Rotation in a plane

Orthogonal transformations preserve the lengths of vectors. If the same orthogonal transformation is applied to two vectors, the angle between them is preserved as well. Because of these restrictions, we can think of orthogonal transfomiations as rotations in a plane (although the formal definition is a little more complicated). 

The type of bearing used in a particular application is determined by the nature of the relative movement and other application constraints. Movement can be grouped into the following categories rotation about a point, rotation about a line, translation along a line, rotation in a plane, and translation in a plane. These movements can be either continuous or oscillating. 

Figure 1.42 A90° pulse brings the equilibrium magnetization to the )i -plane. Itsorien-tation in the x y-plane depends on the direction of the pulse. Applying the pulse along one axis causes the magnetization to rotate in a plane defined by the other two axes.

We begin with a consideration of a classical particle i with mass mt rotating in a plane at a constant distance r, from a fixed center as shown in Figure 5.2. The time r for the particle to make a complete revolution on its circular path is equal to the distance traveled divided by its linear velocity Vi... 

Fig. 13. Angular dependence of the g tensor for the narrow new lines in a sample of germanium lightly doped with phosphorus. The magnetic field is rotated in a plane tilted — 3° from a (110) plane. Inset shows the continuation of the lines for low values of g near (110). T = 2 K, /= 26.06 GHz, dislocation density 104 cm-2. The dashed line shows a portion of one of the four broad lines. (Courtesy Pakulis and Jefferies, reprinted with permission from the American Physical Society, Pakulis, E.J., Jefferies, C.D. (1981). Phys. Rev. Lett. 47,1859.)...

In all cases this variant (Fig. 9 a) implies an independent core actuator rotating in a plane perpendicular to the main flow (direction of extrusion). 

Each of the potentials shown in Figure 12.5 supports at least one bound or quasi-bound state which can be labeled by quantum numbers (j, Cl, J). These zeroth-order states correspond to almost free rotation of HF within the van der Waals complex with quantum numbers j = 0,1,2,... and Cl = 0,1,2,..., min(j, J). In analogy with the nomenclature for electronic states, they are termed E and n for Cl = 0 and 1, respectively. For j = 1 and Cl = 0 the diatom rotates in the plane defined by the three atoms. In contrast, for j = 1 and Cl = 1 it rotates in a plane perpendicular to the intramolecular vector R. As J increases, the centrifugal potential h2[J(J + 1) + j(j + 1) — 2Cl2]/2mR2 increases as well and eventually Veff(R j,Cl,J) becomes purely repulsive and the sequence of bound or quasi-bound states breaks off. 

The carbonyl component can be externally supplied as in the co-ozonolysis reactions (see Section 6.06.8.2) and other dipolarophiles can be used to trap the intermediate CO. Two types of rotations of the carbonyl component can take place relative to the CO <1997JOC2757> one type is in the plane of the heavy atoms which leads to the same stereochemistry as in the original alkene the other type is a rotation in a plane perpendicular to it leading to inversion. The preference of trans-alkenes to furnish in the gas phase the /raar-ozonides indicates a preference for the in-plane rotation and geminate pair recombination within the dipolar complex. At low temperatures this complex appears to be stabilized. 

Higher pre-exponential factors result if the molecule rotates or moves in the transition state. For example, suppose that CO has free rotation in a plane perpendicular to the surface. As rotational partition functions are usually large (see... 

Figure 2.2 A two-dimensional vector s2 remains constant in length on rotation in a plane (z = 0). On rotation about an axis which is not perpendicular to the X — Y plane, s2 moves out of this plane. Its length, s3, remains invariant in 3 dimensions, but its projection along Z, into the X — Y plane, appears to be contracted.

Figure 2 The initial dissociative sticking probability for D2 on Cu(l 1 1) extracted from the state selected measurements of desorbing molecules for various vibrational (A) and rotational (B) states of the molecule [19]. Vibrational energy couples effectively to the reaction coordinate, lowering the translational energy requirement for dissociation. Rotational energy initially hinders and then promoted dissociation. Similar effects of rotational energy are predicted in the trajectory calculations shown in (C) for molecules constrained to rotate in a plane perpendicular to the surface [29].

Diagram to illustrate the allowed proper rotations in a plane. 

It has a specially designed sample rotation holder and a stage controller, which are required for the process of eliminating artifacts from the apparent CD spectra. The stage, which is driven by a pulse motor, can rotate in a plane perpendicular to the transmitted light. 

Fig. 3.4. (a) Angular dependence of the magnetoresistance of (TMTSF)2C104 in 6T and at 1.5 K. The field is rotated in a plane perpendicular to the highly conducting a direction, (b) Second derivative of the upper trace with the minima labeled as described in the text. FVom [260]... 

Brownian motion on circle and sphere. The equilibrium density on a circle or a sphere deviates from uniformity by an amount proportional to cos 6, where 6 is the angle between the dipole and the field. For the circular motion, that is, a dipole confined to rotate in a plane, the cosine retains its form just as in the rectilinear motion discussed above, and relaxes in amplitude to give... 

The wheel sits on a table top. It rotates in a plane that is tilted slightly from the horizontal (unlike an ordinary waterwheel, which rotates in a vertical plane). Water is pumped up into an overhanging manifold and then sprayed out through dozens of small nozzles. The nozzles direct the water into separate chambers around the rim of the wheel. The chambers are transparent, and the water has food coloring in it, so the distribution of water around the rim is easy to see. The water leaks out... 

Given that phase space for highly excited colllnear OCS Is divided. It Is not surprising that energy relaxation would be complicated In planar OCS, where the molecule Is allowed to bend and rotate In a plane but no tumbling rotations are allowed. This expectation Is borne out In Figure 3 which shows an energy relaxation plot for OCS... 

Sometimes J is found in the literature instead of M The Ej are the energies of a rotor confined to rotate in a plane, the axis of rotation being... 

Another idealized problem is the rigid rotator in a plane, for which the wave equation is... 

It is now necessary to resolve the sign of y. This may be done by performing a rotation in a plane other than those three already employed. The results for this rotation may be predicted from the expression... 

The anharmonic potential energy is usually easier to represent in internal coordinates than in normal mode coordinates. However, what restricts the use of internal coordinates is the complicated expression for the vibrational/rotational kinetic energy in these coordinates (Pickett, 1972). It is difficult to write a general expression for the vibrational/rotational kinetic energy in internal coordinates and, instead, one usually considers Hamiltonians for specific molecules. For a bent triatomic molecule confined to rotate in a plane, the internal coordinate Hamiltonian is (Blais and Bunker, 1962) ... 

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