The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations.
Figure 1. Force diagram of a simple gravity pendulum.
Please take the time to consider Figure 1 on the right, showing the forces acting on a simple pendulum. Note that the path of the pendulum sweeps out an arc of a circle. The angle is measured in radians, and this is crucial for this formula. The blue arrow is the gravitational force acting on the bob, and the violet arrows are that same force resolved into components parallel and perpendicular to the bob's instantaneous motion. The direction of the bob's instantaneous velocity always points along the red axis, which is considered the tangential axis because its direction is always tangent to the circle. Consider Newton's second law,
where F is the sum of forces on the object, m is mass, and a is the acceleration. Because we are only concerned with changes in speed, and because the bob is forced to stay in a circular path, we apply Newton's equation to the tangential axis only. The short violet arrow represents the component of the gravitational force in the tangential axis, and trigonometry can be used to determine its magnitude. Thus,
is the acceleration due to gravity near the surface of the earth. The negative sign on the right hand side implies that and always point in opposite directions. This makes sense because when a pendulum swings further to the left, we would expect it to accelerate back toward the right.
This linear acceleration along the red axis can be related to the change in angle by the arc length formulas; is arc length:
change in kinetic energy (body started from rest) is given by
Since no energy is lost, those two must be equal
Using the arc length formula above, this equation can be rewritten in favor of
is the vertical distance the pendulum fell. Look at Figure 2, which presents the trigonometry of a simple pendulum. If the pendulum starts its swing from some initial angle , then , the vertical distance from the screw, is given by
This equation is known as the first integral of motion, it gives the velocity in terms of the location and includes an integration constant related to the initial displacement (). We can differentiate, by applying the chain rule, with respect to time to get the acceleration
which is the same result as obtained through force analysis.
The differential equation given above is not easily solved, and there is no solution that can be written in terms of elementary functions. However adding a restriction to the size of the oscillation's amplitude gives a form whose solution can be easily obtained. If it is assumed that the angle is much less than 1 radian, or
The error due to the approximation is of order θ 3 (from the Maclaurin series for sin θ).
Given the initial conditions θ(0) = θ0 and dθ/dt(0) = 0, the solution becomes,
The motion is simple harmonic motion where θ0 is the semi-amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The period of the motion, the time for a complete oscillation (outward and return) is
which is known as Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude θ0; this is the property of isochronism that Galileo discovered.
Rule of thumb for pendulum length
can be expressed as
If SI units are used (i.e. measure in metres and seconds), and assuming the measurement is taking place on the Earth's surface, then m/s2, and (0.994 is the approximation to 3 decimal places).
Therefore a relatively reasonable approximation for the length and period are,
For amplitudes beyond the small angle approximation, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method (Eq. 2),
Figure 3. Deviation of the "true" period of a pendulum from the small-angle approximation of the period. "True" value was obtained using Matlab to numerically evaluate the elliptic integral.
Figure 4. Relative errors using the power series.
and then integrating over one complete cycle,
or twice the half-cycle
or 4 times the quarter-cycle
which leads to
Note that this integral diverges as approaches the vertical
so that a pendulum with just the right energy to go vertical will never actually get there. (Conversely, a pendulum close to its maximum can take an arbitrarily long time to fall down.)
For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (g = 9.80665 m/s2) at initial angle 10 degrees is . The linear approximation gives . The difference between the two values, less than 0.2%, is much less than that caused by the variation of g with geographical location.
From here there are many ways to proceed to calculate the elliptic integral:
Legendre polynomial solution for the elliptic integral
This yields an alternative and faster-converging formula for the period:
The animations below depict several different modes of oscillation given different initial conditions. The small graph above the pendulums are their phase portraits.
Initial angle of 0°, a stable equilibrium.
Initial angle of 45°
Initial angle of 90°
Initial angle of 135°
Initial angle of 170°
Initial angle of 180°, unstable equilibrium.
Pendulum with just barely enough energy for a full swing
Pendulum with enough energy for a full swing
A compound pendulum (or physical pendulum) is one where the rod is not massless, and may have extended size; that is, an arbitrarily shaped rigid body swinging by a pivot. In this case the pendulum's period depends on its moment of inertiaI around the pivot point.
Physical interpretation of the imaginary period
The Jacobian elliptic function that expresses the position of a pendulum as a function of time is a doubly periodic function with a real period and an imaginary period. The real period is of course the time it takes the pendulum to go through one full cycle. Paul Appell pointed out a physical interpretation of the imaginary period: if θ0 is the maximum angle of one pendulum and 180° − θ0 is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other. This interpretation, involving dual forces in opposite directions, might be further clarified and generalized to other classical problems in mechanics with dual solutions.
^Paul Appell, "Sur une interprétation des valeurs imaginaires du temps en Mécanique", Comptes Rendus Hebdomadaires des Scéances de l'Académie des Sciences, volume 87, number 1, July, 1878
^Adlaj, S. Mechanical interpretation of negative and imaginary tension of a tether in a linear parallel force field , Selected papers of the International Scientific Conference on Mechanics "SIXTH POLYAKHOV READINGS", January 31 - February 3, 2012, Saint-Petersburg, Russia, pp. 13-18.