On Wed, Nov 16, 2011 at 11:12 AM, Luke <hazelnu...@gmail.com> wrote: > Do you mean in my solution that involves differential equations?
Yes (assumedly the solution will be the same no matter what method you use to derive it, so long as you always make the "small angles" approximation). > > I think this dimensional analysis approach may have merit, I just need > to see all the steps and make sure they can all be justified without > referencing a differential equation. > > ~Luke Perhaps this would be easier to see (at least for someone like me who doesn't remember this particular part of physics very well) if you added explicit dimensions to all the numbers in your derivation. It's hard to tell which "1's" are really unit-ed. And by the way, I'm still convinced that the "trick" involves deriving it geometrically somehow, although in my experience, dimensional analysis is a favorite among the types who like to use little tricks to derive things. Aaron Meurer > > On Wed, Nov 16, 2011 at 10:08 AM, Aaron Meurer <asmeu...@gmail.com> wrote: >> But the 2*pi terms all cancel in your solution. >> >> Aaron Meurer >> >> On Wed, Nov 16, 2011 at 10:08 AM, Luke Peterson <hazelnu...@gmail.com> wrote: >>> I think the 2*pi is essential, so without it, I'm not sure. >>> >>> ~Luke >>> >>> On Nov 16, 2011, at 7:04 AM, Ondřej Čertík <ondrej.cer...@gmail.com> wrote: >>> >>>> On Wed, Nov 16, 2011 at 1:04 AM, Luke <hazelnu...@gmail.com> wrote: >>>>>> I thought the general idea of the problem was to not use differential >>>>>> equations and calculus? (or "other fancy mathematical tricks"). I feel >>>>>> like >>>>>> that is the challenge of the problem... >>>>> >>>>> Yes. I just don't know how to separate Newtonian mechanics from >>>>> differential equations.... so I didn't. My solution won't be >>>>> considered for that reason. >>>>> >>>>> I've been discussing this with some other physicists and our >>>>> fundamental stumbling block is that the problem defines the pendulum >>>>> via it's natural period. The natural period of a pendulum depends on >>>>> the amplitude of oscillation, unlike something more intrinsic, like >>>>> the length. So without specifying an amplitude associated with the >>>>> period, this way of characterizing a pendulum is somewhat ambiguous. >>>>> The relationship between period and length that is familiar, namely T >>>>> = 2*pi*sqrt(l/g), I can only obtain from a differential equation. But >>>> >>>> Ah, if this is the only problem, then I know a tricky derivation. It's >>>> super simple: >>>> >>>> 1) assume, that the period depends on the length "l" and "g" in the >>>> following form: >>>> >>>> T = a * l^b * g^c >>>> >>>> where a, b, c are constants, possibly zero. We can try to figure out >>>> some physical basis for it, >>>> I would just say for now, that this is the first order approximation. Why >>>> not. >>>> >>>> From dimensional analysis: >>>> >>>> [T] = s >>>> [l] = m >>>> [g] = m s^-2 >>>> >>>> we get: >>>> >>>> s = a * m^b * m^c s^(-2c) >>>> >>>> so: >>>> >>>> b + c = 0 >>>> 1 = -2*c >>>> >>>> and finally: >>>> >>>> c = -1/2 >>>> b = 1/2 >>>> >>>> in other words: >>>> >>>> T = a * sqrt(l/g) >>>> >>>> it doesn't give you the constant "a", but it gives you the dependence >>>> on "l" and "g". >>>> >>>>> perhaps that relationship is derivable without appealing to >>>>> differential equations. In any event, there must be some way to >>>>> relate period to length, and if it isn't this relationship, I don't >>>>> know what it is or how to rationalize it. >>>> >>>> Given the above, how would you solve the problem? >>>> >>>> Ondrej >>>> >>>> -- >>>> You received this message because you are subscribed to the Google Groups >>>> "sympy" group. >>>> To post to this group, send email to sympy@googlegroups.com. >>>> To unsubscribe from this group, send email to >>>> sympy+unsubscr...@googlegroups.com. >>>> For more options, visit this group at >>>> http://groups.google.com/group/sympy?hl=en. >>>> >>> >>> -- >>> You received this message because you are subscribed to the Google Groups >>> "sympy" group. >>> To post to this group, send email to sympy@googlegroups.com. >>> To unsubscribe from this group, send email to >>> sympy+unsubscr...@googlegroups.com. >>> For more options, visit this group at >>> http://groups.google.com/group/sympy?hl=en. >>> >>> >> >> -- >> You received this message because you are subscribed to the Google Groups >> "sympy" group. >> To post to this group, send email to sympy@googlegroups.com. >> To unsubscribe from this group, send email to >> sympy+unsubscr...@googlegroups.com. >> For more options, visit this group at >> http://groups.google.com/group/sympy?hl=en. >> >> > > > > -- > "Those who would give up essential liberty to purchase a little > temporary safety deserve neither liberty nor safety." > > -- Benjamin Franklin, Historical Review of Pennsylvania, 1759 > > -- > You received this message because you are subscribed to the Google Groups > "sympy" group. > To post to this group, send email to sympy@googlegroups.com. > To unsubscribe from this group, send email to > sympy+unsubscr...@googlegroups.com. > For more options, visit this group at > http://groups.google.com/group/sympy?hl=en. > > -- You received this message because you are subscribed to the Google Groups "sympy" group. 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