Instantaneous axis of rotation

Fix. = fixed axis of rotation Ins. = instantaneous axis of rotation Q-I = quasi-inslantaneous axis of rotation. 

In Eq. (6.2), r" is the moment-arm vector, which is equal to the moment of the musculotendinous force per unit of musculotendinous force. The direction of the moment-arm vector is along the instantaneous axis of rotation of the body B relative to body A whose direction is described by the unit vector... 

Each of these described coupled motions occurs in a single plane and on an instantaneous axis of rotation. If two of these coupled motions occur simultaneously, i.e., lateral flexion accompanying rotatiorr, then multiple planes are involved, and the combined motions occur on a helical axis of rotation... 

Therefore, the vorticity vector at any point in a liquid is equal to twice the angular velocity of the liquid at that point, and the direction of a vortex line is everywhere the instantaneous axis of rotation of a liquid particle. The vorticity is solen-oidal (div co = 0) so that a vortex line cannot begin and end in the liquid but must either form a closed curve or begin and end on the liquid surface. From equation [5.33] it follows that the differential equations of a vortex line are ... 

Here 6 is the instantaneous angle between a given C-D bond vector and the axis of rotational symmetry of the molecules, i.e., the bilayer normal. The brackets denote an average over the time scale of the experiment 10 s) so that Sen is the time-averaged orientation of the particular C—D bond with respect to the bilayer normal. 

The rotation of a macroscopic body can be described classically in terms of angular momentum about an instantaneous rotation axis. The angular momentum P is equal to the angular velocity co multiplied by a quantity I, which is the moment of inertia about the axis of rotation. For a molecule, this depends on the molecular structure, because it is the sum of the products of atomic masses, and the squares of the displacements, r,-, of the atoms from the appropriate axis. 

It is weliknown that all static polarizations of a beam of radiation, as well as all static rotations of the axis of that beam, can be represented on a Poincare sphere [25] (Fig. la). A vector can be centered in the middle of the sphere and pointed to the underside of the surface of the sphere at a location on the surface that represents the instantaneous polarization and rotation angle of a beam. Causing that vector to trace a trajectory over time on the surface of the sphere represents a polarization modulated (and rotation modulated) beam (Fig. lb). If, then, the beam is sampled by a device at a rate that is less than the rate of modulation, the sampled output from the device will be a condensation of two components of the wave, which are continuously changing with respect to each other, into one snapshot of the wave, at one location on the surface of the sphere and one instantaneous polarization and axis rotation. Thus, from the viewpoint of a device sampling at a rate less than the modulation rate, a two-to-one mapping (over time) has occurred, which is the signature of an SU(2) field. 

We can say that such a static device is a U( ) unipolar, set rotational axis, sampling device and the fast polarization (and rotation) modulated beam is a multipolar, multirotation axis, SU(2) beam. The reader may ask how many situations are there in which a sampling device, at set unvarying polarization, samples at a slower rate than the modulation rate of a radiated beam The answer is that there is an infinite number, because from the point of the view of the writer, nature is set up to be that way [26], For example, the period of modulation can be faster than the electronic or vibrational or dipole relaxation times of any atom or molecule. In other words, pulses or wavepackets (which, in temporal length, constitute the sampling of a continuous wave, continuously polarization and rotation modulated, but sampled only over a temporal length between arrival and departure time at the instantaneous polarization of the sampler of set polarization and rotation—in this case an electronic or vibrational state or dipole) have an internal modulation at a rate greater than that of the relaxation or absorption time of the electronic or vibrational state. 

Inside a rectangular well a dipole rotates freely until it suffers instantaneous collision with a wall of the well and then is reflected, while in the field models a continuously acting static force tends to decrease the deflection of a dipole from the symmetry axis of the potential. Therefore, if a dipole has a sufficiently low energy, it would start backward motion at such a point inside the well, where its kinetic energy vanishes. Irrespective of the nature of forces governing the motion of a dipole in a liquid, we may formally regard the parabolic, cosine, or cosine squared potential wells as the simplest potential profiles useful for our studies. The linear dielectric response was found for this model, for example, in VIG (p. 359) and GT (p. 249). 

Figure 2.10 Cylindrically symmetric hydrodynamical model of accretion flow with rotation during the early collapse phase, showing the inflow of matter in the meridional plane and the build-up of a flat rotating disk structure after about 1.05 free-fall times. Arrows indicate matter flow direction and velocity, gray lines indicate cuts of isodensity surfaces with meridional plane. Dark crosses outline locations of supersonic to subsonic transition of inflow velocity this corresponds to the position of the accretion shock. Matter falling along the polar axis and within the equatorial plane arrive within 1600 yr almost simultaneously, which results in an almost instantaneous formation of an extended initial accretion disk [new model calculation following the methods in Tscharnuter (1987), figure kindly contributed by W. M. Tscharnuter],...

We define the order of the singular values as a > a2 > 31. The planar and collinear configurations give a3 0 and a2 a3 = 0, respectively. Furthermore, we let the sign of a3 specify the permutational isomers of the cluster [14]. That is, if (det Ws) = psl (ps2 x ps3) > 0, which is the case for isomer (A) in Fig. 12, fl3 >0. Otherwise, a3 < 0. Eigenvectors ea(a = 1,2,3) coincide with the principal axes of instantaneous moment of inertia tensor of the four-body system. We thereby refer to the principal-axis frame as a body frame. On the other hand, the triplet of axes (u1,u2,u3) or an SO(3) matrix U constitutes an internal frame. Rotation of the internal frame in a three-dimensional space, which is the democratic rotation in the four-body system, is parameterized by three... 

One can also analyze the rotational relaxation of the adsorbed molecules.140 Figure 27a shows a time sequence of a single molecule with an overlay of the unit vector u(t) defined as the direction of the longer principal axis of the gyration tensor. An instantaneous polymer configuration may be described by an ellipse, and therefore, the simplest conformational change is the rotational motion of an ellipse. The time correlation function of u(t) decays exponentially where zr denotes the rotational relaxation time, °c exp(-f/rr). 

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