Pendulum (mechanics) explained

A pendulum is a body suspended from a fixed support such that it freely swings 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 towards 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.

Simple gravity pendulum

A simple gravity pendulum^[1] is an idealized mathematical model of a real pendulum.^{[2] [3] [4]} It is a weight (or bob) on the end of a massless cord suspended from a pivot, without friction. Since in the model there is no frictional energy loss, when given an initial displacement it swings back and forth with a constant amplitude. The model is based on the assumptions:

• The rod or cord is massless, inextensible and always remains under tension.
• The bob is a point mass.
• The motion occurs in two dimensions.
• The motion does not lose energy to external friction or air resistance.
• The gravitational field is uniform.
• The support is immobile.

The differential equation which governs the motion of a simple pendulum iswhere is the magnitude 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

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$$\backslash theta\; \backslash ll\; 1,$$then substituting for into using the small-angle approximation,$$\backslash sin\backslash theta\backslash approx\backslash theta,$$yields the equation for a harmonic oscillator,$$\backslash frac+\backslash frac\; \backslash theta=0.$$

The error due to the approximation is of order (from the Taylor expansion for).

Let the starting angle be . If it is assumed that the pendulum is released with zero angular velocity, the solution becomesThe motion is simple harmonic motion where is the amplitude of the oscillation (that is, the maximum angle between the rod of the pendulum and the vertical). The corresponding approximate period of the motion is thenwhich 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 that Galileo discovered.

Rule of thumb for pendulum length

$$T\_0\; =\; 2\backslash pi\backslash sqrt$$ gives $$\backslash ell\; =\; \backslash frac\backslash frac\; 4.$$

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

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,$$\backslash frac\; =\; \backslash sqrt\backslash frac\backslash ell\backslash frac$$and then integrating over one complete cycle,$$T\; =\; t(\backslash theta\_0\; \backslash rightarrow\; 0\; \backslash rightarrow\; -\backslash theta\_0\; \backslash rightarrow\; 0\; \backslash rightarrow\backslash theta\_0),$$or twice the half-cycle$$T\; =\; 2\; t(\backslash theta\_0\; \backslash rightarrow\; 0\; \backslash rightarrow\; -\backslash theta\_0),$$or four times the quarter-cycle$$T\; =\; 4\; t(\backslash theta\_0\; \backslash rightarrow\; 0),$$which leads to$$T\; =\; 4\backslash sqrt\backslash frac\backslash ell\backslash int^\_0\; \backslash frac\; .$$

Note that this integral diverges as approaches the vertical$$\backslash lim\_\; T\; =\; \backslash infty,$$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$$T\; =\; 4\backslash sqrt\backslash frac\backslash ell\; g\; F\backslash left(\backslash frac\; 2,\; \backslash sin\; \backslash frac\; 2\backslash right)$$where is the incomplete elliptic integral of the first kind defined by$$F(\backslash varphi,\; k)\; =\; \backslash int\_0^\backslash varphi\; \backslash frac\; \backslash ,.$$

Or more concisely by the substitution$$\backslash sin\; =\; \backslash frac$$expressing in terms of,

Here is the complete elliptic integral of the first kind defined by

$$K(k)\; =\; F\; \backslash left(\backslash frac\; \backslash pi\; 2,\; k\; \backslash right)\; =\; \backslash int\_0^\backslash frac\; \backslash frac\backslash ,.$$

For comparison of the approximation to the full solution, consider the period of a pendulum of length 1 m on Earth (=) at an initial angle of 10 degrees is$$4\backslash sqrt\backslash \; K\backslash left(\backslash sin\backslash frac\; \backslash right)\backslash approx\; 2.0102\backslash text.$$The linear approximation gives

$$2\backslash pi\; \backslash sqrt\; \backslash approx\; 2.0064\backslash text.$$

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 solution for the elliptic integral:$$K(k)\; =\backslash frac\backslash sum\_^\backslash infty\; \backslash left(\backslash frack^\backslash right)^$$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:$$\backslash sin\; \backslash frac\; 2\; =\backslash frac12\backslash theta\_0\; -\; \backslash frac\backslash theta\_0^3\; +\; \backslash frac\backslash theta\_0^5\; -\; \backslash frac\backslash theta\_0^7\; +\; \backslash cdots.$$is used in the Legendre polynomial solution above.The resulting power series is:^[5]

$$T\; =\; 2\backslash pi\; \backslash sqrt\backslash frac\backslash ell\; g\; \backslash left(1+\; \backslash frac\backslash theta\_0^2\; +\; \backslash frac\backslash theta\_0^4\; +\; \backslash frac\backslash theta\_0^6\; +\; \backslash frac\backslash theta\_0^8\; +\; \backslash frac\backslash theta\_0^\; +\; \backslash frac\backslash theta\_0^\; +\; \backslash cdots\; \backslash right),$$more fractions available in the On-Line Encyclopedia of Integer Sequences 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:$$K(k)\; =\; \backslash frac,$$where is the arithmetic-geometric mean of and .

This yields an alternative and faster-converging formula for the period:^{[6] [7]} $$T\; =\; \backslash frac\; \backslash sqrt\backslash frac\backslash ell\; g.$$

The first iteration of this algorithm gives$$T\_1\; =\; \backslash frac.$$

This approximation has the relative error of less than 1% for angles up to 96.11 degrees. Since $\backslash frac\backslash left(1+\backslash cos\backslash left(\backslash frac\backslash right)\backslash right)\; =\; \backslash cos^2\; \backslash frac,$ the expression can be written more concisely as$$T\_1\; =\; T\_0\; \backslash sec^2\; \backslash frac.$$

The second order expansion of

\sec ^{2 (\theta} _{0 / 4 )} reduces to $T\; \backslash approx\; T\_0\; \backslash left(1\; +\; \backslash frac\; \backslash right).$

A second iteration of this algorithm gives$$T\_2\; =\; \backslash frac\; =\; \backslash frac.$$

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

T can be determined, for any finite amplitude \theta _{0} < \pi rad, by evaluating the corresponding complete elliptic integral K(k) , where k \equiv \sin(\theta _{0/2)} , 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.^[8] 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

\pi/ 2 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 \pi rad. One of the simplest formulae found in literature is the following one by Lima (2006): $T\; \backslash approx\; -\backslash ,T\_0\; \backslash ,\; \backslash frac$, where a \equiv \cos{(\theta _{0/2)}} .^[9]

• ‘Very large-angle’ formulae, i.e. those which approximate the exact period asymptotically for amplitudes near to

\pi 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:^[10] $T\; \backslash approx\; \backslash frac\backslash ,T\_0\backslash ,\backslash ln$.

Of course, the increase of

T with amplitude is more apparent when \pi/ 2 <\theta _{0<\pi} , as has been observed in many experiments using either a rigid rod or a disc.^[11] 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:$$T\; \backslash approx\; \backslash frac,$$where r = 7 . 17 , which presents a maximum error of only 0.6% (at \theta _{0} = 95 ^{\circ} ).

Arbitrary-amplitude angular displacement

The Fourier series expansion of

\theta(t) is given by^[12]

$$\backslash theta(t)\; =\; 8\backslash sum\_\backslash frac\backslash frac\backslash cos(n\backslash omega\; t)$$

where

q is the elliptic nome, q = \exp\left({ - \pi K l (\sqrt{ style 1- k ^{2}r)} / K(k)}\right), k = \sin (\theta _{0/2),} and \omega =2 \pi/T the angular frequency.

If one defines$$\backslash varepsilon=\backslash frac12\backslash cdot\; \backslash frac$$

q can be approximated using the expansion$$q\; =\; \backslash varepsilon\; +\; 2\backslash varepsilon^5\; +\; 15\backslash varepsilon^\; +\; 150\backslash varepsilon^\; +\; 1707\backslash varepsilon^\; +\; 20910\backslash varepsilon^\; +\; \backslash cdots$$(see). Note that \varepsilon < \tfrac 1 2 for \theta _{0<\pi} , thus the approximation is applicable even for large amplitudes.\operatorname{cd} with modulus k ^[13] $$\backslash theta(t)=2\backslash arcsin\backslash left(k\backslash operatorname\backslash left(\backslash sqrtt;k\backslash right)\backslash right),\backslash quad\; k=\backslash sin\backslash frac.$$

For small

x , \sin x ≈ x , \arcsin x ≈ x and \operatorname{cd}(t; 0 ) = \cos t , so the solution is well-approximated by the solution given in Pendulum (mechanics)#Small-angle approximation.

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.

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 swinging by a pivot

O . In this case the pendulum's period depends on its moment of inertia I _{O} around the pivot point.

The equation of torque gives:$$\backslash tau\; =\; I\; \backslash alpha$$where:

\alpha is the angular acceleration.\tau is the torque

The torque is generated by gravity so:

\tau = - m g r _{⊕} \sin\theta where:m is the total mass of the rigid body (rod and bob)r _{⊕} is the distance from the pivot point to the system's centre-of-mass\theta is the angle from the vertical

Hence, under the small-angle approximation,

\sin\theta ≈ \theta (or equivalently when \theta _{max\ll} 1 ),$$\backslash alpha\; =\; \backslash ddot\; =\; \backslash frac\backslash sin\backslash theta\; \backslash approx\; -\backslash frac\backslash theta$$where I _{O} is the moment of inertia of the body about the pivot point O .

The expression for

\alpha is of the same form as the conventional simple pendulum and gives a period of$$T\; =\; 2\; \backslash pi\; \backslash sqrt$$

And a frequency of$$f\; =\; \backslash frac\; =\; \backslash frac\; \backslash sqrt$$

If the initial angle is taken into consideration (for large amplitudes), then the expression for

\alpha becomes: $$\backslash alpha\; =\; \backslash ddot\; =\; -\backslash frac\backslash sin\backslash theta$$and gives a period of: $$T\; =\; 4\; \backslash operatorname\backslash left(\backslash sin^2\backslash frac\backslash right)\; \backslash sqrt$$where \theta _{max} is the maximum angle of oscillation (with respect to the vertical) and \operatorname{K}(k) is the complete elliptic integral of the first kind.

An important concept is the equivalent length,

\ell ^{eq} , the length of a simple pendulums that has the same angular frequency \omega _{0} as the compound pendulum:{\omega ^{2} _{0 }} = (g)/(\ell ^{eq}) : = (mgr _{⊕})/(I _{O}) \implies \ell ^{eq} = (I _{O})/(mr _{⊕})

Consider the following cases:

• The simple pendulum is the special case where all the mass is located at the bob swinging at a distance

\ell from the pivot. Thus, r _{⊕ =\ell} and I ^{2} _{O = m\ell} , so the expression reduces to: {\omega ^{2} _{0 }} = (mgr _{⊕})/(= I _{O}) (mg\ell)/(= m\ell ^{2}) (g)/(\ell) . Notice \ell ^{eq=\ell} , as expected (the definition of equivalent length).

• A homogeneous rod of mass

m and length \ell swinging from its end has r _{⊕ = (1)/(2)} \ell and I _{O = (1)/(3)} m\ell ^{2} , so the expression reduces to: {\omega ^{2} _{0 }} = (mgr _{⊕})/(= I _{O}) (mg (1)/(\ell 2))/(= (1)/(m\ell ^{2} 3)) (g)/((2)/(\ell 3)) . Notice \ell ^{eq = (2)/(3)} \ell , a homogeneous rod oscillates as if it were a simple pendulum of two-thirds its length.

• A heavy simple pendulum: combination of a homogeneous rod of mass

m _{rod} and length \ell swinging from its end, and a bob m _{bob} at the other end. Then the system has a total mass of m _{bob+m} _{rod} , and the other parameters being m r _{⊕ =m} _{bob\ell+m} ^{rod (\ell)/(2)} (by definition of centre-of-mass) and I _{O=m} ^{2+ (1)/(3)} ^{bob \ell} m ^{2} _{rod \ell} , so the expression reduces to: {\omega ^{2} _{0 }} = (mgr _{⊕})/(= I _{O}) (\left(m _{bob \ell + m} ^{rod (\ell)/(2)} \right)g)/(m ^{2+ (1)/(3)} _{bob \ell} m ^{2} _{rod \ell}) = (g)/(\ell) (m _{bob + (m _{rod})/(2)})/(m _{bob + (m _{rod})/(3)}) = (g)/(\ell) (1+ (m _{rod})/(2 m _{bob}))/(1+ (m _{rod})/(3 m _{bob})) Where \ell ^{eq} = \ell (1+ (m _{rod})/(3 m _{bob}))/(1+ (m _{rod})/(2 m _{bob})) . Notice these formulae can be particularized into the two previous cases studied before just by considering the mass of the rod or the bob to be zero respectively. Also notice that the formula does not depend on both the mass of the bob and the rod, but actually on their ratio, (m _{rod})/(m _{bob}) . An approximation can be made for (m _{rod})/(m _{bob}) \ll 1 :{\omega ^{2} _{0 }} ≈ (g)/(\ell) \left( 1+ (1)/(6) (m _{rod})/(m _{bob}) + … \right)

Notice how similar it is to the angular frequency in a spring-mass system with effective mass.

Damped, driven pendulum

The above discussion focuses on a pendulum bob only acted upon by the force of gravity. Suppose a damping force, e.g. air resistance, as well as a sinusoidal driving force acts on the body. This system is a damped, driven oscillator, and is chaotic.

Equation (1) can be written as

$$ml^2\; \backslash frac\; =\; -mgl\; \backslash sin\; \backslash theta$$

(see the Torque derivation of Equation (1) above).

A damping term and forcing term can be added to the right hand side to get

$$ml^2\; \backslash frac\; =\; -mgl\backslash sin\; \backslash theta\; -\; b\backslash frac\; +\; a\backslash cos(\backslash Omega\; t)$$

where the damping is assumed to be directly proportional to the angular velocity (this is true for low-speed air resistance, see also Drag (physics)).

a and b are constants defining the amplitude of forcing and the degree of damping respectively. $\backslash Omega$ is the angular frequency of the driving oscillations.

Dividing through by $ml^2$:

$$\backslash frac\; +\; \backslash frac\backslash frac\; +\; \backslash frac\; -\; \backslash frac\backslash cos\; (\backslash Omega\; t)\; =\; 0.$$

For a physical pendulum:

$$\backslash frac\; +\; \backslash frac\backslash frac\; +\; \backslash frac\; -\; \backslash frac\backslash cos\; (\backslash Omega\; t)\; =\; 0.$$

This equation exhibits chaotic behaviour. The exact motion of this pendulum can only be found numerically and is highly dependent on initial conditions, e.g. the initial velocity and the starting amplitude. However, the small angle approximation outlined above can still be used under the required conditions to give an approximate analytical solution.

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:^[14] 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 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:$$E\_\backslash text=\backslash fracmL^2\backslash left(\backslash dot\backslash theta\_1^2+\backslash dot\backslash theta\_2^2\backslash right)$$where

m is the mass of the bobs, L is the length of the strings, and \theta _{1} , \theta _{2} are the angular displacements of the two bobs from equilibrium.

The potential energy of the system is:$$E\_\backslash text=mgL(2-\backslash cos\backslash theta\_1-\backslash cos\backslash theta\_2)+\backslash frackL^2(\backslash theta\_2-\backslash theta\_1)^2$$

where

g is the gravitational acceleration, and k is the spring constant. The displacement L(\theta _{2-\theta} _{1)} of the spring from its equilibrium position assumes the small angle approximation.

The Lagrangian is then$$\backslash mathcal=\backslash fracmL^2\backslash left(\backslash dot\backslash theta\_1^2+\backslash dot\backslash theta\_2^2\backslash right)-mgL(2-\backslash cos\backslash theta\_1-\backslash cos\backslash theta\_2)-\backslash frac\; k\; L^2(\backslash theta\_2-\backslash theta\_1)^2$$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

\theta _{1+\theta} _{2} and \theta _{1-\theta} _{2} :with the corresponding solutionswhere

and

A , B , \alpha , \beta are constants of integration.

Expressing the solutions in terms of

\theta _{1} and \theta _{2} alone:

If the bobs are not given an initial push, then the condition

• \theta ^{• 1 ( 0 ) = \theta} _{2(0)=0} requires \alpha = \beta =0 , which gives (after some rearranging):

See also

• Harmonograph
• Conical pendulum
• Cycloidal pendulum
• Double pendulum
• Inverted pendulum
• Kapitza's pendulum
• Rayleigh–Lorentz pendulum
• Elastic pendulum
• Mathieu function
• Pendulum equations (software)

Further reading

• Book: The Pendulum: A Physics Case Study . Gregory L. . Baker . James A. . Blackburn . Oxford University Press . 2005 .
• Karlheinz . Ochs . A comprehensive analytical solution of the nonlinear pendulum . . 32 . 2 . 479–490 . 2011 . 10.1088/0143-0807/32/2/019 . 2011EJPh...32..479O . 53621685 .
• Kenneth L. . Sala . Transformations of the Jacobian Amplitude Function and its Calculation via the Arithmetic-Geometric Mean . SIAM J. Math. Anal. . 20 . 6 . 1514–1528 . 1989 . 10.1137/0520100 .

External links

• Mathworld article on Mathieu Function

Notes and References

1. defined by Christiaan Huygens: Web site: Huygens . Christian . Horologium Oscillatorium . 17centurymaths . 17thcenturymaths.com . 1673 . 2009-03-01., Part 4, Definition 3, translated July 2007 by Ian Bruce
2. Web site: Nave . Carl R. . Simple pendulum . Hyperphysics . Georgia State Univ. . 2006 . 2008-12-10.
3. Web site: Xue . Linwei . Pendulum Systems . Seeing and Touching Structural Concepts . Civil Engineering Dept., Univ. of Manchester, UK . 2007 . 2008-12-10.
4. Web site: Weisstein . Eric W. . Simple Pendulum . Eric Weisstein's world of science . Wolfram Research . 2007 . 2009-03-09.
5. Nelson . Robert . M. G. . Olsson . dead link --> . American Journal of Physics . 54 . 2 . 112–121 . February 1986 . 2012-04-30. 1986AmJPh..54..112N -->,1986AmJPh..54..112N . 10.1119/1.14703 . 121907349 .
6. Book: Jonathan Borwein . J.M. . Borwein. Peter Borwein . P.B. . Borwein . Pi and the AGM . Wiley . New York . 1987 . 0-471-83138-7 . 0877728 . 1–15.
7. A New and Wonderful Pendulum Period Equation . Tom . Van Baak . Horological Science Newsletter . November 2013 . 2013 . 5 . 22–30 .
8. Lima. F. M. S.. 2008-09-10. Simple 'log formulae' for pendulum motion valid for any amplitude . European Journal of Physics. 29. 5. 1091–1098. 10.1088/0143-0807/29/5/021. 121743087 . 0143-0807. IoP journals.
9. Lima. F. M. S.. Arun. P.. October 2006. An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime. American Journal of Physics. 74. 10 . 892–895. 10.1119/1.2215616. 0002-9505. physics/0510206. 2006AmJPh..74..892L. 36304104.
10. Cromer. Alan. February 1995 . Many oscillations of a rigid rod. American Journal of Physics. 63. 2. 112–121 . 10.1119/1.17966. 1995AmJPh..63..112C. 0002-9505.
11. Gil. Salvador. Legarreta. Andrés E.. Di Gregorio. Daniel E.. September 2008. Measuring anharmonicity in a large amplitude pendulum . American Journal of Physics. 76. 9. 843–847. 10.1119/1.2908184. 2008AmJPh..76..843G . 0002-9505.
12. Book: Lawden, Derek F. . Elliptic Functions and Applications. Springer-Verlag . 1989 . 0-387-96965-9. 40. Eq. 2.7.9: $-kk\text{'}\backslash int\backslash operatornameu\backslash ,\; \backslash mathrm\; du=\backslash arcsin\; (k\backslash operatornameu)+C$
13. Web site: A Complete Solution to the Non-Linear Pendulum . 4 December 2021 .
14. Paul . Appell . Sur une interprétation des valeurs imaginaires du temps en Mécanique . On an interpretation of imaginary time values in mechanics . . 87 . 1 . July 1878.