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.
