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The mathematics of pendulums are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allow the equations of motion to be solved analytically for small-angle oscillations.
Simple gravity pendulum
A so-called "simple pendulum" is an idealization of a "real pendulum" but in an isolated system using the following assumptions:
- The rod or cord on which the bob swings is massless, inextensible and always remains taut;
- The bob is a point mass;
- Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
- The motion does not lose energy to friction or air resistance.
- The gravitational field is uniform.
- The support does not move.
The differential equation which represents the motion of a simple pendulum is
- Eq. 1
where g is acceleration due to gravity, l is the length of the pendulum, and θ is the angular displacement.
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
yields the equation for a harmonic oscillator,
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/(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.
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 g ≈ 9.81 m/s2, and g/ ≈ 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.
and then integrating over one complete cycle,
or twice the half-cycle
or four times the quarter-cycle
which leads to
Note that this integral diverges as θ0 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.)
This integral can be rewritten in terms of elliptic integrals as
where F is the incomplete elliptic integral of the first kind defined by
Or more concisely by the substitution
expressing θ in terms of u,
Here K is the complete elliptic integral of the first kind defined by
For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (g = 65 m/s2) at initial angle 10 degrees is 9.806
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
where n!! denotes the double factorial, an exact solution to the period of a pendulum is:
Figure 4 shows the relative errors using the power series. T0 is the linear approximation, and T2 to T10 include respectively the terms up to the 2nd to the 10th powers.
Power series solution for the elliptic integral
Another formulation of the above solution can be found if the following Maclaurin series:
Arithmetic-geometric mean solution for elliptic integral
where M(x,y) is the arithmetic-geometric mean of x and y.
The first iteration of this algorithm gives
The second order expansion of reduces to
A second iteration of this algorithm gives
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
- 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 inertia I around the pivot point.
The equation of torque gives:
- α is the angular acceleration.
- τ is the torque
The torque is generated by gravity so:
- m is the mass of the body
- L is the distance from the pivot to the center of mass of the pendulum
- θ is the angle from the vertical
Hence, under the small-angle approximation sin θ ≈ θ,
where Icm is the moment of inertia of the body about its center of mass.
And a frequency of
If the initial angle is taken into consideration (for large amplitudes), then the expression for becomes:
and gives a period of:
where θ0 is the maximum angle of oscillation (with respect to the vertical) and K(k) is the complete elliptic integral of the first kind.
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.
Coupled pendulums can affect each other's motion, either through a direction connection (such as a spring connecting the bobs) or through motions in a supporting structure (such as a tabletop). The equations of motion for two identical simple pendulums coupled by a spring connecting the bobs can be obtained using Lagrangian Mechanics.
The kinetic energy of the system is:
where is the mass of the bobs, is the length of the strings, and , are the angular displacements of the two bobs from equilibrium.
The potential energy of the system is:
The Lagrangian is then
which leads to the following set of coupled differential equations:
Adding and subtracting these two equations in turn, and applying the small angle approximation, gives two harmonic oscillator equations in the variables and :
with the corresponding solutions
and , , , are constants of integration.
Expressing the solutions in terms of and alone:
If the bobs are not given an initial push, then the condition requires , which gives (after some rearranging):
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- Physical Pendulum
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