Do you mean in my solution that involves differential equations? 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 On Wed, Nov 16, 2011 at 10:08 AM, Aaron Meurer <[email protected]> wrote: > 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. > > -- "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 [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.
