axes with which the position of any object may be specified as it changes with time. The origin of the axes and their spatial di- rections must be specified at every instant of time for the frame to be fully deter- mined.
framework
A collection of light rods that are joined together at their ends to give a rigid structure. In a framework the rods are said to be light if their weights are very much smaller than the loads they are bear- ing. When a framework has external forces acting on it each rod in the framework can either prevent the structure from collapsing or stop the joints connecting the rods from becoming separated. If a rod is stopping a collapse it exerts a push at both ends and is said to be in compression or in thrust. If a rod is stopping the structure from becom- ing separated it exerts a pull at both ends and is said to be in tension. If the whole of the framework is in equilibrium then the forces at each joint have to be in equilib- rium. This means that in equilibrium the external forces on the framework are in equilibrium with the internal forces since the forces in the rods occur as equal and opposite forces.freedom, degrees of
The number of in- dependent quantities that are necessary to determine an object or system. The numberof degrees of freedom is reduced by con- straints on the system since the number of independent quantities necessary to deter- mine the system is also reduced. For exam- ple, a point in space has three degrees of freedom since three coordinates are needed to determine its postion. If the point is con- strained to lie on a curve in space, it has then only one degree of freedom, since only one parameter is needed to specify its posi- tion on the curve.
free oscillation
(free vibration) An oscil- lation at the natural frequency of the sys- tem or object. For example, a pendulum can be forced to swing at any frequency by applying a periodic external force, but it will swing freely at only one frequency, which depends on its length. Compare forced oscillation. See also resonance.free variable
In mathematical logic, a variable that is not within the scope of any quantifier. (If it is within the scope of a quantifier it is bound.) In the formula F(y) → (∃x)G(x), y is a free variable.French curve
A drawing instrument con- sisting of a rigid piece of plastic or metal sheeting with curved edges. The curvature of the edges varies from being almost straight to very tight curves, so that a part of the edge can be chosen to guide a pen or pencil along any desired curvature. An- other instrument that serves the same pur- pose consists of a deformable strip of lead bar cut into short sections and surrounded by a thick layer of plastic. This is bent to form any required curvature.frequency
Symbol: f, ν The number of cy- cles per unit time of an oscillation (e.g. aframe of reference
pendulum, vibrating system, wave, alter- nating current, etc.). The unit is the hertz (Hz). The symbol f is used for frequency, although ν is often employed for the fre- quency of light or other electromagnetic radiation.
Angular frequency (ω) is related to fre- quency by ω = 2πf.
The frequency of an event is the number of times that it has occurred, as recorded in a FREQUENCY TABLE.
frequency curve
A smoothed FREQUENCY POLYGONfor data that can take a continu- ous set of values. As the amount of data is increased and the size of class interval de- creased, the frequency polygon more closely approximates a smooth curve. Rel- ative frequency curves are smoothed rela- tive frequency polygons. See also skewness.frequency function
1. The function thatgives the values of the frequency of each re- sult or observation in an experiment. For a large sample that is representative of the whole population, the observed frequency function will be the same as the probability
DISTRIBUTION FUNCTIONf(x) of a population variable x.
2. See random variable.
frequency polygon
The graph obtained when the mid-points of the tops of the rec- tangles in a HISTOGRAMwith equal class in- tervals are joined by line segments. The area under the polygon is equal to the total area of the rectangles.frequency table
A table showing how often each type (class) of result occurs in a sample or experiment. For example, the daily wages received by 100 employes in a company could be shown as the number in each range from $50.00 to $74.99, $75.00 to $99.99, and so on. In this case the rep- resentative value of each class (the class mark) is $(50 + 74.99)/2, etc. See also his- togram.Fresnel integrals
/fray-nel/ Two integrals C(x) and S(x) defined by:C(x) = ∫x0cos(πu2/2)du
S(x) = ∫x0sin(πu2/2)du.
These integrals occur in the theory of dif- fraction and have been extensively tabu- lated. The Fresnel integrals can be combined to give:
C(x) – i S(x) = ∫x0exp[–i(πu2/2)] du,
C(x) + i S(x) = ∫x0exp[i(πu2/2)] du.
friction
A force opposing the relative mo- tion of two surfaces in contact. In fact, each surface applies a force on the other in the opposite direction to the relative mo- tion; the forces are parallel to the line of contact. The exact causes of friction are still not fully understood. It probably re- sults from minute surface roughness, even on apparently ‘smooth’ surfaces. Frictional forces do not depend on the area of con- tact. Presumably lubricants act by separat- ing the surfaces. For friction between two solid surfaces, sliding friction (or kinetic friction) opposes friction between two moving surfaces. It is less than the force of static (or limiting) friction, which opposes slip between surfaces that are at rest. Rolling friction occurs when a body is rolling on a surface: here the surface in contact is constantly changing. Frictional force (F) is proportional to the force hold- ing the bodies together (the ‘normal reac- tion’ R). The constants of proportionality (for different cases) are called coefficients of friction (symbol: µ):µ = F/R
Two laws of friction are sometimes stated:
1. The frictional force is independent of the
area of contact (for the same force holding the surfaces together).
2. The frictional force is proportional to
the force holding the surfaces together. In sliding friction it is independent of the rel- ative velocities of the surfaces.
frustum
/frus-tŭm/ (pl. frustums or frusta) A geometric solid produced by two parallel planes cutting a solid, or by one plane par- allel to the base.fulcrum
/fûl-krŭm/ (pl. fulcrums or ful-cra) The point about which a lever turns.
function
(mapping) Any defined pro- cedure that relates one number, quantity, etc., to another or others. In algebra, afunction of a variable x is often written as f(x). If two variable quantities, x and y, are related by the equation y = x2+ 2, for ex-
ample, then y is a function of x or y = f(x) = x2+ 2. The function here means ‘square
the number and add two’. x is the indepen- dent variable and y is the dependent vari- able. The inverse function – the one that expresses x in terms of y in this case – would be x = ±√(y –2), which might be written as x = g(y).
A function can be regarded as a rela- tionship between the elements of one set (the range) and those of another set (the domain). For each element of the first set there is a corresponding element of the sec- ond set into which it is ‘mapped’ by the function. For example, the set of numbers {1,2,3,4} is mapped into the set {1,8,27,64} by taking the cube of each element. A func- tion may also map elements of a set into others in the same set. Within the set {all women}, there are two subsets {mothers} and {daughters}. The mapping between them is ‘is the mother of’ and the inverse is ‘is the daughter of’.
functional
A function in which both the domain and range can be sets of functions. Roughly speaking, a functional can be con- sidered to be a function of a function. Functionals are used extensively in analysis and physics, particularly for problems in- volving many degrees of freedom. A func- tional F of a function f is denoted by F[f]. See also functional analysis.functional analysis
A branch of analysis that deals with mappings between classes of functions and the OPERATORSthat bring about such mappings. In functional analy- sis a function can be regarded as a point in an abstract space. Functional analysis wasextensively investigated in the twentieth century, with applications to differential equations, integral equations, and quan- tum mechanics. These enabled the founda- tions of these subjects to be laid on a sound axiomatic basis.