Two-mass system

Suppose we choose a frame of reference where the total momentum is zero, and suppose further that the separation r between the masses is fixed (for example, by a rigid rod). The masses could still be rotating at some frequency co about their center of mass. If the spring is initially pointed along the x-direction, then the axis of rotation can be in the y-direction, the z-direction, or any combination of the two (the middle row of Figure 3.12). Thus there are two degrees of freedom associated with rotation in a two-mass system. 

We have seen that the two-particle system of an electron and a nucleus rotating about a center of mass (COM) can be transformed to the one-particle system of a reduced mass rotating about a fixed point. However, this transformation can be made for any two-mass system, and so it applies also to the case of the nuclei of a rotating diatomic molecule. As we now show, the mathematical outcome for the rotating diatomic molecule is strikingly similar to that for the hydrogenlike ion. 

In cases where two masses (say, two atoms) are connected and rotating in a plane, all of the above equations would apply except that the mass would be replaced by the reduced mass of the two-mass system. This is consistent with earlier treatments of two masses moving relative to each other. A system of two (or more) particles rotating in two dimensions is called a 2-D rigid rotor. 

The second type of mechanical drive is the eccentric mass type, as shown in Figure 6.14. This drive configuration is applied to conveyors and large feeders. The most common arrangement is to use two contra-rotating masses of equal size. This two-mass system is able to produce an oscillating linear motion perpendicular to the axes of the motors. This... 

If we think about two masses connected by a spring, each vibrating with respect to a stationary center of mass Xc of the system, we should expect the situation to be vei similar in form to one mass oscillating from a fixed point. Indeed it is, with only the substitution of the reduced mass p for the mass m... 

Figure 4-2 Two Masses Vibrating Harmonically with Respect to their Center of Mass. The center of mass may be stationary or moving with respect to an external coordinate system.

Can the two-mass N—N system be regarded as a one-mass system having a reduced mass p If so, what is p ... 

Bubbles and Fluidized Beds. Bubbles, or gas voids, exist in most fluidized beds and their role can be important because of the impact on the rate of exchange of mass or energy between the gas and soflds in the bed. Bubbles are formed in fluidized beds from the inherent instabiUty of two-phase systems. They are formed for Group A powders when the gas velocity is sufficient to start breaking iaterparticle forces at For Group B powders, where iaterparticle forces are usually negligible, and bubbles form immediately upon fluidization. Bubbles, which are inherently... 

The main advantages of the ms/ms systems are related to the sensitivity and selectivity they provide. Two mass analyzers in tandem significantly enhance selectivity. Thus samples in very complex matrices can be characterized quickly with Htde or no sample clean-up. Direct introduction of samples such as coca leaves or urine into an ms or even a gc/lc/ms system requires a clean-up step that is not needed in tandem mass spectrometry (28,29). Adding the sensitivity of the electron multiplier to this type of selectivity makes ms/ms a powerhil analytical tool, indeed. It should be noted that introduction of very complex materials increases the frequency of ion source cleaning compared to single-stage instmments where sample clean-up is done first. 

Rubber-Modified Copolymers. Acrylonitrile—butadiene—styrene polymers have become important commercial products since the mid-1950s. The development and properties of ABS polymers have been discussed in detail (76) (see Acrylonitrile polymers). ABS polymers, like HIPS, are two-phase systems in which the elastomer component is dispersed in the rigid SAN copolymer matrix. The electron photomicrographs in Figure 6 show the difference in morphology of mass vs emulsion ABS polymers. The differences in stmcture of the dispersed phases are primarily a result of differences in production processes, types of mbber used, and variation in mbber concentrations. 

This equation may be apphed to a closed, nonreactive, two-phase system. Each phase taken separately is an open system, capable of exchanging mass with the other, and Eq. (4-16) may be written for each phase ... 

Two vacuum systems are used to provide both the high vacuum needed for the mass spectrometer and the differential pumping required for the interface region. Rotary pumps are used for the interface region. The high vacuum is obtained using diffusion pumps, cryogenic pumps, or turbo pumps. 

Two-component systems consist of (1) polyol or polyamine, and (2) isocyanate. The hardening starts with the mixing of the two components. Due to the low viscosities of the two components, they can be used without addition of solvents. The mass ratio between the two components determines the properties of the bond line. Linear polyols and a lower surplus of isocyanates give flexible bond lines, whereas branched polyols and higher amounts of isocyanates lead to hard and brittle bond lines. The pot life of the two-component systems is determined by the reactivity of the two components, the temperature and the addition of catalysts. The pot life can vary between 0.5 and 24 h. The cure at room temperature is completed within 3 to 20 h. 

This describes the motion of a single particle having the reduced mass of the two-particle system, whose position is that of particle two with respect to particle one, and which is acted upon by the interparticle force. 

It is not essential, however, that the unit of heat should be defined in terms of the rise of temperature produced when heat is absorbed by a standard body, say unit mass of water. Any effect of heat absorption which is capable of measurement and numerical expression might be used, and the method of measurement would in all cases be consistent with the axiom that if two identical systems are acted upon by heat in the same way so as to produce two other identical systems, the quantities of heat supplied to the systems are equal. Lavoisier and Laplace (1780-84) took as unit that quantity of heat which must be absorbed by unit mass of ice in order to convert it completely into water. This unit is of course different from the one we adopted, but if a quantity of heat A has been found to raise from lo ° to 16 ° twice as much water as another quantity of heat B, then A will also melt twice as much ice as B. 

Many models in the physical sciences take the form of mathematical relationships, equations connecting some property with other parameters of the system. Some of these relationships are quite simple, e.g., Newton s second law of motion, which says that force = mass x acceleration F = ma. Newton s gravitational law for the attractive force F between two masses m and m2 also takes a rather simple form... 

The combination of the two mass balance equations, together with an explicit form of equilibrium relationship gives a system that is very easily solvable by direct numerical integration, as demonstrated in the simulation example BSTILL. 

The center of mass of the two-particle system is located by the vector R with cartesian components, X, Y, Z... 

Equation (6.11) is the Schrodinger equation for the translational motion of a free particle of mass M, while equation (6.12) is the Schrodinger equation for a hypothetical particle of mass fi. moving in a potential field F(r). Since the energy Er of the translational motion is a positive constant (Er > 0), the solutions of equation (6.11) are not relevant to the structure of the two-particle system and we do not consider this equation any further. 

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