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Using purely gravitational potential energy, what is the highest efficiency one can achieve with a trebuchet counter-weight type of machine? Efficiency defined here as transformation of potential energy in the counterweight to kinetic energy of the trajectory.


Edit: To be more specific, we can use the following idealization:

  • no friction

  • no air resistance

  • no elastic/material losses

But any of the "standard" trebuchet designs are allowed: simple counterweight, hinged counterweight, vertical counterweight, etc.

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Could you be more explicit? You are asking for a maximum of an efficiency over all possible systems imaginable, so the question isn't easily answerable. Are you considering just real systems, or also idealized ones? If real ones, what conditions do they have to satisfy? In any case, efficiency will probably be very close to one with energy loss just due to waves in the material and such, which can be made arbitrarily small if one is free to vary system's parameters. –  Marek Dec 26 '10 at 23:12
    
@Marek: a trebuchet has basically a "parameter" which is the ratio of the sling to the beam. I would assume the OP means what ratio provides the maximum efficiency. en.wikipedia.org/wiki/Trebuchet –  Sklivvz Dec 26 '10 at 23:21
    
@Sklivvz: thanks. But this should be explicitly mentioned in the question because I don't think average person knows how trebuchet works (I didn't, for example). I thought OP was asking for what kind of material minimizes heat losses due to sound waves in the material or something like that :-) –  Marek Dec 26 '10 at 23:31
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This PDF file cited in the Wikipedia article seems to have a pretty detailed discussion of trebuchet mechanics. In light of that I think this is a more interesting question than I first realized. +1 –  David Z Dec 26 '10 at 23:37
    
edited for more clarity –  Jeremy Dec 27 '10 at 0:15
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1 Answer

up vote 7 down vote accepted

The Wikipedia page on trebuchets links to a PDF paper which discusses exactly this question. It considers several models of varying complexity and finds a maximum range efficiency of 83% for a 100 pound counterweight, 1 pound projectile, a 5 foot long beam pivoted 1 foot from the point of attachment of the counterweight, and a 3.25 foot long sling. Here range efficiency is defined as the horizontal range of the realistic trebuchet model relative to the range of a "black box model" which is able to completely convert the gravitational potential energy of the counterweight into kinetic energy of the projectile.

In order to find the energy efficiency, defined as the fraction of the counterweight's gravitational potential energy that actually gets transferred to the projectile, you would need to use the relation

$$\frac{\epsilon_R}{\epsilon_E} = 2\sin\alpha\cos\alpha = \sin 2\alpha$$

where $\alpha$ is the angle of release of the projectile above the horizontal. Unfortunately, the paper doesn't give the value of $\alpha$ corresponding to the simulation that produced the maximum efficiency, so I can't give you a specific number without running the simulations myself. (Perhaps I'll do that when I have time; if anyone else gets to it first, feel free to edit the relevant numbers in.)

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I think the equation you quoted here is the "range efficiency", which is just a different way of saying "what angle should I launch the trajectory to get maximum range?" and the standard answer from HS physics is $45^\circ$, which for your eqn gives 100% range efficiency. the energy efficiency is different, and is listed in the tables near the end of the Wiki pdf file, as 83% (Table I, p 20) –  Jeremy Dec 27 '10 at 15:41
    
@Jeremy: no, the equation I quoted definitely relates the range efficiency $\epsilon_R$ (as defined in the PDF file, which is not just a different way of identifying the angle of maximum range) to the energy efficiency $\epsilon_E$. See the section on efficiency, pages 5-6; this particular formula is explained at the top of page 6. Note that the maximum angle is not actually $45^\circ$ because $v_0$ is not independent of $\alpha$, i.e. the projectile speed increases as the trebuchet falls, as explained near the bottom of page 8. –  David Z Dec 27 '10 at 20:40
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Also, note that the paper mentions that the efficiencies are calculated from the range, e.g. at the bottom of page 18. In fact, if you take the given value for range (282.4 feet), the given projectile and counterweight masses (1 lb and 100 lb respectively), and $h$ calculated using the second formula on page 5 (comes out to 1.707 feet), you get 0.827, or 83%, for the range efficiency $\epsilon_R$. As mentioned on page 6, the energy efficiency is generally greater than the range efficiency, so you couldn't have a range efficiency of 100% and an energy efficiency of 83%. –  David Z Dec 27 '10 at 20:51
    
Well put. I think I can blame the pdf author for confusing me with the whole concept of a "range efficiency." Energy efficiency makes a lot of physical sense, but range doesn't, at least not to me. –  Jeremy Dec 28 '10 at 13:40
    
@Jeremy: yeah, I can see how energy efficiency makes more sense if you're looking at it from a physics perspective. But I think range efficiency would be the more useful metric if you're trying to actually build a trebuchet (i.e. from an engineering perspective), since the range is generally what you want to maximize. –  David Z Dec 28 '10 at 19:52
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