Angular measurement radian

A unit of angular measurement such that there are 2 pi radians in a complete circle. One radian = 180/pi degrees. One radian is approximately 57.3°. radical axis... 

Phase differences may be expressed in angular measure as well as in wavelength two rays, differing in path length by one whole wavelength, are said to differ in phase by 360°, or In radians. If the path difference is d, then the phase difference 0 in radians is given by... 

Static angle of repose of material or Kiln slope in angular measure, (degree, radians)... 

The abbreviation for the SI unit of plane angular measure, the radian, the angle intercepting a circular arc of length equal to its radius (= 360o/27t = 57.3°)... 

Prefixes are used for the basic SI unit with the exception of weight, where the prefix is used with the unit gram (g), not the basic SI unit kilogram (kg). Prefixes are also not used for units of angular measurement (degrees, radians), lime (seconds), or temperature ( C or K). The prefixes are used in a way that the numerical value of a unit lies between 0.1 and 1,000. For example, 56 kN rather than 5.6x 10 N, 11.2kPa rather than 11,200 Pa, and 6.2 mm rather than 0.0062 m. 

A viscoelastic material also possesses a complex dynamic viscosity, rj = rj - - iv( and it can be shown that r = G jiuj-, rj = G juj and rj = G ju), where CO is the angular frequency. The parameter Tj is useful for many viscoelastic fluids in that a plot of its absolute value Tj vs angular frequency in radians/s is often numerically similar to a plot of shear viscosity Tj vs shear rate. This correspondence is known as the Cox-Merz empirical relationship. The parameter Tj is called the dynamic viscosity and is related to G the loss modulus the parameter Tj does not deal with viscosity, but is a measure of elasticity. 

Models for description of liquids should provide us with an understanding of the dynamic behavior of the molecules, and thus of the routes of chemical reactions in the liquids. While it is often relatively easy to describe the molecular structure and dynamics of the gaseous or the solid state, this is not true for the liquid state. Molecules in liquids can perform vibrations, rotations, and translations. A successful model often used for the description of molecular rotational processes in liquids is the rotational diffusion model, in which it is assumed that the molecules rotate by small angular steps about the molecular rotation axes. One quantity to describe the rotational speed of molecules is the reorientational correlation time T, which is a measure for the average time elapsed when a molecule has rotated through an angle of the order of 1 radian, or approximately 60°. It is indirectly proportional to the velocity of rotational motion. 

It cannot be emphasized too much that the resonant frequency in NMR is proportional to the magnetogyric ratio, y, and to the laboratory magnetic field strength, B0. This relationship forms the basis of nearly every phenomenon observed in NMR. There are two ways to measure the precession rate the angular velocity, co0, in units of radians per second and the frequency, v0, in units of cycles per second or hertz. In this book we will use frequencies in hertz. This frequency is sometimes called the Larmor frequency, and the zero subscript refers to this fundamental frequency, which results from the laboratory magnetic field interacting with the nucleus magnetic field. 

In addition to creep and stress relaxation experiments, another type of measurement is quite common. Here the stress or strain, instead of being a step function, is an oscillatory function with an angular frequency a. The standard unit of a is radians per second (rad/s).++ Dynamic modulus values measured using such perturbations are functions of a> rather than time. The problem of putting dynamic experiments on a quantitative level is only slightly more difficult than is the case with step-deformation experiments. 

Fig. 1.4 Definition of the polar coordinates r, 9, f) for a point shown here in pink r is the radial coordinate and 9 and j are angular coordinates. 9 and j are measured in radians (rad). Cartesian axes (x, y and z) are also shown.

The measured data from any experiment include the following A, amplitude of the volume variation P or Pgs, mean system pressure (usually between 5 and 200 torr in exploratory experiments) B. the amplitude of the pressure fluctuation which will vary with frequency of pulsation co, the angular frequency of the volume variation in radians/ minute and cp, the phase difference between pressure and volume. Records of B and 9 as a function of cu can be repeatedly obtained as many times as desired for a given cu and constitute the raw output data of the experiments. 

Small angle neutron diffraction method permits the observation of large scale structures with clusters (particles) size from 1 nm to 100 nm. From the properties of the Fourier transform, [4], it follows that the diffraction intensity from objects of this size is concentrated in a small angle region, 0.2 < (9 < 10 radians, in the so-called zero peak. In conventional diffractometers the zero peak is inseparable from the instrumental broadening of the incident neutron beam. To make the measurement possible, SANS method applies cold neutron sources and filtration of the incident neutron flux by mirror guides or Be filters. In SANS, the angular interval Q < I sin QR/QR) in (2) is practically constant... 

The attractiveness of dynamic analysis is that an accurate determination of the viscoelastic behavior can be made. A common geometry for dynamic measurements is the cone and plate rheometer. In dynamic analysis, the viscosity components can be measured up to an angular frequency of about 500 radians/s. Cox and Merz [72] found empirically that the steady shear viscosity corresponds to the complex viscosity if the shear rate in s is plotted on the same scale as the angular frequency in radians/s. This can be stated as ... 

The derived unit known as the plane angle refers to the angular separation of two lines from a common point in a two-dimensional plane. The SI derived unit for plane angles is the radian. A complete rotation about a point origin is an angular displacement of 2ti radians. The extension of this into three dimensions is known as the solid angle and is measured by the SI derived unit called the steradian. 

The field thus includes a Gaussian distribution of angular frequencies around to , with a FWHM of (8 ln2) /T, or 2.35/t radians/s. In frequency units, the FWHM is (8dn2) /2 rT, or 0.375/t Hz. The shorter the pulse, the broader the spread of frequencies. The measured spectrum of the intensity again is narrower by a factor... 

The coefficient of rolling friction, X, defined as the ratio of the vertical load (WO to the parallel rolling force, depends on the load, the sphere radius, and the viscoelastic properties of the substratum. The rather complicated relationships have been worked out by Atack and Tabor and Flom and Bueche. A measurement of X is roughly equivalent to a measurement of tan 5 at a radian frequency of the order of the angular velocity of the rolling sphere it passes through a maximum as a function of the velocity. The method has been applied to determinations of relative losses in a variety of rubbery and glassy polymers. ... 

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