Pendulum (mathematics): Wikis

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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.

Contents

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

{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 be derived from the conservation of mechanical energy. At any point in its swing, the kinetic energy of the bob is equal to the gravitational potential energy it lost in falling from its highest position at the ends of its swing (the distance h in the diagram). From the kinetic energy the velocity can be calculated.

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

{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

The differential equation given above is not soluble in elementary functions. A further assumption, that the pendulum attains only a small amplitude, that is

\theta \ll 1\,

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

\sin\theta\approx\theta\,

to be made. To first order, the error in this approximation is proportional to θ 3 (from the Maclaurin series for sin θ). Substituting this approximation into (1) yields the equation for a harmonic oscillator:

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

Under the initial conditions θ(0) = θ0 and /dt(0) = 0, the solution is

\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 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.

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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 \scriptstyle g\approx9.81 m/s2, and \scriptstyle g/\pi^2\approx{1} (the exact figure is 0.994 to 3 decimal places).

Therefore

\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 square of the time period (in seconds).

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.
Figure 5. Relative errors using the power series.
{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.

This integral cannot be evaluated in terms of elementary functions. It can be rewritten 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.

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,\varphi) is Legendre's elliptic function of the first kind

F(k,\varphi) = \int^\varphi_0 {1\over\sqrt{1-k^2\sin^2{\theta}}}\,d\theta.

Figure 4 shows the deviation of T\, from T_0\,, the period obtained from small-angle approximation.

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

\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. 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 = ∞.

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}F\left({\sin 10^\circ\over 2},{\pi\over2}\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:

 F \left(x, {\pi \over 2} \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:[1]

\begin{alignat}{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{alignat}

Compound pendulum

A compound 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:

α is the angular acceleration.
τ is the torque

The torque is generated by gravity so:

\tau = - m g L \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

Hence, under the small-angle approximation \scriptstyle \sin \theta \approx \theta\,,

\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}}.

[2]

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:[3] 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

External links

References

  1. ^ Nelson, Robert; M. G. Olsson (February 1986). "The pendulum — Rich physics from a simple system". American Journal of Physics 54 (2): pp. 112–121. http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000054000002000112000001&idtype=cvips&gifs=yes. Retrieved 2008-10-29.  
  2. ^ Physical Pendulum
  3. ^ 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
  • Kenneth L. Sala, “Transformations of the Jacobian Amplitude Function and its Calculation via the Arithmetic-Geometric Mean”, SIAM J. Math. Anal., vol. 20, no. 6, pp. 1514 – 1528, Nov. 1989.

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