The Cosmic Game ( PDFDrive.com )

Cosmic Game © Douglass A. White, 2012 v151207 62
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We can describe the motion of a pendulum with a stiff, but massless arm in two
dimensions with no losses to friction or air resistance using a differential equation:
(d²θ/ dt²) = - (g /R) sin θ .
θ
R
θ
motion of bob
F = - mg
In this equation g is the acceleration caused by gravity and tends downward. R is the
length of the arm, which we will set at 1 meter. The angle of displacement from vertical
is θ. The motion of the bob at any moment is always along the tangent to the circular arc
at the point the bob has reached along the arc, but the arm of the pendulum forces the bob
to follow the circular path rather than continue along the tangent. We can represent the
tangential component of the gravitational force as:
F = -mg sin θ = ma
a = -g sin θ.
Theta (θ) and the acceleration are in opposite directions, for when the bob swings to the
right, it accelerates back to the left. If s is the length of the arc, we represent it in
radians as R θ.
The speed (NOT velocity) along the arc is then v = ds/dt = Rdθ/dt.
The acceleration along the arc is then a = d²θ/ dt² = R d²θ/ dt².
Therefore, d²θ/ dt² = - (g/R) sin θ.
We can also see the relation from the energy conservation principle. As the bob falls, it
loses potential energy and gains kinetic energy. The change in potential energy is mgh,
where h is the change in height of the bob. The change in kinetic energy from the bob at
rest at its highest point when it starts to fall is ½mv². In our frictionless pendulum no
energy is lost, so ½mv² = mgh, and v = √(2gh).