Axis of revolution

Most of the situations encountered in capillarity involve figures of revolution, and for these it is possible to write down explicit expressions for and R2 by choosing plane 1 so that it passes through the axis of revolution. As shown in Fig. II-7n, R then swings in the plane of the paper, i.e., it is the curvature of the profile at the point in question. R is therefore given simply by the expression from analytical geometry for the curvature of a line... 

The general analysis, while not difficult, is complicated however, the limiting case of the very elongated, essentially cylindrical drop is not hard to treat. Consider a section of the elongated cylinder of volume V (Fig. II-18h). The centrifugal force on a volume element is u rAp, where w is the speed of revolution and Ap the difference in density. The potential energy at distance r from the axis of revolution is then w r Apfl, and the total potential energy for the... 

A cylinder is swept out by the rotation of a line parallel to the axis of revolution, so ... 

Another procedure is to rewrite the relation in Equation 2 in terms of quantities which can be accurately measured from a photographic image (1,2,20). At point P we can let the radius of curvature be Rx p of curve Vj, Figure 1. The curve V2, which is perpendicular to Vj, and passes through P, will be such that OP is normal to both curves at P. Further, since OP is on the axis of revolution, P remains on curve Vj when OP rotates about the axis BO. This gives the relation OP = X/sin( ), which is the other radius of curvature of the interface at point P = R p. We can now rewrite Equation 2 as ... 

For the tip geometry, van Eekelen9 approximates the emitter by a hyperloid of revolution with a radius of curvature at the apex given by the tip radius. The field along the axis of revolution is shown to be... 

Figure 14.6—Hollow cathode lamp. Schematic of a typical lamp. The cathode is made from a hollow cylinder whose axis of revolution corresponds to the optical axis of a lamp. On the right is a diagram of the excitation of atoms in the cathode under impact with neon ions.

Figure 7. Scheme of a confocal domain in an homeotropic sample. Lamellar details are not featured. Broken line axis of revolution. 

Low-symmetry LF operators are time-even one-electron operators that are non-totally symmetric in orbit space. They thus have quasi-spin K = 1, implying that the only allowed matrix elements are between 2P and 2D (Cf. Eq. 28). Interestingly in complexes with a trigonal or tetragonal symmetry axis a further selection rule based on the angular momentum theory of the shell is retained. Indeed in such complexes two -orbitals will remain degenerate. This indicates that the intra-t2g part of the LF hamiltonian has pseudo-cylindrical D h symmetry. As a result the 2S+1L terms are resolved into pseudo-cylindrical 2S+1 A levels (/l = 0,1,..., L ). It is convenient to orient the z axis of quantization along the principal axis of revolution. In this way each A level comprises the ML = A components of the L manifold. In a pseudo-cylindrical field only levels with equal A are allowed to interact, in accordance with the pseudo-cylindrical selection rule ... 

Figure 10. Nucleation modes in homogeneous magnets (a) coherent rotation in a sphere, (b) curling in a sphere, and (c) curling in a cylinder. The arrows show the local magnetization M = Mz ez + m, where ez is parallel to the axis of revolution of the ellipsoid (cylinder).

When the radius of the tube is appreciable, equation (2) requires correction, because the meniscus is no longer spherical. In Fig. 57, let b be the radius of curvature of the lowest point O of the meniscus (the two radii will be equal at this point, since the tube is cylindrical and the point is on the axis of revolution). The pressure immediately under the centre of the... 

Let the radius of curvature of the surface at P, in the plane of the paper, be p the radius perpendicular to this is PC, since PC is the normal, and C being on the axis of revolution, P remains on the curve when PC rotates about OC and P moves perpendicular to the paper. The principal radii... 

A prolate spheroid, settling with its axis of revolution parallel to the direction of motion, behaves as a long thin rod when the major diameter a greatly exceeds the equatorial diameter b [7], For this limiting case ... 

The discussion will be restricted to molecules which can be described as ellipsoids of revolution. We shall denote the semi-axis of revolution by a, the equatorial semi-axis by b. The orientation of a molecule may thus be completely described by the orientation of the a semiaxis the frame of reference is shown in Fig. 4. The center of the coordinate system is located in the center of gravity of the ellipsoid. The velocity gradient in the liquid, indicated in Fig. 4, tends to rotate the molecule clockwise. The orientation of the a semi-axis of the molecule is specified by the angles and 0, as defined in the figure and the accompanying legend. 

Fig. 4. Orientation of an ellipsoidal molecule in a flowing liquid of constant velocity gradient. The positive Z axis points perpendicularly upward from the plane of the paper. The projection of the axis of revolution of the ellipsoid on the XV plane is denoted by aa. The movement of the Liquid is parallel to the X axis, and is described by the equation GY, where is the velocity and G is the velocity gradient. The significance of the angle

Although originally introduced for n orbitals, Eq. (20) also applies to a system formed solely by s orbitals, or to a set of orbitals having a common axis of revolution, as in a linear molecule. In the case of s or orbitals, the charges are still located at the nuclei. In the case of hybrids orbitals, they are excentric with respect to the nuclei, but they are situated on the line joining them. 

FIGURE 5.8 Capillary menisci formed around two coaxial cylinders of radii and (I) Meniscus meeting the axis of revolution (II) meniscus decaying at infinity (III) meniscus confined between the two cylinders. h denotes the capillary raise of the liquid in the inner cylinder is the elevation of meniscus II at the contact line r = R2-... 

This method uses centripetal acceleration to control the shape of a liquid in another liquid. If a liquid drop (1) is suspended in an immiscible denser liquid (2) in a horizontal transparent tube which can be spun about its longitudinal axis, this drop will go to the center forming a drop astride the axis of revolution, as shown in Figure 6.7. The drop (i) elongates from a spherical shape to a prolate ellipsoid upon increasing the speed of... 

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