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A pendulum is a body suspended from a fixed support so that it swings freely back and forth under the influence of gravity. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging it back and forth. 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.
A simple gravity pendulum is an idealized mathematical model of a real pendulum. This is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in this model there is no frictional energy loss, when given an initial displacement it will swing back and forth at a constant amplitude. The model is based on these assumptions
The rod or cord on which the bob swings is massless, inextensible and always remains taut;
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 ? 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 ? 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 ? 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 :
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 ?0, 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 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 (?0). 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 ? ? 0 we find sin ? ? ?.
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 ?3 (from Taylor expansion for sin ?).
Given the initial conditions ?(0) = ?0 and (0) = 0, the solution becomes
The motion is simple harmonic motion where ?0 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 ?0; 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 ? 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.
Figure 3. Deviation of the "true" period of a pendulum from the small-angle approximation of the period. "True" value was obtained numerically evaluating the elliptic integral.
Figure 4. Relative errors using the power series for the period.
Figure 5. Potential energy and phase portrait of a simple pendulum. Note that the x-axis, being angle, wraps onto itself after every 2? radians.
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),
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 ?0 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.)
where M(x,y) is the arithmetic-geometric mean of x and y.
This yields an alternative and faster-converging formula for the period:
The first iteration of this algorithm gives
This approximation has the relative error of less than 1% for angles up to 96.11 degrees. Since the expression can be written more concisely as
The second order expansion of reduces to
A second iteration of this algorithm gives
This second approximation has a relative error of less than 1% for angles up to 163.10 degrees.[clarification needed]
Approximate formulae for the nonlinear pendulum period
Though the exact period can be determined, for any finite amplitude rad, by evaluating the corresponding complete elliptic integral , where , this is often avoided in applications because it is not possible to express this integral in a closed form in terms of elementary functions. This has made way for research on simple approximate formulae for the increase of the pendulum period with amplitude (useful in introductory physics labs, classical mechanics, electromagnetism, acoustics, electronics, superconductivity, etc. The approximate formulae found by different authors can be classified as follows:
'Not so large-angle' formulae, i.e. those yielding good estimates for amplitudes below rad (a natural limit for a bob on the end of a flexible string), though the deviation
with respect to the exact period increases monotonically with amplitude, being unsuitable for amplitudes near to rad. One of the simplest formulae found in literature is the following one by Lima (2006): , where .
'Very large-angle' formulae, i.e. those which approximate the exact period asymptotically for amplitudes near to rad, with an error that increases monotonically for smaller
amplitudes (i.e., unsuitable for small amplitudes). One of the better such formulae is that by Cromer, namely:.
Of course, the increase of with amplitude is more apparent when , as has been observed in many experiments using either a rigid rod or a disc. As accurate timers and sensors are currently available even in introductory physics labs, the experimental errors found in 'very large-angle' experiments are already small enough for a comparison with the exact period and a very good agreement between theory and experiments in which friction is negligible has been found. Since this activity has been encouraged by many instructors, a simple approximate formula for the pendulum period valid for all possible amplitudes, to which experimental data could be compared, was sought. In 2008, Lima derived a weighted-average formula with this characteristic:
where , which presents a maximum error of only 0.6% (at ).
Arbitrary-amplitude angular displacement Fourier series
The Fourier series expansion of is given by
where is the elliptic nome, , and the angular frequency.
If one defines
can be approximated using the expansion
(see ). Note that for we have , thus the approximation is applicable even for large amplitudes.
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 inertiaI around the pivot point.
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 ?0 is the maximum angle of one pendulum and 180° - ?0 is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other.
Two identical simple pendulums coupled via a spring connecting the bobs.
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.