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.
Consider Figure 1 on the right, which shows the forces acting on a simple pendulum. Note that the path of the pendulum sweeps out an arc of a circle. The angle θ 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,
where g is the acceleration due to gravity near the surface of the earth. The negative sign on the right hand side implies that θ and a 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 a along the red axis can be related to the change in angle θ by the arc length formulas; s is arc length:
The change in kinetic energy (body started from rest) is given by
Since no energy is lost, the gain in one must be equal to the loss in the other
The change in velocity for a given change in height can be expressed as
Using the arc length formula above, this equation can be rewritten in terms of dθ/dt:
where h 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 θ0, then y0, the vertical distance from the screw, is given by
Similarly, for y1, we have
Then h is the difference of the two
In terms of dθ/dt gives
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 (θ0). 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.
Small-angle approximation for the sine function: For θ ≈ 0 we find sin θ ≈ θ.
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 (often cited as less than 0.1 radians, about 6°), or
The error due to the approximation is of order θ3 (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 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.
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 g ≈ 9.81 m/s2, and g/π2 ≈ 1 (0.994 is the approximation to 3 decimal places).
Therefore, a relatively reasonable approximation for the length and period are,
where T0 is the number of seconds between two beats (one beat for each side of the swing), and l is measured in metres.
The animations below depict the motion of a simple (frictionless) pendulum with increasing amounts of initial displacement of the bob, or equivalently increasing initial velocity. The small graph above each pendulum is the corresponding phase plane diagram; the horizontal axis is displacement and the vertical axis is velocity. With a large enough initial velocity the pendulum does not oscillate back and forth but rotates completely around the pivot.
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
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.
^Appell, Paul (July 1878). "Sur une interprétation des valeurs imaginaires du temps en Mécanique" [On an interpretation of imaginary time values in mechanics]. Comptes Rendus Hebdomadaires des Scéances de l'Académie des Sciences. 87 (1).
^Adlaj, S. (2012), "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", Saint Petersburg, p. 13–18