Precessional velocity

In one variation of this gyroscope flowmeter, the angular precessional velocity is applied about the x axis as a small oscillatory angular velocity... 

We notice first that the end of the vortex is precessing around the inner wall of the separator at some precessional frequency / and precessional velocity vector Vp. For the case illustrated, this motion is counterclockwise (ccw) as viewed from above. However, superimposed on this motion is the vortex core spin or rotational velocity vector, vcs- This core spin velocity adds to the precessional velocity Vp at its top-most position (position A in illustration) producing the resultant velocity v. On the other hand, the core spin vector opposes the precessional velocity at the bottom-most position (position B). 

In Fig. 9.2.7 we plot the velocity at which the vortex end or core trans-verses around the inside wall of the hopper (from Eq. 9.2.1) versus the estimated maximum tangential velocity of the vortex core (from Eq. 9.2.2). This plot strongly suggests that the precessional velocity is directly related to the maximum spin velocity. Thus, as stated earlier, precession frequency can be estimated by simply dividing the core s maximum spin velocity by the circumference of the inner wall to which it is attached. If this is true, then the end of the vortex acts much like a rubber wheel that rotates around the inner walls at a velocity equal to the maximum spin velocity of the vortex core. 

This velocity is called the Larmor precessional frequency. 

There are many experiments which determine only specific frequency components of the power spectra. For example, a measurement of the diffusion coefficient yields the zero frequency component of the power spectrum of the velocity autocorrelation function. Likewise, all other static coefficients are related to autocorrelation functions through the zero frequency component of the corresponding power spectra. On the other hand, measurements or relaxation times of molecular internal degrees of freedom provide information about finite frequency components of power spectra. For example, vibrational and nuclear spin relaxation times yield finite frequency components of power spectra which in the former case is the vibrational resonance frequency,28,29 and in the latter case is the Larmour precessional frequency.8 Experiments which probe a range of frequencies contribute much more to our understanding of the dynamics and structure of the liquid state than those which probe single frequency components. 

A two-dimensional rotation of a dipole we may represent as a superposition of (1) a deflection relative the symmetry axis and (2) a precession about this axis. The square of the polar velocity i) presents in Eqs. (42a) and (42b) the axial component of a kinetic energy and (1/ sini )2—the precessional component. Below we consider two limiting cases. 

But no fine structure - yet - until in 1915 Bohr considered the effect of relativistic variation of mass with velocity in elliptical orbits under the inverse square law of binding, and pointed out that the consequential precessional motion of the ellipses would introduce new periodicities into the motion of the electron, whose consequences would be satellite lines in the spectra. The details of the dynamics were worked out independently by SOMMERFELD [38] and WILSON [39] in 1915/16 based on a generalisation of Bohr s quantization, namely, the quantization of action the values of the phase integrals Jf = fpj.d, - of classical mechanics should be constrained to assume only integral multiples of h. 

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