First of all, a spherical bob (on a massless string) centered L below the
pivot is a
compound pendulum. Its period, for a given amplitude, depends on its radius
R! It acts
like a point mass centered at L+(3/5)*R^2/L rather than L. Unless it's
hollow, when it's
just L+R^2/L. So you can tell if it's hollow by swinging it. For a unit
n-cube hanging
by a vertex, this "center of oscillation" is 2/3 down the main diagonal, for
all n.
Wikipedia claims the center of oscillation is also the center of percussion.
So if you
hang a bat by the knob, and match oscillations with a ball on an adjustably
long string,
the ball will hang down to the sweet spot. If you're the sort of person who
swings bats
by their knob.
Mathematica 7 clams on the ODE of motion:
{\[Theta]''[t] == -g*Sin[\[Theta][t]]/L, \[Theta][0] == 0}
but Macsyma returns an integral for t[\Theta] which Mma knows is an
EllipticF. Testing by taking the limit for small \[Theta]'[0] insists
the period approaches 149/30 Sqrt[L/g] instead of 2 pi Sqrt[L/g]! The bug
is in Limit.
Mma then solves for Theta,
\[Theta][t] -> 2*JacobiAmplitude[
(1/2)*t*Derivative[1][\[Theta]][0],(4*g)/(L*Derivative[1][\[Theta]][0]^2)]
with period
\[CapitalLambda] == 4*Sqrt[L/g]*EllipticK[Sin[Subscript[\[Theta], max]/2]^2]
and amplitude
Subscript[\[Theta], max] ==
2*ArcSin[(1/2)*Sqrt[L/g]*Derivative[1][\[Theta]][0]]
The differentiator and simplifiers eventually confess that these solve the
ODE.
But had I thought to search "pendulum" in the Mma doc, it would have led me
to an
"application" of JacobiAmplitude where it is claimed, without derivation, to
describe
an overswung pendulum. This would have been a clue.
I'm surprised this all isn't more obvious in, e.g., Wikipedia or MathWorld.
Or DSolve.
--rwg
