Rotating-frame angular velocity

Fig. 4. Magnetic fields acting on the nuclear spins in the laboratory frame (to the left) and in the frame rotating with angular velocity (o (to the ri t). In the laboratory frame, we have time-dependent fields H, = (2Hi cos

The preceding argument shows that observers in dUfer-ent coordinate systems perceive different accelerations. Mathematically, this is expressed in the following way. First of all, the relationship between the derivative of the position vector r in an inertial reference ftame denoted by subscript 1 and its derivative in a reference frame rotating with angular velocity n relative to the inertial frame is... 

Coriolis Acceleration The Coriohs acceleration arises in a rotating frame, which has no parallel in an inertial frame. When a body moves at a linear velocity u in a. rotating frame with angular speed H, it experiences a Coriolis acceleration with magnitude ... 

Frame of reference rotating with a constant angular velocity (two-dimensional case)... 

Assume that origins of two Cartesian systems of coordinates are located at the same point and the frame of reference P rotates about a point 0 of the frame P with constant angular velocity co. Let us imagine two planes, one above another, so that the upper plane P rotates and, correspondingly, unit vectors iiand ji change their direction, Fig. 2.2b. Consider an arbitrary point p, which has coordinates x, y on the plane P and xi, yi on P, and establish relationships between these pairs of coordinates. For the radius vector of the point p in both frames we have... 

We see that the acceleration in the inertial frame P can be represented in terms of the acceleration, components of the velocity and coordinates of the point p in the rotating frame, as well as the angular velocity. This equation is one more example of transformation of the kinematical parameters of a motion, and this procedure does not have any relationship to Newton s laws. Let us rewrite Equation (2.37) in the form... 

The first pseudo force, Fi, is called the Coriolis force, and its magnitude is directly proportional to the angular velocity of the rotating frame of reference and the linear velocity of the particle in this frame. By definition, this force is perpendicular to the plane where vectors Vi and o are located, Fig. 2.3a, and depends on the mutual position of these vectors. The second fictitious force, F2, is called the centrifugal force. Its magnitude is directly proportional to the square of the angular velocity and the distance from the particle to the center of rotation. It is directed outward from the center and this explains the name of the force. It is obvious that with an increase of the angular velocity the relative contribution of this force... 

As was pointed out earlier, when we have considered the physical principles of the ballistic gravimeter and the pendulum an influence of the Coriolis force was ignored. Now we will try to take into account this factor and consider the motion of a particle near the earth s surface. With this purpose in mind let us choose a non-inertial frame of reference, shown in Fig. 3.5a its origin 0 is located near the earth s surface and it rotates together with the earth with angular velocity a>. The unit vectors i, j, and k of this system are fixed relative to the earth and directed as follows i is horizontal, that is, tangential to the earth s surface and points south, j is also horizontal and points east, k is vertical and points upward. As is shown in Fig. 3.5a SN is the earth s axis, drawn from south to north, I is the unit vector along OiO, and K is a unit vector parallel to SN. 

Equations (3.75 and 3.76) describe the motion of a free particle in a rotating frame of reference, when the angular velocity is relatively small and the z-axis is directed along the plumb line. Certainly, a presence of terms with a> is related to the rotation of the earth. Besides, the gravitational field also contains a term with this frequency. It may be proper to emphasize again that in deriving these formulas we assumed... 

Since historically the dissipation is evaluated using the local velocity at the boundary and the shear stress is evaluated as the product of the viscosity and the shear rate at the boundary, it follows that if the velocity is not frame indifferent then the dissipation will not be frame indifferent. As discussed previously in this chapter, rotation of the barrel at the same angular velocity as the screw are the conditions that produce the same theoretical flow rate as the rotating screw. Because the flow rate is the same and the dissipation is different, it follows that the temperature increase for barrel and screw rotation is different. This section will demonstrate this difference from both experimental data and a theoretical analysis. 

The prime indicates that one has switched to the so-called rotating frame in order to remove any precession effect at the angular velocity coq = 2tivo, Vo being the resonance frequency. 

These simple product operators precess in the x -y plane of the rotating frame at a frequency corresponding to the chemical shift in hertz relative to the center of the spectral window (the resonance offset Av = v0 — vr). The chemical shift frequency Av can also be represented as the angular velocity 2 in units of rad/s ( 2 = 2ttAv). Using 2 allows us to skip all the 2tt terms. 

This is an important result. It tells us that the total angular momentum determined in the laboratory frame is the sum of a centre-of-mass contribution and the total angular momentum tiJ measured in the centre-of-mass frame. In a rigid rotor the velocity vector of the kW particle in the centre-of-mass frame is related to the angular velocity, u>, of the rotating particle by... 

An important issue in reaction stereodynamics must be eonsidered. Experiment and vector preparations are usually performed in the laboratory reference frame (space fixed) whereas the important preparation for the reaction is the molecular reference frame (body fixed) which rotates during the collisions at a non-constant angular velocity. This leads to numerous difficulties, which have motivated an important literature reviewed in Ref [18], As far as stereodynamics considerations are... 

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