# Pendulum (mathematics)

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Template:Dynamics The mathematics of pendulums 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

A simple pendulum is an idealization of a real pendulum using the following assumptions:

• The rod or cord on which the bob swings is massless, inextensible and always remains taut;
• Motion occurs only in two dimensions, i.e. the bob does not trace an ellipse but an arc.
• The motion does not lose energy to friction or air resistance.

The differential equation which represents the motion of a simple pendulum is

${d^{2}\theta \over dt^{2}}+{g \over \ell }\sin \theta =0\quad \quad \quad \quad \quad (1)$ where g is acceleration due to gravity, $\ell$ is the length of the pendulum, and θ is the angular displacement.

## Small-angle approximation

The differential equation given above is not easily solved. 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, or

$\theta \ll 1\,$ ,

then substituting for sin θ into (1) using the small-angle approximation,

$\sin \theta \approx \theta \,$ ,

yields the equation for a harmonic oscillator

${d^{2}\theta \over dt^{2}}+{g \over \ell }\theta =0.$ The error due to the approximation is of order θ 3 (from the Maclaurin series for sin θ).

Given the initial conditions θ(0) = θ0 and /dt(0) = 0, the solution becomes,

$\theta (t)=\theta _{0}\cos \left({\sqrt {g \over \ell \,}}\,t\right)\quad \quad \quad \quad \theta _{0}\ll 1.$ The motion is simple harmonic motion where θ0 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

$T_{0}=2\pi {\sqrt {\frac {\ell }{g}}}\quad \quad \quad \quad \quad \theta _{0}\ll 1$ 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 θ0; this is the property of isochronism that Galileo discovered.

### Rule of thumb for pendulum length

$T_{0}=2\pi {\sqrt {\frac {\ell }{g}}}$ can be expressed as $\ell ={\frac {g}{\pi ^{2}}}\times {\frac {T_{0}^{2}}{4}}.$ 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 $g\approx 9.81$ m/s2, and $g/\pi ^{2}\approx {1}$ (the exact figure is 0.994 to 3 decimal places).

Therefore a relatively reasonable approximation for the length and period are,

$\ell \approx {\frac {T_{0}^{2}}{4}},$ $T_{0}\approx 2{\sqrt {\ell }}$ ## 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,

${dt \over d\theta }={\sqrt {\ell \over 2g}}{1 \over {\sqrt {\cos \theta -\cos \theta _{0}}}}$ and then integrating over one complete cycle,

$T=t(\theta _{0}\rightarrow 0\rightarrow -\theta _{0}\rightarrow 0\rightarrow \theta _{0}),$ or twice the half-cycle

$T=2t\left(\theta _{0}\rightarrow 0\rightarrow -\theta _{0}\right),$ or 4 times the quarter-cycle

$T=4t\left(\theta _{0}\rightarrow 0\right),$ $T=4{\sqrt {\ell \over 2g}}\int _{0}^{\theta _{0}}{1 \over {\sqrt {\cos \theta -\cos \theta _{0}}}}\,d\theta .$ This integral can be re-written in terms of elliptic functions as

$T=4{\sqrt {\ell \over g}}F\left({\theta _{0} \over 2},\csc {\theta _{0} \over 2}\right)\csc {\theta _{0} \over 2}$ $F(\varphi ,k)=\int _{0}^{\varphi }{1 \over {\sqrt {1-k^{2}\sin ^{2}{u}}}}\,du\,,$ $T=4{\sqrt {\ell \over g}}\,K\left(\sin {\theta _{0} \over 2}\right)$ $K(k)=F\left({\pi \over 2},k\right)=\int _{0}^{\pi /2}{1 \over {\sqrt {1-k^{2}\sin ^{2}{u}}}}\,du\,.$ Figure 3 shows the deviation of T from T0, the period obtained from small-angle approximation.

Value for the complete elliptic function can be computed using the following series:

{\begin{alignedat}{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 {(2n)!}{(2^{n}\cdot n!)^{2}}}\right)^{2}\cdot \sin ^{2n}\left({\frac {\theta _{0}}{2}}\right)\right].\end{alignedat}} Figure 4 shows the relative errors using the power series. T0 is the linear approximation, and T2 to T10 include respectively the terms up to the 2nd to the 10th powers.

For a swing of exactly 180° the bob is balanced over its pivot point and so T = ∞. Figure 5. 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 pendulum of length 1 m on Earth (g = 9.80665 m/s2) at initial angle 10 degrees is $4{\sqrt {1\ \mathrm {m} \over g}}K\left({\sin {10^{\circ } \over 2}}\right)\approx 2.0102\ \mathrm {s}$ , where the linear approximation gives $2\pi {\sqrt {1\ \mathrm {m} \over g}}\approx 2.0064\ \mathrm {s}$ .
The difference (less than 0.2%) is much less than that caused by the variation of g with geographical location.

By using the following Maclaurin series:

$K\left(x\right)={\pi \over 2}\left(1+{\frac {1}{4}}x^{2}+{\frac {9}{64}}x^{4}+{\frac {25}{256}}x^{6}+{\frac {1225}{16384}}x^{8}+\cdots \right)$ $\sin \left({\theta _{0} \over 2}\right)=\left({\frac {1}{2}}\theta _{0}-{\frac {1}{48}}\theta _{0}^{3}+{\frac {1}{3840}}\theta _{0}^{5}-{\frac {1}{645120}}\theta _{0}^{7}+\cdots \right)$ The equivalent power series is:

{\begin{alignedat}{2}T&=2\pi {\sqrt {\ell \over g}}\left(1+{\frac {1}{16}}\theta _{0}^{2}+{\frac {11}{3072}}\theta _{0}^{4}+{\frac {173}{737280}}\theta _{0}^{6}+{\frac {22931}{1321205760}}\theta _{0}^{8}+{\frac {1319183}{951268147200}}\theta _{0}^{10}+{\frac {233526463}{2009078326886400}}\theta _{0}^{12}+...\right)\end{alignedat}} A much simpler and better converging formula for the period is discussed on pp. 1096-1097 of the September 2012 issue of the Notices of the AMS:

$T={\frac {2\pi }{M(\cos(\theta _{0}/2))}}{\sqrt {\frac {L}{g}}},$ ## Examples

The animations below depict several different modes of oscillation given different initial conditions. The small graph above the pendulums are their phase portraits.

## 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. In this case the pendulum's period depends on its moment of inertia I around the pivot point.

The equation of torque gives:

$\tau =I\alpha \,$ where:

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

The torque is generated by gravity so:

$\tau =-mgL\sin \theta \,$ where:

L is the distance from the pivot to the center of mass of the pendulum
θ is the angle from the vertical
$\alpha \approx -{\frac {mgL\theta }{I}}$ This is of the same form as the conventional simple pendulum and this gives a period of:

$T=2\pi {\sqrt {\frac {I}{mgL}}}$ And a frequency of:

$f={\frac {1}{T}}={\frac {1}{2\pi }}{\sqrt {\frac {mgL}{I}}}$ ## 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.