Motion frictionless

The most commonly used method for applying constraints, particularly in molecula dynamics, is the SHAKE procedure of Ryckaert, Ciccotti and Berendsen [Ryckaert et a 1977]. In constraint dynamics the equations of motion are solved while simultaneous satisfying the imposed constraints. Constrained systems have been much studied in classics mechanics we shall illustrate the general principles using a simple system comprising a bo sliding down a frictionless slope in two dimensions (Figure 7.8). The box is constrained t remain on the slope and so the box s x and y coordinates must always satisfy the equatio of the slope (which we shall write as y = + c). If the slope were not present then the bo... 

The basic concepts of shock and particle velocities are well illustrated by an example first introduced by Duvall and Band (1968). Here we assume that a string of beads of diameter d, mass m, and spaced a fixed distance / apart on a smooth (frictionless) wire is impacted by a rigid, massive piston at velocity v. Each bead is assumed to undergo perfectly elastic, rigid-body motion upon impact with its neighbor. 

Euler expressed Newton s second law of motion for a frictionless (inviscid) fluid along a streamline as... 

The spherical pendulum, which consists of a mass attached by a massless rigid rod to a frictionless universal joint, exhibits complicated motion combining vertical oscillations similar to those of the simple pendulum, whose motion is constrained to a vertical plane, with rotation in a horizontal plane. Chaos in this system was first observed over 100 years ago by Webster [2] and the details of the motion discussed at length by Whittaker [3] and Pars [4]. All aspects of its possible motion are covered by the case, when the mass is projected with a horizontal speed V in a horizontal direction perpendicular to the vertical plane containing the initial position of the pendulum when it makes some acute angle with the downward vertical direction. In many respects, the motion is similar to that of the symmetric top with one point fixed, which has been studied ad nauseum by many of the early heroes of quantum mechanics [5]. 

When the particle concentration is high, the shear motion of particles leads to interparticle collisions. The transfer of momentum between particles can be described in terms of a pseudoshear stress and the viscosity of particle-particle interactions. Let us first examine the transfer of momentum in an elastic collision between two particles, as shown in Fig. 5.8(a). Particle 1 is fixed in space while particle 2 collides with particle 1 with an initial momentum in the x-direction. Assume that the contact surface is frictionless so that the rebound of the particle is in a form of specular reflection in the r-x-plane. The rate of change of the x-component of the momentum between the two particles is given by... 

When the velocity-time data have been obtained it is a simple matter to plot them and obtain distance-time values by graphical integration, and hence the trajectory of the particle. This procedure has been used to compile Table 3 and Figure 4, and the result is compared with the trajectory obtained by considering the motion of the particles as if they were moved in a frictionless fluid, i.e., CR = 0. The equations for this condition are... 

A reversible process is the thermodynamic analog of frictionless motion in mechanics, When a process has been conducted reversibly, we can, by performing the inverse process in reverse, set the system back in precisely its initial state, with zero net expenditure of work in the overall process. The system and its surroundings are once again exactly as they were at the beginning. A reversible process is an idealization which constitutes a limit that may he approached but not attained in real processes.)... 

Toward the end of the nineteenth century the science of fluid mechanics began to develop in two branches, namely theoretical hydrodynamics and hydraulics. The first branch evolved from Euler s equations of motion for a frictionless, non-... 

Defining the mass of an object as the quantity of matter it possesses is not a very good scientific definition. A better one can be found in N vton s second law of motion. If a constant force is applied to an object on a frictionless, horizontal surface, the object accelerates—its velocity increases uniformly with time. If a force twice as large is applied to the same object, its acceleration doubles as well. The object s acceleration is proportional to the force applied to it. We might write Fa a where F is the force applied to the object and a is the acceleration of the object while the force acts. The symbol a means that the two quantities, force and acceleration, are proportional that is, if the force doubles the acceleration doubles. 

In the latter case, the work done by the expanding gas and the piston will be W = / F d x. The work of the gas would be Vf = p dV if the process were reversible, that is, if the force F divided by the piston area were differentially less at all times than the pressure of the gas, and if the piston were frictionless but in a real process some of the work done by the gas is dissipated by viscous effects, and the piston will not be frictionless, so that the work as measured by J F dx will be less than / p dV. For these two integrals to be equal, none of the available energy of the system could be degraded to heat or internal energy. To achieve such a situation we have to ensure that the movement of the piston is frictionless and that the motion of the piston proceeds under only a differential imbalance of forces so that no shock or turbulence is present. Naturally, such a process would take a long time to complete. 

VELOCITY OF SOUND. The velocity of sound through a continuous material medium, also called the acoustical velocity, is the velocity of a very small compression-rarefaction wave moving adiabatically and frictionlessly through the medium. Thermodynamically, the motion of a sound wave is a constant-entropy, or isentropic, process. The magnitude of the acoustical velocity in any medium is... 

Superconductors have been applied to develop train systems that operate with magnetic-levitation (MAGLEV) in which the train effectively travels sslOmm above its tracks, i.e. virtually frictionless motion. The first commercial train came into service in Shanghai in 2003 and can reach speeds of 440 km h. ... 

In the geostrophic layer it may be assumed that the atmosphere is inviscid (frictionless). The equations of continuity and motion for such a fluid are... 

In the dumbbell model, a polymer chain in a solvent is pictured as two massless spheres of equal size connected by a frictionless spring. The spheres experience a hydrodynamic drag proportional to their size, characterized by the Flory radius. Assume that the displacement of the spring generated by the thermal energy is also characterized by the Flory radius. Write the equation of motion for the dumbbell and show that the characteristic relaxation time for the chain deformation is that given by Eq. (9.2.1). 

What physical meaning should one attach to the velocity potential For the flow of an ideal, frictionless fluid, the velocity potential has no physical meaning whatever. To illustrate this, consider the steady flow of a frictionless, constant-density fluid in a horizontal pipe see Fig. 10.3. (Such a frictionless fluid, once started in motion by some external force, would continue moving forever, because there is no force to stop it.) For such a frictionless fluid, the velocity is uniform over the cross section perpendicular to the flow. From Bernoulli s equation we can see that there is no change with distance of pressure, velocity, or elevation, and by straightforward arguments we can show that there is no change of temperature, refractive index, dielectric constant, or any other measurable property. But from Eq. 10.7 we know that, because is constant, there is a steady decrease of (f> in the x direction. Thus the velocity potential for a perfect fluid (f> is not a function of any measurable physical property of the fluid. 

Movement of a soluble chemical throughout a water body such as a lake or river is governed by thermal, gravitational, or wind-induced convection currents that set up laminar, or nearly frictionless, flows, and also by turbulent effects caused by inhomogeneities at the boundaries of the aqueous phase. In a river, for example, convective flows transport solutes in a nearly uniform, constant-velocity manner near the center of the stream due to the mass motion of the current, but the friction between the water and the bottom also sets up eddies that move parcels of water about in more randomized and less precisely describable patterns where the instantaneous velocity of the fluid fluctuates rapidly over a relatively short spatial distance. The dissolved constituents of the water parcel move with them in a process called eddy diffusion, or eddy dispersion. Horizontal eddy diffusion is often many times faster than vertical diffusion, so that chemicals spread sideways from a point of discharge much faster than perpendicular to it (Thomas, 1990). In a temperature- and density-stratified water body such as a lake or the ocean, movement of water parcels and their associated solutes will be restricted by currents confined to the stratified layers, and rates of exchange of materials between the layers will be slow. 

Beware These figures represent water horsepower only. No allowance has been made for any losses. These figures assume frictionless pipe, a 100 percent pump efficiency, and a perfect conversion of the energy (from whatever source) into the mechanical motion required by the pump mechanism. 

Fignre 9.2. Mass-spring string network with frictionless guide rods to restrict motion to one dimension. 

Perpetual motion of the third kind. A form of motion that continues indefinitely but without doing any useful work. An example is the random molecular motion in a substance. This type postulates the complete elimination of friction. A mechanism consisting of frictionless bearings maintained in a vacuum could turn indefinitely, once started, without contravening the first or second laws of thermodynamics, provided it did no external work. Experience indicates that on the macroscopic scale such a condition cannot be achieved. On the microscopic scale, however, a superconducting ring of wire will apparently sustain a perpetual current flow without the application of an external force. This could be considered a form of... 

For this example, we will initially assume that the tip of the manipulator is already in motion relative to the contact surface (slipping). The coefficient of friction is finite. In this case, we may assume that the contact forces applied in the directions of motion are already laige enough to overcome static friction. We will also examine the same ccxitact when the coefficient of friction is negligible (frictionless surface), and when the manipulatcx tip is not slipping on the surface. 

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