![Oscillating pendulum](https://upload.wikimedia.org/wikipedia/commons/2/24/Oscillating_pendulum.gif)
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
pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. 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 th ...
s are in general quite complicated. Simplifying assumptions can be made, which in the case of a
simple pendulum
A pendulum is a weight suspended from a pivot so that it can swing freely. 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 th ...
allow the equations of motion to be solved analytically for small-angle oscillations.
Simple gravity pendulum
A ''simple gravity pendulum'' is an idealized mathematical model of a real pendulum.
This is a weight (or
bob
Bob, BOB, or B.O.B. may refer to:
Places
* Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname ...
) on the end of a massless cord suspended from a
pivot
Pivot may refer to:
*Pivot, the point of rotation in a lever system
*More generally, the center point of any rotational system
*Pivot joint, a kind of joint between bones in the body
*Pivot turn, a dance move
Companies
*Incitec Pivot, an Austra ...
, without
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
. 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 amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
. The model is based on these 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
In mathematics, a plane is a Euclidean ( flat), two-dimensional surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. Planes can arise as ...
, i.e. the bob does not trace an
ellipse but an
arc.
* The motion does not lose energy to
friction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material elements sliding against each other. There are several types of friction:
*Dry friction is a force that opposes the relative lateral motion of ...
or
air resistance
In fluid dynamics, drag (sometimes called air resistance, a type of friction, or fluid resistance, another type of friction or fluid friction) is a force acting opposite to the relative motion of any object moving with respect to a surrounding flu ...
.
* The gravitational field is uniform.
* The support does not move.
The
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
which represents the motion of a simple pendulum is
where is the
magnitude
Magnitude may refer to:
Mathematics
*Euclidean vector, a quantity defined by both its magnitude and its direction
*Magnitude (mathematics), the relative size of an object
*Norm (mathematics), a term for the size or length of a vector
*Order of ...
of the
gravitational field, is the length of the rod or cord, and is the angle from the vertical to the pendulum.
Small-angle approximation
![Small-angle approximation for sine function](https://upload.wikimedia.org/wikipedia/commons/d/d0/Small-angle_approximation_for_sine_function.svg)
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
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
(often cited as less than 0.1 radians, about 6°), or
then substituting for into using the
small-angle approximation
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
:
\begin
\sin \theta &\approx \theta \\
\cos \theta &\approx 1 - \ ...
,
yields the equation for a
harmonic oscillator,
The error due to the approximation is of order (from the
Taylor expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor seri ...
for ).
Let the starting angle be . If it is assumed that the pendulum is released with zero
angular velocity, the solution becomes
The motion is
simple harmonic motion
In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of periodic motion of a body resulting from a dynamic equilibrium between an inertial force, proportional to the acceleration of the body away from the ...
where is the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The corresponding approximate
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
of the motion is then
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 ; this is the property of
isochronism
A sequence of events is isochronous if the events occur regularly, or at equal time intervals. The term ''isochronous'' is used in several technical contexts, but usually refers to the primary subject maintaining a constant period or interval ( ...
that
Galileo discovered.
Rule of thumb for pendulum length
gives
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 , and (0.994 is the approximation to 3 decimal places).
Therefore, relatively reasonable approximations for the length and period are:
where is the number of seconds between ''two'' beats (one beat for each side of the swing), and is measured in metres.
Arbitrary-amplitude period
![Pendulum phase portrait](https://upload.wikimedia.org/wikipedia/commons/4/49/Pendulum_phase_portrait.svg)
For amplitudes beyond the
small angle approximation
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
:
\begin
\sin \theta &\approx \theta \\
\cos \theta &\approx 1 - \ ...
, one can compute the exact period by first inverting the equation for the angular velocity obtained from the energy method (),
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 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 integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s as
where is the
incomplete elliptic integral of the first kind defined by
Or more concisely by the
substitution
expressing in terms of ,
Here 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 ( = ) 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 with geographical location.
From here there are many ways to proceed to calculate the elliptic integral.
Legendre polynomial solution for the elliptic integral
Given and the
Legendre polynomial
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ...
solution for the elliptic integral:
where denotes the
double factorial, an exact solution to the period of a simple pendulum is:
Figure 4 shows the relative errors using the power series. is the linear approximation, and to 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:
is used in the Legendre polynomial solution above.
The resulting power series is:
more fractions available in the
On-Line Encyclopedia of Integer Sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching at AT&T Labs. He transferred the intellectual property and hosting of the OEIS to t ...
with having the numerators and having the denominators.
Arithmetic-geometric mean solution for elliptic integral
Given and the
arithmetic–geometric mean solution of the elliptic integral:
where is the arithmetic-geometric mean of and .
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.
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.
Examples
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.
File:Pendulum_0deg.gif, Initial angle of 0°, a stable equilibrium
File:Pendulum_45deg.gif, Initial angle of 45°
File:Pendulum_90deg.gif, Initial angle of 90°
File:Pendulum_135deg.gif, Initial angle of 135°
File:Pendulum_170deg.gif, Initial angle of 170°
File:Pendulum_180deg.gif, Initial angle of 180°, unstable equilibrium
File:Pendulum_190deg.gif, Pendulum with just barely enough energy for a full swing
File:Pendulum_220deg.gif, Pendulum with enough energy for a full swing
Compound pendulum
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
In physics, a rigid body (also known as a rigid object) is a solid body in which deformation is zero or so small it can be neglected. The distance between any two given points on a rigid body remains constant in time regardless of external fo ...
swinging by a pivot. In this case the pendulum's period depends on its
moment of inertia around the pivot point.
The equation of
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
gives:
where:
* is the angular acceleration.
* is the torque
The torque is generated by gravity so:
where:
* is the mass of the body
* is the distance from the pivot to the object's center of mass
* is the angle from the vertical
Hence, under the small-angle approximation ,
where is the moment of inertia of the body about the pivot point.
The expression for is of the same form as the conventional simple pendulum and gives a period of
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 is the maximum angle of oscillation (with respect to the vertical) and 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 In mathematics, a doubly periodic function is a function defined on the complex plane and having two "periods", which are complex numbers ''u'' and ''v'' that are linearly independent as vectors over the field of real numbers. That ''u'' and '' ...
with a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
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 is the maximum angle of one pendulum and is the maximum angle of another, then the real period of each is the magnitude of the imaginary period of the other.
Coupled pendula
![Coupled pendulums](https://upload.wikimedia.org/wikipedia/commons/e/ef/Coupled_pendulums.svg)
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
In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph- ...
.
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:
where
is the
gravitational acceleration
In physics, gravitational acceleration is the acceleration of an object in free fall within a vacuum (and thus without experiencing drag). This is the steady gain in speed caused exclusively by the force of gravitational attraction. All bodi ...
, and
is the
spring constant
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of th ...
. The displacement
of the spring from its equilibrium position assumes the
small angle approximation
The small-angle approximations can be used to approximate the values of the main trigonometric functions, provided that the angle in question is small and is measured in radians:
:
\begin
\sin \theta &\approx \theta \\
\cos \theta &\approx 1 - \ ...
.
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
where
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):
See also
*
Blackburn pendulum
*
Conical pendulum
A conical pendulum consists of a weight (or bob) fixed on the end of a string or rod suspended from a pivot. Its construction is similar to an ordinary pendulum; however, instead of swinging back and forth, the bob of a conical pendulum moves at ...
*
Cycloidal pendulum
*
Double pendulum
*
Inverted pendulum
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and without additional help will fall over. It can be suspended stably in this inverted position by using a control system to monitor the angle ...
*
Kapitza's pendulum
*
Rayleigh–Lorentz pendulum Rayleigh–Lorentz pendulum (or Lorentz pendulum) is a simple pendulum, but subjected to a slowly varying frequency due to an external action (frequency is varied by varying the pendulum length), named after Lord Rayleigh and Hendrik Lorentz. This ...
*
Spring pendulum
*
Mathieu function
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation
:
\frac + (a - 2q\cos(2x))y = 0,
where a and q are parameters. They were first introduced by Émile Léonard Mathieu ...
*
Pendulum equations (software)
References
Further reading
*
*
*{{cite journal , first=Kenneth L. , last=Sala , title=Transformations of the Jacobian Amplitude Function and its Calculation via the Arithmetic-Geometric Mean , journal=SIAM J. Math. Anal. , volume=20 , issue=6 , pages=1514–1528 , year=1989 , doi=10.1137/0520100
External links
Mathworld article on Mathieu Function
Differential equations
Dynamical systems
Horology
Mathematical physics
Mathematics