Usual and Unusual Pendulums

A pendulum can be defined as a bob hanging from a string whose other end is attached to a fixed point. When the bob is displaced from its equilibrium position and set free, it will move back and forth along a circular path.
Huygens showed that if the string of a moving pendulum is forced to bend along a cycloid, then the bob moves along a congruent cycloid. In a similar way, the string can be bent along other curves. The curve along which the string of a pendulum is bent is the evolute of the orbit of the pendulum.
Such pendulums work only if the evolute of the orbit is concave upward. Otherwise, the motion of the bob can be simulated by a bead that slides without friction along a wire that has the shape of the desired orbit. In this case, the endpoints of the orbit must be at the same height and they must be located higher than any other point of the orbit.
This Demonstration compares the motion of five pendulums that follow different orbits: a circle, a cycloid, an ellipse, a parabola, and a hyperbola. The starting, ending, and lowest points of all orbits coincide. For each orbit, the corresponding evolute is also shown. Since the evolutes of the parabola and the hyperbola are concave downward, these pendulums follow the bead model.
The periods of the pendulums vary considerably; the shortest is that of the cycloidal pendulum, which is the well-known brachistochrone property of the cycloid.


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The evolute of a curve is defined as the locus of the centers of curvature of the curve. The evolute of a curve is important in the theory of the pendulum because of the property expressed by the following theorem (see [1], p. 312).
If and are the centers of curvature corresponding to the points and of a curve, the length of the arc of the evolute between and is equal to the difference between the radii of curvature at and .
The evolute of the orbit of a pendulum can be calculated in the following way (see e.g. [2], p. 79). If is a parametric equation of the orbit of a pendulum, the unit tangent vector of at any is given by:
The curvature of at any is given by the magnitude of the vector:
while for each the center of curvature can be located by the vector:
It is interesting to note that from this definition we find that the evolute of a circle consists of a single point, the center of the circle.
To describe the motion of a pendulum, we need to derive a function that gives the position of the pendulum at any time . Since the motion of a pendulum is periodic, we only need to know the period of the pendulum and the function for . The function can be derived on the basis of the principle of conservation of mechanical energy (see e.g. [1], p. 309). The calculations for this Demonstration are done numerically using standard Mathematica functions.
The principle of conservation of mechanical energy requires that the kinetic energy of the pendulum at any instant be equal to the difference between its potential energy at the initial position and its potential energy at the current position. Assuming that a parametric equation of the orbit of a pendulum is and that the endpoints of the orbit are and , we find:
where is the acceleration of gravity.
If gives the arc length traveled by the bob as a function of time, then
From this equation we find that the period of the pendulum is
We also find that the time it takes the bob to move from the initial position to a position is
Since the motion of the bob is symmetric, we only need to establish a function that gives the time for any only for the first half of the orbit, that is, for the first quarter of the period. By inverting this function we can obtain the functions and for . Because of the symmetries in the motion of the bob, this function can be extended so that the function gives the position of the bob for any time in . Since the period of the bob can also be calculated as we indicate above, its position can be determined at any time .
[1] G. F. Simmons, Calculus Gems: Brief Lives and Memorable Mathematics, New York: McGraw-Hill, 1992.
[2] B. Davis, H. Porta, and J. Uhl, Calculus & Mathematica: Vector Calculus: Measuring in Two and Three Dimensions, Reading, MA: Addison-Wesley Publishing Company, 1994.
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