Circular motion

The velocity vector will be parallel to the displacement only if the particle is moving in a straight line. In Fig. 11.8, Ar = r(t + At) - r(t) can clearly be in a different direction. Thus, v(t) need not, in general, be parallel to r(t). Acceleration is the time derivative of velocity. We find 

FIGURE 11.9 Displacement vector r(f) for particle moving in a circular path with angular velocity co. The x and y components of r are shown. 

As we have seen, the axial vector representing angular velocity is given by ) = k i). Thus, 

To find the acceleration of a particle in uniform circular motion, note that is a time-independent vector, so that 

In this case, acceleration represents a change in the direction of the velocity vector, while the magnitude v remains constant. Using the right-hand rule, the acceleration is seen to be directed toward the center of the circle. This is known as centripetal acceleration, centripetal meaning center seeking. The magnitude of the centripetal acceleration is 

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When high-efficiency (which requires small cyclone diameter) and large throughput are both desired, a number of cyclones can be operated in parallel. In a multiple tube cyclone, the housing contains a large number of tubes that have a common gas inlet and outlet in the chamber. The gas enters the tubes through axial inlet vanes which impart a circular motion (AWMA, 1992). 

Vortex formation is a condition that arises from centrifugal acceleration acting on gravitational acceleration. The circular motion of the entire contents of the tank predominates over the flow of the liquid from the impeller. Flow orientation thus is important not only in cases of noticeable vortex formation, but... 

Kreis-ausschnitt. m. sector, -bahn, /. circular path orbit, -bewegung,/. circular motion, rotation, revolution, -blende, /. (Photog.) iris diaphragm, -bogen, m. arc of a circle circular arch. 

Ill 1759 Smeaton presented to the Royal Society a paper entitled An Experimental Enquiry concerning the natural Power of Water and Wind to turn mills, and other machines, depending on a circular motion. This paper was the result of a series of experiments carried out in 1752 and 1753, Smeaton having delayed its publication until he had put his deductions into practice. At the time there was a lively debate as to the merits of undershot and overshot waterwheels, but little published data to substantiate various claims. Smeaton s solution to this problem depended solely on experiments made with working... 

Smeaton, J. (17.S9). An Experimental Enquiiy Concerning the Natural Power of Water and Wind to Turn Mills and Other Machinei-y, Depending on a Circular Motion. Philosophical Transactions of the Royal Society 51 100-174. 

Mr. Conti emptied the martini glass, and strained his cocktail into it in a circular motion, pouring on the sides of the glass, so that the liquid wouldn t splash, then dropped a pitted green olive into it. 

After selecting the site for injection, cleanse the skin. Most hospitals have a policy regarding the type of skin antiseptic used for cleansing the skin before parenteral drug administration. Cleanse the skin with a circular motion, starting at an inner point and moving outward. 

Figure 2. Synchrotron X-ray source (schematic). The electrons execute circular motions in the storage ring and emit intense X-rays along the tangent of the orbit. This radiation is enhanced by undulator magnets that are often placed inside the vacuum vessel for enhanced performance. The storage ring has a number of straight sections for undulators and wigglers (not shown).

To understand this simplification, imagine a man sitting on a merry-go-round with a large placard on which several sentences are written. As long as the merry-go-round is rotating rapidly, a stationary observer may not be able to read the words on the placard as it whirls round. However, if the observer were to jump onto the merry-go-round, then the apparent circular motion of the man with the placard would disappear to the observer since the two would now have the same angular velocity, and the words on the placard would be readable. 

The amplitude represents a circular motion of the magnetization vector along the Larmor frequency of the nucleus is inherent to it and would therefore remain unchanged throughout the FID. 

Shifting screens, operated with a circular motion in the plane of the screening surface. The actual motion may be circular, gyratory, or circularly vibrated. Used for the wet and... 

Gluck, S.E. Gyratory, Circular Motion and Special Action Screens, Chemical Engineering, Oct. 25, 1965, p. 131. 

This accounts for the fact that the line spectrum of hydrogen shows only lines having certain wavelengths. In order for the electron to move in a stable orbit, the electrostatic attraction between it and the proton must be balanced by the centrifugal force that results from its circular motion. As shown in Figure 1.7, the forces are actually in opposite directions, so we equate only the magnitudes of the... 

The composite vector is seen to spiral around the z-axis and in projection moves anti-clockwise in a circle around the z-axis. The other component which is the mirror image of the first, performs a clockwise circular motion in projection along z. The decomposition into circularly polarized components can also be formulated in complex notation, Er = E0e t6. 

In the ion cyclotron resonance (ICR) analyzer, ions are trapped by a strong magnetic field. The magnetic field will cause the ions to move in a circular motion with a frequency that depends on their m/z.. Ions to be detected are excited to make them move closer to the detection plates. Then a small current will be induced in the plate each time an ion passes by. Since the ions with different m/z have different ICR frequencies, each generated current frequency will correspond to a certain m/z value. 

Image current detection is (currently) the only nondestructive detection method in MS. The two mass analyzers that employ image current detection are the FTICR and the orbi-trap. In the FTICR ions are trapped in a magnetic field and move in a circular motion with a frequency that depends on their m/z. Correspondingly, in the orbitrap ions move in harmonic oscillations in the z-direction with a frequency that is m/z dependent but independent of the energy and spatial spread of the ions. For detection ions are made... 

Most examples of flow in nature and many in industry are turbulent. Turbulence is an instability phenomenon caused, in most cases, by the shearing of the fluid. Turbulent flow is characterized by rapid, chaotic fluctuations of all properties including the velocity and pressure. This chaotic motion is often described as being made up of eddies but it is important to appreciate that eddies do not have a purely circular motion. 

The angular momentum, L, of a circulating particle is defined as its linear momentum multiplied by the radius of its circular motion. Bohr s second postulate requires that the angular momentum of a rotating electron be an integral multiple of h ... 

In the distillation, evaporated gas flows to the condensation tube with circular motion to remove centrifiigally large particles formed when bubbles on the water surface were broken. 

Upon rearrangement we obtain the radius r of this circular motion... 

In natural convection, the temperature gradient in the fluid creates variations in density within the fluid. Gravity then causes the colder, denser fluid to sink, and the hotter less-dense fluid to rise. This movement results in the circular motion of warm fluid (i.e., this motion is convective heat transfer). In space where there is no gravity, natural convection does not occur. 

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