But the 2*pi terms all cancel in your solution. Aaron Meurer
On Wed, Nov 16, 2011 at 10:08 AM, Luke Peterson <[email protected]> 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 <[email protected]> wrote: > >> On Wed, Nov 16, 2011 at 1:04 AM, Luke <[email protected]> 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 [email protected]. >> To unsubscribe from this group, send email to >> [email protected]. >> 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 [email protected]. > To unsubscribe from this group, send email to > [email protected]. > 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 [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/sympy?hl=en.
