Rotating local frame

If the origin of the local frame moves with nonconstant velocity in the lab frame, or if the local frame rotates with respect to the lab frame, then the local frame has finite acceleration and is noninertial. In this case the motion of particle i in the local frame does not obey Newton s second law as it does in an inertial frame. We can, however, define an effective net force whose relation to the particle s acceleration in the local frame has the same form as Newton s second law ... 

A simple example may make these statements clear. Consider a small unattached object suspended in the weightless environment of an orbiting space station. Assume the object is neither moving nor spinning relative to the station. Let the object be the system, and fix the local frame in the space station. The local frame rotates with respect to local stars as the station orbits around the earth the local frame is therefore noninertial. The only true force exerted on the object is a gravitational force directed toward the earth. This force explains the object s acceleration relative to local stars. The fact that the object has no acceleration in the local frame can be explained by the presence of a fictitious centrifugal force having... 

We will make several simplifying assumptions. The rotating local frame has the same origin and the same z axis as the lab frame, as shown in Fig. G.5. The z axis is vertical and is the axis of rotation for the local frame. The local frame rotates with constant angular velocity co = dd/ dt, where i is the angle between the x axis of the lab frame and the... 

We have seen by Eqs. [38] and [39] that the normal waves are linearly polarized in the local frame of reference for 0. As the local frame rotates d = 2ttzIP), so do the polarization vectors of the normal waves. This is the waveguide regime discussed by de Gennes and is also the regime in which twisted nematic field-effect devices operate (values typical of such devices are P 50 Aim, 1.65 and X 0.5 A m, leading to a value of X 0.006). 

Suppose that a gradient, G=G u is applied in the /-direction. In a local frame of reference rotating about the... 

As before, we note that the resonance frequency of a nucleus at position r is directly proportional to the combined applied static and gradient fields at that location. In a gradient G=G u, orthogonal to the slice selection gradient, the nuclei precess (in the usual frame rotating at coq) at a frequency ciD=y The observed signal therefore contains a component at this frequency witli an amplitude proportional to the local spin density. The total signal is of the fomi... 

The local frame for rigid molecules, once chosen, will always be clearly defined. The necessary transformation of the separation vector from the laboratory to the local frame is usually accomplished by multiplication by the rotation matrix of the central molecule. The construction of this rotation matrix is usually a straightforward task. In fact, it will already be available in any program that describes molecular orientations in terms of quaternions (coordinate-transformed eulerian angles) [3,24]. 

But the Bi variation with distance from the coil centre can also be exploited for spatial localization by rotating-frame methods (cf. Section 6.3) [Botl, Sty 1]. If surface coils are used for excitation and detection the space dependence of signal excitation and signal detection enters into the acquired signal in a multiplicative fashion. 

Suppose that a gradient, G=G u, is applied in the z-direction. In a local frame of reference rotating about the combined polarizing and gradient fields at the frequency co=coQ+Y((j. r), an excitation pulse fij(t) applied at the central resonance... 

Fig. 21imL Schematic drawing to illustrate how the reorientation of the local frame, represented by a, b, and c axes, is decomposed into two components, one representing axial reorientation by % t) and the other a uthal rotation by gc(t)...

If the system as a whole does not move or rotate in the laboratory, a lab frame is an appropriate local frame. Then U is the same as the system energy Esys measured in the lab frame. 

If the system consists of the contents of a rigid container that moves or rotates in the lab, as in the illustration above, it may be convenient to choose a local frame that has its origin and axes fixed with respect to the container. 

If the Cartesian axes of the local frame do not rotate relative to the lab frame, then the heat is the same in both frames q = giab-" ... 

The expressions for dwiab and w are the same as those for dw and w in Eqs. 3.1.1 and 3.1.2, with dx interpreted as the displacement in the lab frame. There is an especially simple relation between w and wiab when the local frame is a center-of-mass frame—one whose origin moves with the center of mass and whose axes have no rotational motion ... 

A more general relation can be written for any local frame that has no rotational motion and whose origin has negligible acceleration in the lab frame ... 

Simple relations such as these between q and qiab, and between w and wiab, do not exist if the local frame has rotational motion relative to a lab frame. 

Hereafter in this book, thermodynamic work w will be called simply wort For all practical purposes you can assume the local frames for most of the processes to be described are stationary lab frames. The discussion above shows that the values of heat and work measured in these frames are usually the same, or practically the same, as if they were measured in a local frame moving with the system s center of mass. A notable exception is the local frame needed to treat the thermodynamic properties of a liquid solution in a centrifuge cell. In this case the local frame is fixed in the spinning rotor of the centrifuge and has rotational motion. This special case will be discussed in Sec. 9.8.2. 

Figure 9.12 (a) Sample cell of a centrifuge rotor (schematic), with Cartesian axes x, V, z of a stationary lab frame and axes x, y, z of a local frame fixed in the spinning rotor. (The rotor is not shown.) The axis of rotation is along the z axis. The angular velocity of the rotor is o) = di / dt. The sample cell (heavy lines) is stationary in the local frame. 

The system is the solution. The rotor s angle of rotation with respect to a lab frame is not relevant to the state of the system, so we use a local reference frame fixed in the rotor as shown in Fig. 9.12(a). The values of heat, work, and energy changes measured in this rotating frame are different from those in a lab frame (Sec. G.9 in Appendix G). Nevertheless, the laws of thermodynamics and the relations derived from them are obeyed in the local frame when we measure the heat, work, and state functions in this frame (page 498). 

Note that an equilibrium state can only exist relative to the rotating local frame an observer fixed in this frame would see no change in the state of the isolated solution over time. While the rotor rotates, however, there is no equihbrium state relative to the lab frame, because the system s position in the frame constantly changes. 

We assume the centrifuge rotor rotates about the vertical z axis at a constant angular velocity (O. As shown in Fig. 9.12(a), the elevation of a point within the local frame is given by z and the radial distance from the axis of rotation is given by r. 

In the rotating local frame, a body of mass m has exerted on it a centrifugal force centr directed horizontally in the outward +r radial direction (Sec. G.9). The... 

There is also a Coriolis force that vanishes as the body s velocity in the rotating local frame approaches zero. The centrifugal and Coriohs forces are apparent or fictitious forces, in the sense that they are caused by the acceleration of the rotating frame rather than by interactions between particles. When we treat these forces as if they are real forces, we can use Newton s second law of motion to relate the net force on a body and the body s acceleration in the rotating frame (see Sec. G.6). 

As explained in Sec. 2.6.2, a lab frame may not be an appropriate reference frame in which to measure changes in the system s energy. This is the case when the system as a whole moves or rotates in the lab frame, so that Esys depends in part on external coordinates that are not state functions. In this case it may he possible to define a local frame moving with the system in which the energy of the system is a state function, the internal energy U. 

We continue to treat the earth-fixed lab frame as an inertial frame, although this is not strictly true (Sec. G.IO). If the origin of the local frame moves at constant velocity in the lab frame, with Cartesian axes that do not rotate with respect to those of the lah frame, then the... 

APPENDIX G FORCES, ENERGY, AND WORK G.9 Rotating Local frame... 

A rotating local frame is the most convenient to use in treating the thermodynamics of a system with rotational motion in a lab frame. A good example of such a system is a solution in a sample cell of a spinning ultracentrifuge (Sec. 9.8.2). 

Here n is the radial distance of the particle from the axis of rotation, and et is a unit vector pointing from the particle in the direction away from the axis of rotation (see Fig. G.5). The direction of ei in the local frame changes as the particle moves in this frame. 

In a rotating local frame, the work during a process is not the same as that measured in a lab frame. The heats q and are not equal to one another as they are when the local frame is nonrotating, nor can general expressions using macroscopic quantities be written... 

Let R. be the rotation matrix which transforms an arbitrary vector V from the laboratory frame to the vector v in the local frame defined as the principal axes of the second... 

Next we need to define the form of the time evolution operator (Liouvillian) for the density matrix described by the SLE. The molecule being partitioned in two fragments, as described above, we have (i) two local frames respectively fixed on the pahnitate chain (CF) and on the tempo probe (PB) these are chosen with their respective z axes directed along the rotating bond, for convenience (ii) the molecular frame (MF), fixed on the pahnitate chain this is the frame which diagonalizes the... 

In contrast to nematics, a helical twist of the molecular director takes place in the chiral nematic phase. Studies of the spin-lattice relaxation in chiral nematics have shown that the relaxation mechanisms are essentially the same as in pure nematics [141, 142]. At high Larmor frequencies the relaxation is diminished by molecular self-diffusion and by local molecular rotations, whereas director fluctuations determine the relaxation rate at low Larmor frequencies. This can be easily understood because the spin-lattice relaxation rate in the MHz region is dominated by orientational fluctuations with wavelength much smaller than the period of the helix. The influence upon the rotating frame spin-lattice relaxation time Tip of the rotation of the molecules due to diffusion along the helix, an effect specific for twisted structures, has not been observed in COC [143]. 

In a laboratory frame, the dielectric tensor can be expressed with the help of the rotation tensors. The local frame in the SmC phase can be reached from the laboratory system by rotating the director about the two-fold symmetry axis c X fc by an angle 6, then by rotating the c-director around the layer normal k by the azimuth angle [Pg.224]

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