Pendulum (mathematics)

The mathematics of pendula are in general quite complicated. Simplifying assumptions can be made, which in the case of a simple pendulum allows the equations of motion to be solved analytically for small-angle oscillations.

Simple gravity pendulum

Trigonometry of a simple gravity pendulum.

A simple pendulum is an idealisation, working on the assumption that:

• The rod or cord on which the bob swings is massless, inextensible and always remains taut;
• Motion occurs in a 2-dimensional plane, i.e. the bob does not trace an ellipse.
• The motion does not lose energy to friction.

The differential equation which represents the motion of the pendulum is

Failed to parse (Missing texvc executable; please see math/README to configure.): {d^2\theta\over dt^2}+{g\over \ell} \sin\theta=0 \quad\quad\quad\quad\quad(1)

      see derivation

This is known as Mathieu's equation. It can also be obtained via the conservation of mechanical energy principle: any given object which fell a vertical distance Failed to parse (Missing texvc executable; please see math/README to configure.): 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.

The first integral of motion found by integrating (1) is

Failed to parse (Missing texvc executable; please see math/README to configure.): {d\theta\over dt} = \sqrt{{2g\over \ell}\left(\cos\theta-\cos\theta_0\right)} \quad\quad(2)

      see derivation

It gives the velocity in terms of the angle and includes the initial displacement (θ₀) as an integration constant.

Small-angle approximation

It is not possible to integrate analytically the full equations of a simple pendulum. A further assumption, that the pendulum attains only a small amplitude, that is

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta \ll 1

is sufficient to allow the system to be solved easily. Making the assumption of small angle allows the approximation

Failed to parse (Missing texvc executable; please see math/README to configure.): \sin\theta\approx\theta

to be made. To first order, the error in this approximation is proportional to Failed to parse (Missing texvc executable; please see math/README to configure.): \theta^3

(from the Maclaurin series for Failed to parse (Missing texvc executable; please see math/README to configure.): \sin \theta

). Substituting this approximation into (1) yields the equation for a harmonic oscillator:

Failed to parse (Missing texvc executable; please see math/README to configure.): {d^2\theta\over dt^2}+{g\over \ell}\theta=0.

Under the initial conditions Failed to parse (Missing texvc executable; please see math/README to configure.): \theta(0)=\theta_0

and Failed to parse (Missing texvc executable; please see math/README to configure.): {d\theta\over dt}(0)=0

, the solution is

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta(t) = \theta_0\cos\left(\sqrt{g\over \ell\,}\,t\right) \quad\quad\quad\quad \theta_0 \ll 1.

The motion is simple harmonic motion where Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_0

is the semi-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

Failed to parse (Missing texvc executable; please see math/README to configure.): T_0 = 2\pi\sqrt{\frac{\ell}{g}} \quad\quad\quad\quad\quad \theta_0 \ll 1

which is Christiaan Huygens's law for the period. Note that under the small-angle approximation, the period is independent of the amplitude Failed to parse (Missing texvc executable; please see math/README to configure.): \theta_0

this is the property of isochronism that Galileo discovered.

Rule of thumb for pendulum length

Failed to parse (Missing texvc executable; please see math/README to configure.): T_0 = 2\pi\sqrt{\frac{\ell}{g}}

can be expressed as Failed to parse (Missing texvc executable; please see math/README to configure.): \ell = {\frac{g}{\pi^2}}\times{\frac{T_0^2}{4}}.

If SI units are used (i.e. measure in metres and seconds), and an assumption is made the measurement is taking place on the earth's surface, then g = 9.80665 m/s², and Failed to parse (Missing texvc executable; please see math/README to configure.): {\frac{g}{\pi^2}}\approx{1}

(the exact figure is 0.994 to 3 decimal places).

Therefore Failed to parse (Missing texvc executable; please see math/README to configure.): \ell\approx{\frac{T_0^2}{4}} , or 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)

Figure 4. Deviation of the period from small-angle approximation.

Failed to parse (Missing texvc executable; please see math/README to configure.): {dt\over d\theta} = {1\over\sqrt{2}}\sqrt{\ell\over g}{1\over\sqrt{\cos\theta-\cos\theta_0}}

and integrating over one complete cycle,

Failed to parse (Missing texvc executable; please see math/README to configure.): T = \theta_0\rightarrow0\rightarrow-\theta_0\rightarrow0\rightarrow\theta_0,

or twice the half-cycle

Failed to parse (Missing texvc executable; please see math/README to configure.): T = 2\left(\theta_0\rightarrow0\rightarrow-\theta_0\right),

or 4 times the quarter-cycle

Failed to parse (Missing texvc executable; please see math/README to configure.): T = 4\left(\theta_0\rightarrow0\right),

which leads to

Failed to parse (Missing texvc executable; please see math/README to configure.): T = 4{1\over\sqrt{2}}\sqrt{\ell\over g}\int^{\theta_0}_0 {1\over\sqrt{\cos\theta-\cos\theta_0}}\,d\theta.

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 little advantage since that form is also insoluble.

Failed to parse (Missing texvc executable; please see math/README to configure.): T = 4\sqrt{\ell\over g}F\left({\theta_0\over 2},\csc^2{\theta_0\over2}\right)\csc {\theta_0\over 2}

or more concisely,

Failed to parse (Missing texvc executable; please see math/README to configure.): T = 4\sqrt{\ell\over g}F\left(\sin{\theta_0\over 2}, {\pi \over 2} \right)

where Failed to parse (Missing texvc executable; please see math/README to configure.): F(k,\phi)

is Legendre's elliptic function of the first kind

Failed to parse (Missing texvc executable; please see math/README to configure.): F(k,\phi) = \int^{\phi}_0 {1\over\sqrt{1-k^2\sin^2{\theta}}}\,d\theta.

Figure 4 shows the deviation of Failed to parse (Missing texvc executable; please see math/README to configure.): T\,

from Failed to parse (Missing texvc executable; please see math/README to configure.): T_0\,

, the period obtained from small-angle approximation.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{alignat}{2} T & = 2\pi \sqrt{\ell\over g} \left( 1+ \left( \frac{1}{2} \right)^2 \sin^2\left(\frac{\theta_0}{2}\right) + \left( \frac{1 \cdot 3}{2 \cdot 4} \right)^2 \sin^4\left(\frac{\theta_0}{2}\right) + \left( \frac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \right)^2 \sin^6\left(\frac{\theta_0}{2}\right) + \cdots \right) \\ & = 2\pi \sqrt{\ell\over g} \cdot \sum_{n=0}^\infty \left[ \left ( \frac{(2 n)!}{( 2^n \cdot n! )^2} \right )^2 \cdot \sin^{2 n}\left(\frac{\theta_0}{2}\right) \right]. \end{alignat}

Figure 5 shows the relative errors using the power series. Failed to parse (Missing texvc executable; please see math/README to configure.): T_0\,

is the linear approximation, and Failed to parse (Missing texvc executable; please see math/README to configure.): T_2
to Failed to parse (Missing texvc executable; please see math/README to configure.): T_{10}
include respectively the terms up to the 2nd to the 10th powers.

For a swing of exactly Failed to parse (Missing texvc executable; please see math/README to configure.): 180^\circ

the bob is balanced over its pivot point and so Failed to parse (Missing texvc executable; please see math/README to configure.): T=\infty

.

For example, the period of a 1m pendulum on Earth (g = 9.80665 m/s²) at initial angle 10 degrees is Failed to parse (Missing texvc executable; please see math/README to configure.): 4\sqrt{1\over g}F\left({\sin 10\over 2},{\pi\over2}\right) = 2.0102

seconds, whereas the linear approximation gives Failed to parse (Missing texvc executable; please see math/README to configure.): 2\pi \sqrt{1\over g} = 2.0064

.

Physical pendula

A physical pendulum is one where the rod is not massless, and the mass may have extended size; in this case the pendulum and rod have a moment of inertia Failed to parse (Missing texvc executable; please see math/README to configure.): I

around the pivot point.

The equation of torque gives:

Failed to parse (Missing texvc executable; please see math/README to configure.): T = I a

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): a

is the angular acceleration.

Failed to parse (Missing texvc executable; please see math/README to configure.): T

is the torque

The torque is generated by gravity so:

Failed to parse (Missing texvc executable; please see math/README to configure.): T = - m g L sin(\theta)

where:

Failed to parse (Missing texvc executable; please see math/README to configure.): L

is the distance from the pivot to the center of mass of the pendulum

Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

is the angle from the vertical

Hence, under the small-angle approximation Failed to parse (Missing texvc executable; please see math/README to configure.): \sin \theta \approx \theta ,

Failed to parse (Missing texvc executable; please see math/README to configure.): a \approx \frac{mgL \theta} {I}

This is of the same form as the conventional simple pendulum and this gives a period of:

Failed to parse (Missing texvc executable; please see math/README to configure.): T = 2 \pi \sqrt{\frac{I} {mgL}}.

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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.

See also

• Mathieu function
• Double pendulum

External links

• Mathworld article on Mathieu Function

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.
• The Pendulum: A Physics Case Study, Gregory L. Baker and James A. Blackburn, Oxford University Press, 2005cs:Matematické kyvadlo

de:Mathematisches Pendel pt:Equação do pêndulo