Laws of motion

FIGURE 9.1 Sir Isaac Newton (1642-1727). In 1687, he published Principia Mathematica, in which his three laws of motion were first stated. They are still the most widespread way to describe the motion of objects. Knighted in 1705, Newton received this honor not for his scientific achievements, as is usually assumed, but for his political activities. 

Newton s second law should be considered more closely, because it is perhaps the most familiar of the laws. Force, F, is a vector quantity, having magnitude and direction. For a single object of mass m, Newton s second law is usually expressed in the form  

Equation 9.1 can be written in a different way using calculus. Acceleration is the change of the velocity vector with respect to time, or dv/dt. But velocity v is the change in position with respect to time. If we represent the position by its onedimensional coordinate x, then we can write acceleration as the time derivative of the time derivative of position, or 

This means that Newton s second law can be written 

It is not uncommon to ignore the vector character of force and position and express equation 9.3 as 

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In molecular dynamics, successive configurations of the system are generated by integrating Newton s laws of motion. The result is a trajectory that specifies how the positions and velocities of the particles in the system vary with time. Newton s laws of motion can be stated as follows ... 

It is helpful to distinguish three different types of problem to which Newton s laws of motion may be applied. In the simplest case, no force acts on each particle between collisions. From one collision to the next, the position of the particle thus changes by v,5f, where v, is the (constant) velocity and 6t is the time between collisions. In the second situation, the particle experiences a constant force between collisions. An example of this type of motion would be that of a charged particle moving in tr uniform electric field. In the third case, the force on the particle depends on its position relative to the other particles. Here the motion is often very difficult, if not impossible, to describe analytically, due to the coupled nature of the particles motions. 

The equation of motion is based on the law of conservation of momentum (Newton s second law of motion). This equation is written as... 

F(t)=Zk QcVk exp(-itEk/fe). The relative amplitudes Ck are determined by knowledge of the state at the initial time this depends on how the system has been prepared in an earlier experiment. Just as Newton s laws of motion do not fully determine the time evolution of a elassieal system (i.e., the eoordinates and momenta must be known at some initial time), the Sehrodinger equation must be aeeompanied by initial eonditions to fully determine T(qj,t). 

An appropriate set of iadependent reference dimensions may be chosen so that the dimensions of each of the variables iavolved ia a physical phenomenon can be expressed ia terms of these reference dimensions. In order to utilize the algebraic approach to dimensional analysis, it is convenient to display the dimensions of the variables by a matrix. The matrix is referred to as the dimensional matrix of the variables and is denoted by the symbol D. Each column of D represents a variable under consideration, and each tow of D represents a reference dimension. The /th tow andyth column element of D denotes the exponent of the reference dimension corresponding to the /th tow of D ia the dimensional formula of the variable corresponding to theyth column. As an iEustration, consider Newton s law of motion, which relates force E, mass Af, and acceleration by (eq. 2) ... 

The dimensional matrix associated with Newton s law of motion is obtained as (eq. 3)... 

In its most simplistic form, Newton s equation of motion (also known as Newton s second law of motion) states that... 

An alternative method, proposed by Andersen [23], shows that the coupling to the heat bath is represented by stochastic impulsive forces that act occasionally on randomly selected particles. Between stochastic collisions, the system evolves at constant energy according to the normal Newtonian laws of motion. The stochastic collisions ensure that all accessible constant-energy shells are visited according to their Boltzmann weight and therefore yield a canonical ensemble. 

When analysing meehanieal systems, it is usual to identify all external forees by the use of a Free-body diagram , and then apply Newton s seeond law of motion in the form ... 

To determine the equation of motion, a foree balanee on the partiele is applied to the partiele. From Newton s seeond law of motion, the rate of ehange of partiele momentum, M, is equal to the net foree aeting. 

The result that Archimedes discovered was the first law of hydrostatics, better known as Archimedes Principle. Aixhimedes studied fluids at rest, hydrostatics, and it was nearly 2,000 years before Daniel Bernoulli took the next step when he combined Archimedes idea of pressure with Newton s laws of motion to develop the subject of fluid dynamics. 

When a driver commands an increase in vehicle velocity, that vehicle obeys Newton s first law of motion, which states that when a force (F) acts on a body of mass (M) and initially at rest, that body tvill experience an acceleration (a). For an automobile, typical units for acceleration, which is the rate of change of velocity, would be miles per hour per sec-... 

Hydrodynamic marked the beginning of fluid dynamics—the study of the way fluids and gases behave. Each particle in a gas obeys Isaac Newton s laws of motion, but instead of simple planetary motion, a much richer variety of behavior can be observed. In the third century B.C.E., Archimedes of Syracuse studied fluids at rest, hydrostatics, but it was nearly 2,000 years before Daniel Bernoulli took the next step. Using calculus, he combined Archimedes idea of pressure with Newton s laws of motion. Fluid dynamics is a vast area of study that can be used to describe many phenomena, from the study of simple fluids such as water, to the behavior of the plasma in the interior of stars, and even interstellar gases. 

Law of gravitation announced by English mathematician and physicist Isaac Newton five years later his Philosophiae Naturalis Principia Mathcinadca is published, setting forth the laws of motion as well as gravitation. 

Two of the laws of thermodynamics and two of the laws of motion will cover almost all needs. In the simplest terms, it can be said that ... 

The basic statement covering inertia is Newton s first law of motion. His first law states A body at rest tends to remain at rest, and a body in motion tends to remain in motion at the same speed and direction, unless acted on by some unbalanced force. This simply says what you have learned by experience - that you must push an object to start it moving and push it in the opposite direction to stop it again. 

We next consider the consequences of Newton s second law of motion i.e. the consequences of the fact that the rate of change of momentum of a fluid cell must equal the total force that is acting on it. 

In equations 12.19 and 12.20, Ry represents the momentum transferred per unit area and unit time. This momentum transfer tends to accelerate the slower moving fluid close to the surface and to retard the faster-moving fluid situated at a distance from the surface. It gives rise to a stress Ry at a distance y from the surface since, from Newton s Law of Motion, force equals rate of change of momentum. Such stresses, caused by the random motion in the eddies, are sometimes referred to as Reynolds Stresses. 

A force, F, is an influence that changes the state of motion of an object. For instance, we exert a force to open a door—to start the door swinging open—and we exert a force on a ball when we hit it with a bat. According to Newton s second law of motion, when an object experiences a force, it is accelerated. The acceleration, a, of the object, the rate of change of its velocity, is proportional to the force that it experiences ... 

As soon as we start this journey into the atom, we encounter an extraordinary feature of our world. When scientists began to understand the composition of atoms in the early twentieth century (Section B), they expected to be able to use classical mechanics, the laws of motion proposed by Newton in the seventeenth century, to describe their structure. After all, classical mechanics had been tremendously successful for describing the motion of visible objects such as balls and planets. However, it soon became clear that classical mechanics fails when applied to electrons in atoms. New laws, which came to be known as quantum mechanics, had to be developed. 

To calculate the force exerted by a single molecule, we use Newton s second law of motion force is equal to the rate of change of momentum of a particle (Section A). Momentum is the product of mass and velocity so, if a molecule of mass m is traveling with a velocity vx parallel to the side of the box that we are calling x, then its linear momentum before it strikes the wall on the right is mvx. Immediately after the collision, the momentum of the molecule is mvx because the velocity has changed from vx to —vx. 

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