Glossary for Wonder site

The work of a curious fellow
A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z


Absolute Value
Not an agreement reached by buyer and seller...
The absolute value of a number is -1 times the number if it is negative or +1 times the number if it is positive. The absolute value makes all numbers positive regardless of sign.
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An ancestor, twice removed, of displacement...
Acceleration is the rate of change of velocity with respect to time. Acceleration is a vector quantity. Acceleration is related to the force on an object and its mass by Newton's second law, stating that acceleration equals force divided by mass. A=F/m. The rate of change of acceleration is called jerk. Jerk is equal to the rate of change of the force on an object. See the notes on acceleration for more information.
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Active State Change
Contrary to Mother Nature's wishes...
Changes in the state of a object that causes the object to depart from the path dictated by inertia. Active state changes involve action by a player outside the object whose state we are tracking.
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A tricky subject, fraught with uncertainty...
Age is the current temporal extent of an object. The age of an object could be read from a clock that had accompanied that object on its path through spacetime, its world line. See the age notes for more.
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Angular Frequency
Time, stood on its head...
The frequency of a periodic system, multiplied by 2π The units are in radians per unit time but since radians are unitless, it comes out to be t-1, the same as angular velocity of circular motion. Angular frequency is symbolized by the Greek letter omega ω.
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Angular Momentum
Angular momentum is the product of a vector quantity, angular velocity, and a scalar quantity, moment of inertia.
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Angular Velocity
Entry text
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Applied Force
As that of the hammer on a nail...
We make a somewhat arbitrary distinction among the forces acting in a dynamical system. The categories are applied force, centering force and drag or friction force. The applied force is taken to be that force which is applied to the moving parts of a system by an outside agent as for example the force applied to a pendulum by someone pushing it or the force applied to a piece of metal by the magnetic field of an electromagnet. An applied force results in an energy transfer across the system boundary.
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Where a system is contented...
An attractor is a particular state or set of states of a dynamical system that the system seeks as time passes. For instance a pendulum with friction will eventually come to rest hanging straight down with zero velocity. The point (0,0) then is an attractor for an undriven pendulum. Attractors may also be periodic, consisting of a set of states that the system visits, one after another, repeatedly. Periodic attractors may be finite or infinite sets. Another category of attractor is chaotic. Chaotic attractors consist of an infinite set of states that never repeat but are all contained in a finite volume of phase space.
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Basin of Attraction
The collection of starting places for things ending together...
If a dynamical system eventually settles down to no motion, periodic motion or bounded chaotic motion, it is said to have settled on its attractor. It is possible that a system have multiple attractors and which long-term state of motion the systems seeks depends on its initial state. The set of all initial states that lead to a certain attractor is called that attractor's basin of attraction.
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The point where one becomes two...
In general a bifurcation is a splitting or branching of an object. With regard to an attractor, it refers to a doubling of the number of points in a finite periodic attractor as a control variable changes.
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Big Bang
Marking the absolute zero of time...
The Big Bang is the name given to the event that seems to have originated the universe. I don't think I can do better describing it than to refer you to the Wikipedia article on it. See Big Bang Timeline
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Causal Relationship
The relationship between cause and effect...
Two events have a causal relationship if one caused the other. The caused event is often called the effect. The Spacetime Interval between cause and effect must be time-like or light-like.

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Centering Force
The poor thing is hardly ever satisfied...
We make a somewhat arbitrary distinction among the forces acting in a dynamical system. The categories are applied force centering, sometimes called restoring, force and drag or friction force. The centering force is taken to be that force which tends to restore the system to some equilibrium state as for example the force applied to a pendulum by gravity. A centering force does not result in an energy transfer across the system boundary since the centering agent is customarily taken to be part of the system.
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This is highly irregular...
Chaotic is an adjective describing a dynamical system or mathematical function in which future states or values are not related in any simple way to the current state or value. It is not to imply that there is no connection between past and future states or values, only that the connection is so complex as to make prediction of the future from the past a practical impossibility. If the set of future states or values, though unpredictable, is limited to a finite range of values, the system or function is said to be bounded as well as chaotic.
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Comoving Frame
A reference frame favored by Nature...
The comoving reference frame is the frame in which the cosmic microwave background (See the Wikipedia article on CMB.) appears isotropic, unvarying with respect to direction. For details I refer you to the Wikipedia article on Comoving Coordinates.
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Complex Number
Not so much complex as complete...
Ordinary numbers that we use for accounting and simple calculations are called real numbers. There is no real number such that the square of that number is -1 since the product of any real number with itself is positive. To remedy this situation, the square root of -1 was defined and given the symbol i. Then a new class of numbers was invented, called imaginary numbers, made up of i times the set of real numbers. A complex number is a number comprised of a real part added to an imaginary part like a+b*i. See the notes on complex numbers for more.
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Complex Plane
Where the imaginary is joined to the real...
The complex plane contains the set of complex numbers. It is a plane spanned by the set of real numbers, normally along the horizontal axis, and the set of imaginary numbers, normally along the vertical axis. See the notes on complex numbers for more.
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An element of a composition...
Literally components are things from which something else is composed. For our purposes it usually applies to a vector. Any vector can be expressed as the sum of a pair of perpendicular vectors. These perpendicular vectors are called the components of the initial vector. See the vector arithmetic page for more detail.
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Conservative System
Having nothing to do with politics...
A dynamical system in which no energy is either lost or gained by the system. These are systems where friction is negligible.
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A pair of objects that have an opportunity to interact...
In common usage, contemporaries are objects whose lifetimes overlap. In the heading above, I propose a modified definition taking into account the effects of relativity. See the notes on contemporaries for more.
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Everywhere dense and smooth...
Continuum – a continuous thing with no gaps or jumps however closely examined.
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Control Parameter
Sort of like the volume control on your stereo...
In a dynamical system or mathematical function, the defining equation may contain parameters that are either constant, or subject to change. Those that are subject to change are called control parameters. For example in the equation Y=A*exp(-(b-X)2) if the parameter A can be adjusted between runs through the domain of X, A would be a control parameter.
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Closing in on something...
If the dependent variable of a function under iteration gets closer and closer to a fixed value the function is said to converge under iteration.
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Conversion Factors
e.g Horsepower to stoneweight furlongs/fortnight...
A conversion factor is a number by which we multiply one unit, a meter of length for example, to get that same length expressed in other units, centimeters for example. In this case the conversion factor is 100. Multiply 10 meters by 100 centimeters per meter to get 1000 centimeters. The conversion factor may be decided by tradition, committee work or Nature. See Converting time to distance for another example.
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Companion to the sine...
In a right triangle, the cosine is the ratio of the adjacent side to the hypotenuse. See the trigonometric functions page for more.
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Everything from goes around to comes around...
The set of all the states or values visited by a periodic system or function during one period. In other words one cycle of anything that is repetitive is everything it does before repeating.
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Makes small objects heavy...
The mass per unit volume of an object. It would be measured in kilograms per cubic meter in the SI system of units.
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Extent or manifoldness...
The word dimension is commonly used in two ways. The size of a thing is given by its dimensions as in, "This is how you are to make it: the length of the ark three hundred cubits, its breadth fifty cubits, and its height thirty cubits." The other usage identifies the number of dimensions of a space as in, "We live in a three-dimensional space." It is normally clear from the context which usage is intended.
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Start subtracted from finish...
Displacement is the difference between a final position and an initial one. Displacement is a vector quantity, not necessarily the path length.
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Not taking time into account...
Distance is the separation in space between two particles or events.
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Receding, to be seen no more...
If absolute value of the dependent variable of a function under iteration increases without limit the function is said to diverge under iteration.
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The home of independent variables...
The domain is the set of values that the independent variable of a function may take on. A domain may be finite as in the set of numbers {1, 2, 3..n} or infinite as in all the numbers between 0 and 1.
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Dynamical System
Different today than yesterday...
A system that changes with the passage of time. Basically that is any system with moving parts.
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Not statics...
The study of motion and the forces which cause it.
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Effective Mass
A great simplification of reality...
The mass in a dynamical system that must be included when we treat the moving parts of the system as though they were a particle , using the free body analysis in applying Newton's laws of motion .
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Elastic Scattering
Not rubber bands scurrying off in all directions...
An interaction where two particles collide and the total kinetic energy of the two particles remains constant. The direction and speed of both particles will in general be different after the collision.
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Enables us to get up and do things...
Classically energy is defined as the ability of an object to do work on its surroundings. It may be in the form of kinetic energy or of potential energy . In relativity mass is included in energy according to E=mc2. See the energy notes for more.
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Hinders us from getting up and doing things...
Entropy may be thought of as a way to quantify the inability to get things done. The greater the entropy of a system, the less work it can accomplish. A collection of objects at rest relative to each other, with no forces acting between them and at the same temperature as their surroundings could not have much impact on their universe. That would be a high entropy system. Entropy and energy are in a sense inverses of each other. High entropy systems are characterized by uniformity. If nothing distinguishes one location in space from another, there is no motivation for objects to change positions or otherwise get involved in the goings on of their universe. High energy systems are characterized by distinction. High is different than low, fast is different than slow and hot is different than cold.
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An eternal truth...
An equation is a mathematical expression with an equal sign (=) in it. It signifies that the numerical or vector value on one side of the = is the same as the numerical or vector value on the other side. An equation may include variables and parameters. If any of the variables are rates of change, the equation is called a differential equation.
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A happening thing...
An event is an occurrence that may be considered localized in spacetime, identified with specific values of the space and time coordinates. See notes on events for more.
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Shorthand for multiple multiplications...
Raising a number to a power. The expression be means multiply b (the base) by itself e (the exponent) times.
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The reach of a thing...
The size of a thing in spacetime, spanning a certain range in each of the three spatial dimensions and a certain duration in time.
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Making a career of shapelessness...
Fluid is material that takes the shape of its container. It may be a gas which expands to fill the entire volume of its container or liquid which settles into the bottom of its container. An alternative definition of fluid is that it is material that does not support shear forces.
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Causing immediate acceleration...
Quite simply a force is a push or a pull. Force is a vector quantity.
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Free Body
Having nothing to do with scandalous behavior...
An object that is unconstrained so that it may respond to forces in accordance with Newton's laws of motion.
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Rapidity of recurrence...
The number of cycles per unit time that a periodic system or function completes. The frequency (f) is related to the period (T) by f=1/T.
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Allowing us to get a grip...
We consider two kinds of friction, sliding friction and turbulent friction. Sliding friction occurs when two solid objects maintain contact with one another while in relative motion. The small hills and valleys on the surfaces tend to get caught on one another as they pass, requiring force to be applied to keep the motion going. The reaction to this applied force is called the force of sliding friction. The work done by the applied force raises the temperature of the objects. The force perpendicular to the motion, holding the surfaces of the objects together, is called the normal force. Turbulent friction occurs when an object moves through a fluid medium, stirring it up and losing some of its kinetic energy to the medium. This situation also requires the application of a force to keep the motion going and the reaction to that force is the force of turbulent friction. The work done by the applied force raises the temperature of the object and the surrounding fluid.
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Mathematicians have moved on, but for us...
A mathematical function is a rule relating two sets of objects. Here we will restrict ourselves to objects that are numbers or vectors. One of the sets is called the domain of the function, the other is called the range of the function. Functions are frequently expressed as equations as for example Y=X+2. This function is interpreted as follows. For every X in the domain, add 2 to it to get the corresponding Y in the range. Because we are free to choose any X we want, X is called the independent variable. Because once X is chosen Y is fixed, we call Y the dependent variable.
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Different from the past and not what we think...
Future – the continuum of time later than now.
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Why no one is normal...
The bell shaped curve that is used to describe the distribution of quantities around some normal value, named in honor of Mr. Gauss we believe. This function is expressed as Y=A*exp(-(b-X)2) , where "A" is the amplitude or height of the curve and "b" is the location of the peak of the curve on the X axis. The exp() symbol represents the number "e" (approximately equal to 2.7182818284), raised to the power of the stuff in its parentheses. For example exp(0)=1, exp(1)=e, exp(2)=e2, exp(-1)=1/e, exp(-2)=1/(e2), and so on. As you can see when X=b, Y=A in the Gaussian function. As X departs from b in either direction, the value of the exp() approaches zero, forcing Y to approach zero as well. See the Non-Linear Rate of Change display in the Rate of Change lesson for an illustration
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Nature's preferred path...
A geodesic is a curve in space that minimizes or maximizes a certain parameter along its length. In flat 3-dimensional space geodesics are straight lines that minimize the distance between two points. On the curved 2-dimensional surface of a globe the geodesics are "great circles", like lines of longitude. A string stretched tight between two points on a globe, minimizing the distance between them, will lie along a great circle. In 4-dimensional spacetime, the geodesics are curves that maximize aging for an object traveling that path. In flat spacetime geodesics between events are straight lines
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With the endless task of keeping it all together...
Gravity is one of the four fundamental forces of Nature. It acts on all mass over all distances and is a purely attractive force proportional to the product of the two masses involved divided by the square of the distance between them. It is, by a factor of about 1040 the weakest of the fundamental forces. It is so weak that it requires an object the mass of the Earth to make a pound of butter weigh a pound. See the notes on gravity for more information.
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No entry here yet.
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Impulse Force
Having nothing to do with rash behavior...
A force applied for a time which is short compared to the observation time, as for example the force between a bat and ball where the observation is over the entire flight of the ball from leaving the pitcher's hand to landing in the bleachers.
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A manifestation of the Nature's laziness...
Inertia is the resistance of any physical object to a change in its state of motion or rest. It is proportional to an object's mass. The path an object follows under the influence of inertia is a spacetime curve called a geodesic. In flat spacetime geodesics are straight lines.
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The one thing in spacetime on which one may depend...
Interval – the separation in spacetime between two events. The spacetime interval between two events is invariant among reference frames. See the interval notes for more.
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A short enough span of time that shorter makes no difference...
Instant – an extent of time so short that in the context its duration may be neglected.
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Getting infinite variety from simple repetition ...
Iteration is the process of taking the value of the dependent variable of a function and feeding it back into the function as the independent variable.
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Julia Set
Gaston Julia had some sets...
A Julia set, named for Gaston Julia, is the set of all points in the complex plane such that the iterated function Zn=Z(n-1)2+k, where k is a fixed complex number, does not diverge as n approaches infinity. It differs from the Mandelbrot set in that there is a different Julia set for each value of k. There is only one Mandelbrot set.
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Kinetic Energy
The faster things go the harder they hit you...
The energy an object has as a result of its motion. Numerically the kinetic energy is equal to 1/2*m*v2 where m is the mass of the object and v is the magnitude of its velocity .
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An example of the relentless urge to name things...
The study of objects in motion without explicit consideration for the forces which produced the motion.
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The time allotted to do the necessary...
Lifetime is the total temporal extent of an object. The amount of time run off in the lifetime of a clock depends on some effects of relativity.
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Disregards possibly important information...
The size of a thing, without regard for its sign (+ or -) or direction. Similar to the absolute value of a number but applies to vectors as well.
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Mandelbrot Set
Benoit Mandelbrot had a set...
The Mandelbrot set, named for Benoit Mandelbrot, is the set of all points c in the complex plane such that the iterated function Zn=Z(n-1)2+c does not diverge as n approaches infinity.
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Causes objects to hate to change speed...
The property of an object which determines its resistance to changes in velocity. In the presence of a gravitational field, as near the surface of a planet, the mass of an object is proportional to its weight, the force exerted on the object by the planet.
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Having not much to do with automobile repair...
The study of objects in motion. Mechanics is normally limited to a small number of large slow objects, as opposed to statistical mechanics which deals with large numbers of objects, relativistic mechanics which deals with objects moving near the speed of light and quantum mechanics which deals with objects more or less the size of atoms. Mechanics encompasses the topics of kinematics and dynamics.
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Moment of Inertia
To rotation what mass is to linear motion.
The moment of inertia is the resistance an object offers to changes in rotational speed. It depends on the mass of the object and the distribution of that mass about the axis of rotation.
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Sticking straight out...
Another word for perpendicular, normal in this sense is usually used in referring to a vector's orientation relative to some surface. For example a vertical vector is normal to a horizontal surface. I would not suggest that on recovering from mental disease you tell people that you feel perfectly perpendicular.
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The boundary between future and past where chance is converted to certainty...

Now – the current instant. The duration of now is either zero or so close to it that the difference does not concern us in the present context.

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A thing, an element of stuff...
The term "object" is the most general form of thingness. There are physical objects like baseballs and uranium atoms, and mathematical objects like numbers and vectors . It will be clear from the context what sort of object we are talking about. Also the particle like aspect of fields, like the photon of electromagnetic radiation or gauge bosons in general may be considered objects. See the notes on objects for more.
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Where occupants of reference frames normally sit...
The point in a reference frame from which measurements are made. It is the location of the zero value for each axis in the frame.
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Sort of controlling the shape of a function...
In an equation those elements that are not variables are parameters. If a parameter is multiplied times a variable it may be called a coefficient. Parameters may be fixed or adjustable. Fixed parameters are called constants. Adjustable parameters are called control parameters. In the function Y=A*exp(-(b-c*X)2) A, b and c are parameters. If A takes on different values during different runs through the values of X, it is considered a control parameter. If b is fixed during all runs through the values of X, it is a constant. The parameter c is a coefficient of X.
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Objects of trifling size...
An object whose size is negligible in the context of our observation of it. For example the Earth might be considered a particle if we were studying its orbit around the Sun, but not if we want to know anything about its rotation about its axis. The nucleus of an atom might be a particle in an experiment on elastic scattering , but not in considering nuclear fission.
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Passive State Change
Requiring minimum effort on the part of Nature...
A passive state change is the change in state of an object that results from the object's interaction only with the spacetime it inhabits.
Entry text
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Having nothing to do with punctuation...
The interval of time between the occurrence of identical states in a periodic system.
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Where some of the possible has become what happened...
The past is the continuum of time earlier than the now.
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Exactly repetitive...
A dynamical system or function which at some point returns to the same state or value. If a system or function ever revisits the identical state or value it will continue to come back to it again and again in equal intervals of time. That is why we call such a system periodic.
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Phase Angle
Slides a periodic function along the axis...
The offset from the origin of a periodic function like the sine or cosine. For example in the function x=A*sin(w*t + f), f is the phase angle. The units on f are radians.
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Phase-Control Space
Spanned by parameter and variable...
If in a phase space representation of a dynamical system we exchange the roles of a control parameter and the independent variable, allowing the former to vary while holding the latter fixed, the resulting graph is plotted in a mixed space called phase-control space. See the Order online course for more information.
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Phase Space
A space without measure...
Sometimes as an aid in understanding what is happening in a dynamical system it is useful to mentally create a new kind of space. This space has dimensions of the state variables for our dynamical system. In the case of the pendulum, for example, it would be one dimension measured in position and one dimension measured in velocity. Time might be considered a third dimension of this "phase space", just as it might be considered a fourth dimension of ordinary space.
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Chunks of radiation...
A photon is the particle -like aspect of light or other electromagnetic radiation, a bundle of energy localizable in space. Photons travel at the speed of light regardless of the reference frame in which they are observed, as predicted by relativity theory. It appears that the electrical forces between charged particles may be due to an exchange of photons that never are detected except by their action in producing the force. Such photons are called virtual photons.

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Of pie are square fame...
The ratio of the circumference of a circle to its diameter is named by the Greek letter pi (π). The numerical value of pi is approximately 3.1415926.
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Not of the sort preferred to a job...
The location of an object relative to some point we have chosen to be the reference point. Position is a vector quantity.
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Potential Energy
Energy unrealized...
The potential energy of an object is the energy that object has as a result of its position relative to other objects. The numerical value of potential energy depends on the nature of the interaction of the object with its surroundings and the choice of a position to be the zero energy point.
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An illusory slowing of the sweep of time...
The present is the slice of the recent past that we perceive as our current moment. Perception being a rather sluggish process, it is hopelessly outpaced by the advance of now.
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Proper Time
Time displayed on an observer's wrist watch...
On account of the effects of relativity time flows at varying rates for different observers. This requires that we be careful in specifying whose time we are talking about. The time displayed on the t-axis of a spacetime diagram is the reference frame time. Each observer in that reference frame experiences her "own time", "eigenzeit" in Einstein's native language. In English that is usually called "proper time". An observer's proper time is that displayed on a clock that accompanies observer throughout her relativistic adventures.

Of course the proper time will depend on the time set on the clock initially so in preparing a spacetime diagram it is customary to re-zero all clocks at the time origin of the diagram. From that time on, the duration of each observer's proper time unit as seen from the reference frame will depend on that observer's relativistic status.

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Generally not sold or traded...
A characteristic that is inherently associated with the object which is said to have that property. For example the mass of an object is one of its properties. So also might be color, density and many other characteristics. Properties are classified as extensive or intensive. Extensive properties increase in proportion to the size of the object, as mass does for example. Intensive properties are independent of the size of the object. The density for examples remains the same if I cut an object in half and throw half of it away. Things like an object's position or velocity are not considered to be properties of the object. They are not a characteristic of the object only but are also dependent on the reference frame in which the object is located.
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Sort of square-like...
A function involving the second and lower power, and none higher, of the independent variable. A quadratic function may contain x2 explicitly or it may contain terms like x*(1-x), where the second power of x is implied. In general a quadratic may be written as y=a*x2+b*x+c . For an illustration of a quadratic function, see the Quadratic Derivative display in the Rate of Change lesson.
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Addressing the "How much?" question...
A numerical value either scalar or vector , which describes some attribute of an object like its position or its velocity . We sometimes speak of physical quantities to signify that we are talking about an object's properties or attributes as opposed to a purely mathematical quantity.
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Quantum Mechanics
A small scale sort of science...
Early in the twentieth century it became evident that the state of very small systems did not change in a continuous way. Atoms jump from one allowed state to another, skipping over the forbidden states in between. Newton's laws of classical mechanics are not equipped to handle situations like this so a new approach was required. Quantum Mechanics evolved to extend classical mechanics to very small systems. See the notes on quantum mechanics for more.
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A radius out, along and back...
An angular unit of measure. A radian is an angle subtended by an arc whose length equals one radius. Since the circumference of a circle is 2* π *radius and a radian spans an arc of one radius, there are 2*π radians in a complete circle. So 1 radian equals 360/(2*π) degrees. This is illustrated below.
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Radius of Curvature
A fitting measure...
The radius of the largest circle containing the point at which the radius of curvature is to be determined and fitting within the curve. See the illustration below.
radius of curvature
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Where the dependent variable is at home...
The range is the set of values that the dependent variable of a function may take on. A range may be finite as in the set of numbers {1, 2, 3..n} or infinite as in all the numbers between 0 and 1.
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Reference Frame
A hypothetical construct of great usefulness...
A mathematical object which is used to allow comparison of the positions in space of physical objects like particles, or the comparison of one particle's positions at different times. The reference frame may be made up of any set of coordinates which uniquely specify a point in space. See the reference frame notes for more.
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Einstein's way of keeping pyhsicists up at night...
The special theory of relativity deals with very fast moving objects. The general theory deals with accelerated reference frames and gravity. A lot of physics can be done without concern for these theories but where the effects are felt, the results are profoundly different than what classical physics would predict. I recommend that you at least see the relativity notes if not read a good text on the subject.
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Like plain vanilla numbers...
A scalar quantity is one having only magnitude , not direction information. This is as opposed to a vector quantity which has both magnitude and direction.
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Second Law
Time's arrow...

There is a law of nature that seems to be universal, applying to all sorts of situations. It was initially formulated with respect to thermodynamics, the study of heat flow. In that context the second law states that heat flows from hot objects to cold objects with a consequent decrease in the hot object's temperature and increase in the cold object's temperature. The temperature changes reduce the distinction between the hot and cold objects, increasing the entropy of the system. It is the tendency for entropy to increase over time that is the universal aspect of the second law. This tendency is so ubiquitous that it has been taken to define the direction of time's flow from past to future. See the second law notes for more.

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One of the collective nouns of math...
In mathematics a set is a collection of related objects. The mathematical usage is similar to the ordinary English meaning of the word. The objects that make up a set are called the elements of the set. If a set contains an unlimited number of elements it is an infinite set. Otherwise it is a finite set.
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Shear Force
Not a filmy translucent thing at all...
Imagine a solid rectangular block of material with the bottom face held fixed on the surface on which the block was resting and a force applied along the top face... Sort of the same force you might apply to an Oreo cookie to slide the top cookie off the bottom one to get at the cream filling. The applied force in this situation is called a shear force. A solid block subject to this shear force would be deformed so that its front face that was a rectangle becomes a parallelogram.
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Significant Figures
No, no. Not that kind of figures...
The number of digits in a numerical value that are reliably known. If the numbers being used in a calculation are measured values, there will always be a limit on the accuracy of the measurement. The results of any calculations based on those numbers should not be reported with more significant figures than the least accurate of the measured values. For example if the length of a rectangle is measured to within 0.1 cm to be 25.3 cm and its width to within 0.1 cm to be 6.6 cm, multiplying shows the area to be 166.98 cm2. In significance arithmetic the result however should be reported only to 2 significant figures since the width is only known to that accuracy, giving an area of 170 cm2.
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O Lord, give us a sine...
In a right triangle, the ratio of the opposite side to the hypotenuse. See the trigonometric functions page for more.
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Invented by God to keep everything from being in the same place...
It is thought that at a very small scale space may be found to be quantized, being made up of discrete steps. If this is true the length of the steps is thought to be smaller than 10-34 meters. For our purposes we may consider space a non-temporal, three-dimensional continuum. occupied by objects and events – that part of spacetime where there is freedom of motion.
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Space and time as a single entity...
An extension of the concept of space to include an additional dimension perpendicular to the normal axes spanning our familiar three-dimensional space. This additional dimension measures time so that a point in spacetime locates an event. Since full four-dimensional spacetime is difficult to picture, we frequently work in spacetime consisting of one or two spatial dimensions and the time dimension. Spacetime is integral to Einstein's relativity theories. See the notes on spacetime
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Spacetime Diagram
A convenient tool for exploring the effects of special relativity is the spacetime diagram. It is a plot of one dimension of space along the horizontal and the time dimension along the vertical. The worldlines of objects and individual events may be displayed on such a plot. See the spacetime diagram notes for more.
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What state of affairs is this...
Dynamical systems evolve over the course of time. The state of the system at any instant may be identified by the values of certain variables at that instant. For example specifying the angle from the vertical and the velocity of a frictionless pendulum allows us to predict its position and velocity at any future time. Therefore the state of the pendulum at any instant is its position and velocity. In this example the position and velocity are known as state variables. See the state of a system article for more.
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State Variable
The keys to the future...
An observable quantity which must be specified in order to determine how a Dynamical systems changes over the course of time. In conservative systems if all the state variables are known at any instant, the state of the system is determined for all future time.
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A collection of objects and interactions...

For our purposes a system is a set of objects interacting in accordance with certain laws of nature. There is a degree of arbitrariness in defining a system. We get to choose what to consider to be the system and everything left over is the system's environment. If our system behaves in unexpected ways the first place to look for problems is in the system definition. Perhaps we have not included all the pertinent pieces in our system definition. See the system notes for more.

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Systéme International (International System) (SI) Units
Lightening the load on conversion factors...
The most commonly accepted system of units in scientific work. The fundamental units in this system are the meter, kilogram and second. See the Wikipedia article for more.
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That which one sometimes goes off on...
A straight line which touches a curve in one and only one point. The slope of a tangent is the slope of the curve at that point. Slope is the change in the vertical coordinate divided by the corresponding change in the horizontal coordinate. See the Rate of Change lesson for more on slopes.
Also, in a right triangle, the ratio of the opposite side to the adjacent. See the trigonometric functions page for more.
Tangential Velocity
The slingshot trick...
Tangential velocity is a term often applied to rotating wheels or orbiting satellites. It is the component of velocity tangent to the rim of the wheel or the satellite orbit.
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The timewise aspect of a thing...
Temporal – of or relating to time.
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Invented by God to keep everything from happening at once...
It is thought that at a very small scale time may be found to be quantized, being made up of discrete steps. If this is true the duration of the steps is thought to be smaller than 10-34 seconds. For our purposes we may consider time a non-spatial continuum in which events along a worldline occur in irreversible succession.
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A path best avoided...
The path an object takes through space. Frequently associated with a projectile like a bullet or a missile.
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There is one of these critters under every rock...
Universality refers to the fact that a certain number, approximately 4.669, discovered by Mitchell Feigenbaum and called Fiegenbaum's number, keeps popping up in what appear to be totally unrelated circumstances. See the Order online course for more information.
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Are these the arrows of outrageous fortune?...
A quantity having both magnitude and direction. The direction may be expressed as an angle from a single axis in two dimensions. In three dimensions, the direction must be a pair of angles measured from different axes. On free body diagrams vectors are represented by arrows. See the vector arithmetic page for ways in which vectors may be combined.
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Lying between acceleration and displacement...
The speed of an object in a given direction. Velocity is a vector quantity.
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An important consideration in interference...
The distance covered by a travelling wave in one period . It is the distance between points of the same phase angle in a travelling wave. For example the distance between peaks, or the distance between valleys of a wave train. The wavelength is frequently symbolized by the Greek letter lambda l .
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All work and no play makes Jack...
Work is defined as the application of force over some displacement . Numerically the work done is the product of the force and the distance moved in the direction of that force. This may be calculated as force times displacement times the cosine of the angle between force and displacement. The angle gets involved because things do not always move in the direction in which you push them.
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A track on the sands of time...
A worldline is path through spacetime connecting the events in which a particular object participates including the sort of null event "object found here". The worldline is a record of the object's history. The time scale along an object's worldline will vary depending on the relativistic history of the object.
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