Often used in scrying. Pendulums can be made of almost anything, often though a stone of some kind on the end of some cord. If they are attuned to the owner, they will answer 'yes' or 'no' questions. Not cosmic magic 8 balls, they will correctly answer, although sometimes they return a firm 'maybe, i don't *know*'. A pendulum not properly attuned is useless; you can tell almost the moment you pick it up whether it will work for you or not. They respond differently to different people; mine circles clockwise for "yes", counterclockwise for no, and jerks up and down or back and forth for 'maybe'. I've heard of pendulums that go north-south for yes and east-west, not circling just swinging; it varies. The best way to have a "key" is to ask it questions you *know* the answers to, a set of yes and a set of no, and from this pattern you should be able to discern the correct response.

The period of a pendulum is the amount of time it takes for it to move along its entire distance. (From one end to the other and back, for instance.) This is described by:

T = 2pi(l/g)1/2

Where T is Time, l is legth of pendulum in meters and g is gravity in m/sec2 (9.8 on earth, give or take for altitude/air pressure/your professors anal-retentiveness.) The reciprocal of T is the frequency (how often the pendulum completes per second.)

However,


the fun continues! A similar formula can be used to describe the frequency and period of an oscillating body, say a spring: T = 2pi(m/k)1/2

yadda yadda ya, m is mass and k is our given spring coefficient.

2pi is taken from the formula for the circumference of a circle, the square-root of length over gravity creates a description for the circumference of our hypothetical circle.

An exact solution to the equations of motion for a pendulum requires the use of elliptic integrals and elliptic functions, and the pendulum problem is one of the most basic illustrations of the usefulness of elliptic integrals and functions in physics and applied mathematics.

As everyone is no doubt aware, a pendulum may easily be idealized as simply a mass m suspended from a rigid rod (of negligible mass) of constant length L:

///////////////////////////////////
-----------------------------------
                  |\
                  | \
                  |θ \
                  |   \
                  |    \
                  |     \ L
                  |      \
                  |       \
                           \
                        __- o
               mg sin θ     |\
                            | \
                            v  v
                           mg mg cos θ

Summing forces, we see that the weight component mg cos θ (where g is the acceleration due to gravity of 9.8 m/s2 or thereabouts) cancels out due to the restraint of the rod. The only weight component that contributes to the overall motion of the pendulum is mg sin θ, that is tangent to the path of motion. If we take s as the arc length of the path, then Newton's Second Law of motion F = ma gives us:

  d2s
m ___ = -mg sin θ

  dt2

The arc length s of a circle of radius L is in terms of the central angle θ obviously s = Lθ (where θ is in radians of course), so in terms of the angular position θ the equation of motion for the pendulum becomes:

d2θ   g2
___ + __ sin θ = 0
dt2   L2

Note that the mass of the pendulum bob has cancelled out (so it's still true that the weight of the hanged man doesn't influence his motion, as Umberto Eco would put it). This differential equation is obviously nonlinear, as sin θ is a nonlinear function of θ, but we can integrate the equation to yield an equation in terms of the energy of the pendulum:

1  dθ      g2
_ (__)2  - __ cos θ = C
2  dt      L2

where C is a constant related to the initial potential energy of the pendulum, which we immediately see is given by g2/L2 cos θM, where θM is the maximum angular displacement of the pendulum (what Webster 1913 calls the arc of vibration). By solving for dθ/dt in that equation, we obtain:

dθ     2g
__ = ±(__)1/2 sqrt(cos θ - cos θm)
dt      L

We take t to be 0 when θ = 0, and dθ/dt > 0 (so the negative solution above is considered extraneous). integrating the equation from 0 to θM we get:

 θM
∫  (cos θ - cos  θM)-1/2 dθ = sqrt(2g/L) t
 0

This is a quarter of a cycle, so the time t we get once we evaluate the left hand integral is a quarter of the pendulum's period T. Since θ is always less than or equal to θM by definition, we can make the half-angle substitution sin(θ/2) = sin(θM/2) sin φ, so the integral above reduces to:

                 π/2         dφ
T = 4 sqrt(L/g) ∫    ___________________
                 0   sqrt(1 - k2 sin2 φ)

where k = sin(θM/2). This integral we recognize as one of Legendre's elliptic integral standard forms, specifically the (complete) elliptic integral of the first kind, so we finally get:

T = 4 sqrt(L/g) F(sin(θM/2)|π/2)

As an example, say we had a pendulum with a rod which is 0.600 m long, and we started the pendulum at a maximum angular displacement of 80.0 degrees. T = 0.990 F(sin 80.0/2 | π/2) = 0.990*1.78677 = 1.77 seconds (from tables in Abramowitz and Stegun). The approximation in piq's wu on the other hand yields 1.55 seconds, which is quite inaccurate. However, that approximation, which was first derived by Christiaan Huygens (1629-1695) in the 17th century (by linearizing the nonlinear equation of motion above to d2θ/dt2 = -(g/L)2θ), is pretty good when displacements are small, as the smaller θM is the closer F(k|π/2) approaches π/2. Using π/2 as an approximation for the elliptic integral produces 2π sqrt(L/g).

On the other hand, in order to solve for the angle of motion as a function of time, for instance to make a simulation of the movement of the pendulum, it becomes necessary to look into inverse functions of the elliptic integrals, the Jacobi elliptic functions.

References

Abramowitz, Milton and Irene A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.

Andrews, Larry C., Special Functions of Mathematics for Engineers, McGraw-Hill, 1992.

Arfken, George, Mathematical Methods for Physicists, Academic Press, 1985

Eco, Umberto. Foucault's Pendulum, Ballantine Books, 1990.

Greenhill, Alfred George, The Applications of Elliptic Functions, Macmillan and Co., 1892.

Pendulum

Pendulum are an electronic/drum and bass band from Perth, Australia. The group consists of Rob Swire, Gareth McGrillen and Paul Harding. They currently live in the United Kingdom.

Their sound is generally drum and bass with electronic synthesizers. Their music can be pretty trippy, especially if you're pilling. In 2005, they released a remix of Voodoo people with The Prodigy. Their biggest hit so far has been Slam which reached #34 on the UK Singles Chart. Their first album was Hold Your Colour which was one of the biggest selling drum and bass releases of all time. They are currently recording a new album under the Breakbeat Kaos Records label, which will be released in late 2007.

Their songs feature a variety of different instruments and effects. They use synthesizers and electronic sounds along with drums and some spoken word vocals and some distorted vocals. Their music is often played at clubs as the drops in the songs come on a lot stronger when on ecstasy. The band is influenced by Aphex Twin and other techno/dance artists. The songs vary from having a lot of vocals to being totally instrumental, and almost always have a fast tempo with a lot of drums.

Albums

Hold Your Colour
Junglesound vol. 2 (mixed by Pendulum)
Jungle Sound Gold (mixed by Pendulum)

Singles

Vault
Spiral / Ulterior Motive
Back To You / Still Grey
Another Planet / Voyager
Submarines (Pendulum Remix)/ Submarines, DJ Fresh
Bacteria (Pendulum Remix), Ed Rush & Optical
Guns At Dawn, DJ Baron featuring Pendulum
Tarantula / Fasten Your Seatbelt, featuring The Freestylers
Slam / Out Here
Voodoo People (Pendulum Remix), The Prodigy
Hold Your Colour (Bi-Polar Remix) / Streamline
Painkiller / Jump N Twist, Pendulum vs The Freestylers feat. Sir Real
Blood Sugar/Axle Grinder
Distress Signal

Pen"du*lum (?), n.; pl. Pendulums (#). [NL., fr. L. pendulus hanging, swinging. See Pendulous.]

A body so suspended from a fixed point as to swing freely to and fro by the alternate action of gravity and momentum. It is used to regulate the movements of clockwork and other machinery.

The time of oscillation of a pendulum is independent of the arc of vibration, provided this arc be small.

Ballistic pendulum. See under Ballistic. -- Compensation pendulum, a clock pendulum in which the effect of changes of temperature of the length of the rod is so counteracted, usually by the opposite expansion of differene metals, that the distance of the center of oscillation from the center of suspension remains invariable; as, the mercurial compensation pendulum, in which the expansion of the rod is compensated by the opposite expansion of mercury in a jar constituting the bob; the gridiron pendulum, in which compensation is effected by the opposite expansion of sets of rodsof different metals. -- Compound pendulum, an ordinary pendulum; -- so called, as being made up of different parts, and contrasted with simple pendulum. -- Conical or Revolving, pendulum, a weight connected by a rod with a fixed point; and revolving in a horizontal cyrcle about the vertical from that point. -- Pendulum bob, the weight at the lower end of a pendulum. -- Pendulum level, a plumb level. See under Level. -- Pendulum wheel, the balance of a watch. -- Simple or Theoretical, pendulum, an imaginary pendulum having no dimensions except length, and no weight except at the center of oscillation; in other words, a material point suspended by an ideal line.

© Webster 1913.

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