Pendulum (mathematics)

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The mathematics of pendulums can be quite complex, but some formulas and proofs are given below.

1 Simple gravity pendulum
2 Small-angle approximation
- 2.1 Further approximation
3 Arbitrary-amplitude period
4 Physical interpretation of the imaginary period
5 See also
6 External links
7 References

[edit] Simple gravity pendulum

Trigonometry of a simple gravity pendulum.

A simple pendulum is an ideality involving these two assumptions:

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 represents the approximate motion of the pendulum is

${d^2\theta\over dt^2}+{g\over \ell} \sin\theta=0.$ ^{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 $h$ 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 is

${d\theta\over dt} = \sqrt{{2g\over \ell}\left(\cos\theta-\cos\theta_0\right)}.$ ^{see derivation}

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

[edit] 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

$\sin\theta\approx\theta$

if and only if

$|\theta|\ll 1.$

Substituting this approximation into (1) yields

${d^2\theta\over dt^2}+{g\over \ell}\theta=0.$

Under the initial conditions $θ(0) = θ 0$ and ${d\theta\over dt}(0)=0$ , the solution to this equation is a well-known, and quite expected, oscillatory function

$\theta(t) = \theta_0\cos\left(\sqrt{g\over \ell\,}\,t\right) \quad\quad\quad\quad |\theta_0| \ll 1.$

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 $\frac{2\pi}{T_0}$ ,

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

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

the period of a complete oscillation can be easily found, and we have obtained Huygens's law:

$T_0 = 2\pi\sqrt{\frac{\ell}{g}}$

$T_0 = 2\pi\sqrt{\ell\over g}\quad\quad\quad\quad |\theta_0| \ll 1.$

[edit] Further approximation

$T_0 = 2\pi\sqrt{\frac{\ell}{g}}$ 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).

Therefore $\ell\approx{\frac{T_0^2}{4}}$

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.

[edit] Arbitrary-amplitude period

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

${dt\over d\theta} = {1\over\sqrt{2}}\sqrt{\ell\over g}{1\over\sqrt{\cos\theta-\cos\theta_0}}$

and integrating over one complete cycle,

$T = \theta_0\rightarrow0\rightarrow-\theta_0\rightarrow0\rightarrow\theta_0,$

or twice the half-cycle

$T = 2\left(\theta_0\rightarrow0\rightarrow-\theta_0\right),$

or 4 times the quarter-cycle

$T = 4\left(\theta_0\rightarrow0\right),$

which leads to

$T = 4{1\over\sqrt{2}}\sqrt{\ell\over g}\int^{\theta_0}_0 {1\over\sqrt{\cos\theta-\cos\theta_0}}\,d\theta.$

Figure 4. Deviation of the period from small-angle approximation. Even at relatively large amplitudes, approximation is accurate within 5%-7%.

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

$T = 4\sqrt{\ell\over g}F\left({\theta_0\over 2},\csc^2{\theta_0\over2}\right)\csc {\theta_0\over 2}$

or more concisely,

$T = 4\sqrt{\ell\over g}F\left({\sin\theta_0\over 2}, {\pi \over 2} \right)$

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

$F(k,\phi) = \int^{\phi}_0 {1\over\sqrt{1-k^2\sin^2{\theta}}}\,d\theta.$

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

$T = 2\pi \sqrt{\ell\over g} \left( 1+ \left( \frac{1}{2} \right)^2 \sin^2\left(\frac{\theta_0}{2}\right) + \left( \frac{1 \cdot 3}{2 \cdot 4} \right)^2 \sin^4\left(\frac{\theta_0}{2}\right) + \left( \frac {1 \cdot 3 \cdot 5}{2 \cdot 4 \cdot 6} \right)^2 \sin^6\left(\frac{\theta_0}{2}\right) + \cdots \right).$

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 $4\sqrt{1\over g}F\left({\sin 10\over 2},{\pi\over2}\right) = 2.0102$ seconds, whereas the approximation $2\pi \sqrt{1\over g} = 2.0064$ that's about 1 second per swing (both examples use g = 9.80665 m/s²).

[edit] 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.