The not-so-simple pendulum

A comprehensive analytical solution of the nonlinear pendulum

“…The product on the righthand side of this equation is the periodically repeated, highlighted part of the sn function in figure A1. The angular velocity can be found by inserting (19) in (9) and considering definition (12):…”

“…In recent years, some effort has been made to solve this differential equation by means of approximations, JACOBI elliptic functions and hypergeometric functions; see e.g. [4][5][6][7][8][9][10][11][12][13][14][15][16][17]. Whereas most papers are limited to a pendulum that does not swing over, one can find in [16] and [17] also a solution for a pendulum that swings over.…”

Figuring the Acceleration of the Simple Pendulum

“…1,2 The physics to calculate the acceleration of a simple pendulum has been around for more than 300 years, and a fairly complete treatise has been given by C. Schwarz in this journal. 3 But sentences like "the acceleration is always directed towards the equilibrium position" beside the picture of a swing on a circular arc can still be found in textbooks, as e.g. in Ref.…”

“…A dimensionless factor f is defined as: (4) With this convention, the centripetal acceleration is a c = (f − 2)g + 2g cos j (5) and the amplitude of the oscillation is Ref. 3. Focusing on the curve whose intercept with the left axis is 2, we see that the acceleration at the lowest point of that orbit is 2g and is a maximum, and we also see that the acceleration decreases to the value g at its turning points of ±90°.…”

Ubiquitous Drawing Errors for the Simple Pendulum

“…Derivations of the angular dependence of the forces and the acceleration for the simple pendulum appear in textbooks and have been addressed in detail in this journal. 2 Measurements of acceleration and tension that help clarify the physics have also been described. 3,4 At any point in the trajectory of the swinging mass, the weight (W) and the string tension (T) are the only forces acting.…”