NationMaster - Encyclopedia: Pendulum (mathematics)

The mathematics of pendulums can be quite complex, but some formula and proofs are given below. Simple gravity pendulum assumes no air resistance and no friction of/at the nail/screw. ...

1 Simple gravity pendulum
2 Small-angle approximation
- 2.1 Further approximation
3 Arbitrary-amplitude period
4 Physical interpretation of the imaginary period
5 References

Simple gravity pendulum

Trigonometry of a simple gravity pendulum.

A simple pendulum is an ideality involving these two assumptions: Image File history File links Simple_pendulum_height. ... Image File history File links Simple_pendulum_height. ...

The rod/string/cable on which the bob is swinging is massless and always remains taut;
Motion occurs in a 2 dimensional plane, i.e. the bob does not trace an ellipse.

The differential equation which can yield the motion of the pendulum is The ellipse and some of its mathematical properties. ... Graph of a differential equation In mathematics, a differential equation is an equation in which the derivatives of a function appear as variables. ...

${d^2thetaover dt^2}+{gover ell} sintheta=0.$ ^{see derivation}

It can also be obtained via the conservation of mechanical energy principle: any given object which fell a vertical distance $h$ would have acquired kinetic energy equal to that which it lost to the fall. In other words, gravitational potential energy is converted into kinetic energy. Figure 2. ... In physics, mechanical energy is one of several forms of energy. ... Kinetic energy is the energy that a body possesses as a result of its motion. ... In physics, gravitational potential is the measure of potential energy an object possesses due to its position in a gravitational field. ...

The first integral of motion is

${dthetaover dt} = sqrt{{2gover ell}left(costheta-costheta_0right)}.$ ^{see derivation}

It gives the velocity in terms of the location and includes an integration constant related to the initial displacement (θ₀). Figure 2. ...

Small-angle approximation

The problem with the equations developed in the previous section is that they are unintegrable. To shed some light on the behavior of the pendulum we shall make another approximation. Namely, we restrict the motion of the pendulum to a relatively small amplitude, that is, relatively small $θ$ . How small? Small enough that the following approximation is true within some desirable tolerance

$sinthetaapproxtheta$

if and only if

$|theta|ll 1.$

Substituting this approximation into (1) yields

${d^2thetaover dt^2}+{gover ell}theta=0.$

Under the initial conditions $θ(0) = θ 0$ and ${dthetaover dt}(0)=0$ , the solution to this equation is a well-known, and quite expected, oscillatory function A harmonic oscillator is a system which, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement : where is a positive constant. ...

$theta(t) = theta_0cosleft(sqrt{gover ell,},tright) quadquadquadquad |theta_0| ll 1.$

where $θ 0$ is the semi-amplitude of the oscillation, that is the maximum angle between the rod of the pendulum and the vertical.

The term $sqrt{frac{g}{ell}}$ is a pulsation, which is equal to $frac{2pi}{T_0}$ , Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ...

where $T 0$ is the period of a complete oscillation (outward and return).

Since $omega = sqrt{frac{g}{ell}} = frac{2pi}{T_0},$

the period of a complete oscillation can be easily found, and we have obtained Huygens's law: Christiaan Huygens Christiaan Huygens (pronounced in English (IPA): ; in Dutch: )(April 14, 1629â€“July 8, 1695), was a Dutch mathematician, astronomer and physicist; born in The Hague as the son of Constantijn Huygens. ...

$T_0 = 2pisqrt{frac{ell}{g}}$

$T_0 = 2pisqrt{ellover g}quadquadquadquad |theta_0| ll 1.$

Further approximation

$T_0 = 2pisqrt{frac{ell}{g}}$ can be expressed as $ell = {frac{g}{pi^2}}times{frac{T_0^2}{4}}.$

If we use SI units (i.e. measure in metres and seconds), and assume the measurement is taking place on the earth's surface, then g = 9.80665 m/s², and ${frac{g}{pi^2}}approx{1}$ (the exact figure is 0.994 to 3 decimal places). Cover of brochure The International System of Units. ...

Therefore $ellapprox{frac{T_0^2}{4}}$

or to put it in words:

On the surface of the earth, the length of a pendulum (in metres) is approximately one quarter of the time period (in seconds) squared.

Arbitrary-amplitude period

For amplitudes beyond the small angle approximation, one can compute the exact period by inverting equation (2)

${dtover dtheta} = {1oversqrt{2}}sqrt{ellover g}{1oversqrt{costheta-costheta_0}}$

and integrating over one complete cycle,

$T = theta_0rightarrow0rightarrow-theta_0rightarrow0rightarrowtheta_0,$

or twice the half-cycle

$T = 2left(theta_0rightarrow0rightarrow-theta_0right),$

or 4 times the quarter-cycle

$T = 4left(theta_0rightarrow0right),$

which leads to

$T = 4{1oversqrt{2}}sqrt{ellover g}int^{theta_0}_0 {1oversqrt{costheta-costheta_0}},dtheta.$

Figure 4. Deviation of the period from small-angle approximation. Even at relatively large amplitudes, approximation is accurate within 10%-15%.

Alas, this integral cannot be evaluated in terms of elementary functions. It can be re-written in the form of the elliptic function of the first kind (also see Jacobi's elliptic functions), which gives one very little advantage for it is a redundant exercise of expressing one insoluble integral in terms of another Image File history File links Period of a simple gravity pendulum divided by the period from the small-angle approximation as a function of the amplitude. ... Image File history File links Period of a simple gravity pendulum divided by the period from the small-angle approximation as a function of the amplitude. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ...

$T = 4sqrt{ellover g}Fleft({theta_0over 2},csc^2{theta_0over2}right)csc {theta_0over 2}$

or more concisely,

$T = 4sqrt{ellover g}Fleft({sintheta_0over 2}, {pi over 2} right)$

where $F (k,φ)$ is Legendre's elliptic function of the first kind

$F(k,phi) = int^{phi}_0 {1oversqrt{1-k^2sin^2{theta}}},dtheta.$

The value of the elliptic function can be also computed using the following series:

$T = T_0 left( 1+ left( frac{1}{2} right)^2 sin^2left(frac{theta_0}{2}right) + left( frac{1 cdot 3}{2 cdot 4} right)^2 sin^4left(frac{theta_0}{2}right) + left( frac {1 cdot 3 cdot 5}{2 cdot 4 cdot 6} right)^2 sin^6left(frac{theta_0}{2}right) + cdots right).$

Figure 4 shows the deviation of $T$ from $T 0$ , the period obtained from small-angle approximation.

For a swing of $180^circ$ the bob is balanced over its pivot point and so $T=infty$ (keep in mind the pendulum is made of a rigid rod).

Potential energy and phase portrait of a simple pendulum. Note that the x-axis being angle, wraps onto itself after every 2π radians.

For example, the period of a 1m pendulum at initial angle 10 degrees is $4sqrt{1over g}Fleft({sin 10over 2},{piover2}right) = 2.0102$ seconds, whereas the approximation $2pi sqrt{1over g} = 2.0064$ that's about 1 second per swing (both examples use g = 9.80665 m/s²). Image File history File links Download high resolution version (1200x900, 20 KB) Summary Phase portrait of a simple pendulum Licensing File links The following pages link to this file: Pendulum User:Deeptrivia/Album Wikipedia:Reference desk archive/Mathematics/February 2006 ... Image File history File links Download high resolution version (1200x900, 20 KB) Summary Phase portrait of a simple pendulum Licensing File links The following pages link to this file: Pendulum User:Deeptrivia/Album Wikipedia:Reference desk archive/Mathematics/February 2006 ...

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 period of another, then the real period of each is the magnitude of the imaginary period of the other. In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e. ... In mathematics, a doubly periodic function is a function f defined at all points x in a plane and having two periods, which are linearly independent vectors u and v such that See elliptic function for an account of doubly periodic functions that are meromorphic on the complex plane, and... In mathematics, the set of real numbers, denoted R, or in blackboard bold , is the set of all rational and irrational numbers. ... Paul Ã‰mile Appell (September 27, 1855 in Strasbourg â€“ October 24, 1930 in Paris) was a French mathematician and Rector of the University of Paris. ...

References

Paul Appell, "Sur une interprétation des valeurs imaginaires du temps en Mécanique", Comptes Rendus Hebdomadaires des Scéances de l'Académie des Sciences, volume 87, number 1, July, 1878.

Category: Applied mathematics Paul Ã‰mile Appell (September 27, 1855 in Strasbourg â€“ October 24, 1930 in Paris) was a French mathematician and Rector of the University of Paris. ...

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