The Cosmic Game ( PDFDrive.com )

Cosmic Game © Douglass A. White, 2012 v151207 67
given unit of time and the angle reduces efficiency. Somewhere in between is a
maximum lift for the amount of push. We might guess it would be at 45°, but let's
calculate.
We only give the pendulum one initial push, and the average ratio changes with different
pushes. To find the sweet spot in that ratio (the maximum lift per push) we will
calculate the derivative of the ratio and set the result to zero.
d(L/P)/d θ = d[(1-cos θ)/tan θ]/d θ =
tan θ d(1-cos θ) - (1-cos θ)d(tan θ)/tan² θ.
(sin² θ cos θ/cos² θ) - (1/cos² θ) + (cos θ / cos² θ) = 0
sin² θ cos θ + cos θ = 1
sin² θ + cos² θ = 1
sin² θ (cos θ -1) = cos θ (cos θ - 1).
sin² θ = cos θ.
This equation is satisfied approximately at 51° 49' 48" (51.83°) where sin θ = .78618 and
cos θ = .618.
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Here is another way of reaching the same result that will also clarify the mechanics. We
will use the half angle A = θ/2, which is the angle used to calculate the accurate period of
a pendulum. Vector P indicates the distance the bob would move tangent to the lowest
point if uninfluenced by the pendulum arm. The chord which we will label c will be the
hypotenuse of a small right triangle with a run of x and a rise of L (or y on the y-axis if
we place it on an x-y grid) that is similar to either of the two halves of the isosceles
triangle that forms between the two radii (fulcrum F to bob at starting point O and
fulcrum F to bob at stopping point H) when the chord and the angle between those two
radii are bisected. (On the above diagram the triangles have sides R, c/2, B and B, c/2,
H-R).
As the bob swings to the right, the pendulum arm pulls the bob upward parallel to the y
axis OF and backward along the bob's straight tangential path on the x axis. As the
distance of the bob from perpendicular to the center fulcrum F increases along the
original tangent path, the pendulum arm acts as a brake, slowing down the bob's
horizontal component of progress and translating part of the motion into upward lift
against gravity, so that gravity slows the upward progress until the bob stops at the
furthest point it reaches along the arc defined by the chord c. The triangle L, w, z shows
the force vectors that bring the bob to its furthest point away from OF along the arc in
this analysis. The force vectors counteract the horizontal force of the push P and the
downward pull of gravity and lead to the resultant lift (L = y) at the distance x along the
tangent to the lowest point.