theorems. Open Problems What are some of your favorite levers? Is this notion helpful to make explicit? Share this: Reddit Facebook. Like this: Like Loading Bhupinder Singh Anand permalink. Allen Knutson permalink.
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