The Cosmic Game ( PDFDrive.com )

Cosmic Game © Douglass A. White, 2012 v151207 66
The kinematics of this situation are clearly different from those of uniform circular
motion, such as when a rock tied by a string to a rotating pole swings in a circle in a
horizontal plane. In that case the tangential speed is constant and varies in direction at a
constant speed. The centripetal acceleration is also constant and varies in direction at a
constant speed. This makes such motion much simpler.
Clockwise Uniform Circular Motion in a Horizontal Plane
To calculate the lift L of a pendulum when the bob is set in motion by an impulse, we do
as follows.
For simplicity we set R = 1 m. The tangent of the arc at the bob's lowest point represents
the direction of the initial velocity given to the bob by the push. If the bob were a free
particle, it would continue at a constant velocity in the original tangent direction
indefinitely. Thus the length of that tangent would simply increase at a constant pace
over time. However, the arm lifts the bob as it travels forward horizontally along the
tangent of the circle described by the bob on the pendulum arm translating a portion of
the horizontal impulse into vertical lift. The standard approach analyzes from the
viewpoint of angle θ at the fulcrum and the right angle the arm forms with the tangent at
the location of the bob wherever it happens to be on its arc path. If we draw a line
through the bob at its highest point that is parallel to the original horizontal tangent that
passes through the bob's lowest point in the swing, we generate a smaller triangle that is
similar to the larger triangle formed by the original horizontal tangent, a line from the
fulcrum through the bob's lowest point, and the line along the arm that passes through the
bob at its highest point. The vertical side of the smaller triangle has a length of R cos θ,
because the hypotenuse is equal to the radius R. The height of the lift L that the bob
experiences from the swing is thus L = R - (R cos θ) = (1-cos θ). We know that the
tangent P of the large triangle will equal tan θ. Thus the ratio of the lift from the arm's
influence to the horizontal progress of the bob along the tangent line without the pull of
the pendulum arm is L / P = (1 - cos θ) / tan θ. And this ratio tells us the ratio of the
horizontal impulse of acceleration to the opposing downward acceleration of gravity.
Ordinarily the two accelerations would not interfere, except that the pendulum arm
translates a portion of the horizontal acceleration into upward lift against gravity as the
numerator in the ratio shows.
By inspection we know that when the pendulum is vertical, there is no lift, and when it is
horizontal, there is also no lift. For soft pushes the bob goes far horizontally, but rises
only a tiny bit as the angle starts to shift. For hard pushes, the bob soon goes almost
directly upward against gravity, which requires much more push per amount of lift in a