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.
