Motion orthogonal components

The equations of Table 2-5 can be used to define orthogonal components of motion in space, and these components are then combined vectorally to give the complete motion of the particle or point in question. 

Since hydrodynamic lubrication depends on the behavior of real fluids with the property of viscosity, we cannot overlook the influence of shear stresses in the lubricant fluid even though they may be small compared to the normal pressures in the fluid. Each orthogonal component of tension, o, will differ from -p by quantities depending on the motion of distortion, which, as we have seen, are functions of a, fa, and c only. Let us postulate that these functions are linear and write out the following relations [4] ... 

The equations of Table 2-5 are all scalar equations representing discrete components of motions along orthogonal axes. The axis along which the component to or a acts is defined in the same fashion as for a couple. That is, the direction of to is outwardly perpendicular to the plane of counterclockwise rotation (Figure 2-7). 

Because of its oscillatory component wave motion requires a related, but more complicated description than linear motion. The methods of particle mechanics use vectors to describe displacements, velocities and other quantities of motion in terms of orthogonal unit vectors, e.g. 

If, in a vector space of an infinite number of dimensions the components Ai and Bi become continuously distributed and everywhere dense, i is no longer a denumerable index but a continuous variable (x) and the scalar product turns into an overlap integral f A(x)B(x)dx. If it is zero the functions A and B are said to be orthogonal. This type of function is more suitable for describing wave motion. 

The shielding factor is a property of the molecule, but as we see in later examples, the ability of the magnetic field to influence the motion of electrons depends on the orientation of the molecule relative to B0. Hence, O is a second-rank tensor, not a simple scalar quantity. It is always possible to define three mutually orthogonal axes within a molecule such that o may be expressed in terms of three principal components, on, molecular symmetry requires that two of the components of o be equal (and in other instances it is possible to assume approximate equality), so that the components may be expressed relative to the symmetry axis as chemical shielding anisotropy defined (ct — cr,). 

Tetrahedral distortions corresponding to the lattice shear distortion of Fig. 8-5. (There is also a rotation of the tetrahedron.) Note that there are radial components to the displacement though the elastic deformation is pure shear. In addition, the hybrids can accommodate to this distortion without disrupting the orthogonality by modifying the cunleiU of p,> and s> orbitals. Notice finally that such a motion of the atoms a, b, c, and d tends to cause displacement of the central atom relative to them in the x-dircction. 

Figure 3.24 Deconvolution of the same symmetry components of the motion analysis displayed in Figure 3.34 into orthogonal combinations of the local vectors which distinguish pure vibrations from pure translations.

The second of these components, q, describes the displacement orthogonal to the motion in the orbit, and its equation of motion gives most of the information which we will need. Let us also suppose that there is a single first integral of the motion, say... 

Initially, students are encouraged to think in terms of orthogonal velocity components. They are given tasks, such as the one illustrated in figure 19.4, where one student has to control the motion of the horizontal arrow and another student has to control the motion of die vertical arrow. By performing this task, students should induce two principles. The first is that the law induced in the first microworld, concerning the additivity of impulses, applies to this new vertical dimension of motion. The second is that impulses plied in the vertical dimension have no effect on the dot s horizontal velocity, and vice versa -- hi other words, the dot s horizontal and vertical velocity components are independent of one another. 

Later in the progression of tasks for microworld 2, students go on to thinking in terms of the speed and direction representation of motion, and to understanding how orthogonal velocity components combine to determine the speed and direction of the dot s motion. For example, in... 

According to this definition, a tensor of the first rank is simply a vector. As examples of second rank tensors within classical mechanics one might think of the inertia tensor 6 = (0y) describing the rotational motion of a rigid body, or the unity tensor 5 defined by Eq. (2.11). A tensor of the second rank can always be expressed as a matrix. Note, however, that not each matrix is a tensor. Any tensor is uniquely defined within one given inertial system IS, and its components may be transformed to another coordinate system IS. This transition to another coordinate system is described by orthogonal transformation matrices R, which are therefore not tensors at all but mediate the change of coordinates. The matrices R are not defined with respect to one specific IS, but relate two inertial systems IS and IS. ... 

The generation term is stated in terms of forces acting on the system. This is a consequence of Newton s second law, which states that the rate of change of momentum is equal to a force, that is, f = d mv)/dt. It is important to realize that the momentum balance is a vector equation with components in each of three mutually orthogonal coordinate directions. We will consider in this derivation only the x component of the equation of motion in rectangular coordinates. 

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