# dynamics

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## dynamics

**dynamics,**branch of mechanics that deals with the motion of objects; it may be further divided into kinematics, the study of motion without regard to the forces producing it, and kinetics, the study of the forces that produce or change motion. Motion is caused by an unbalanced force acting on a body. Such a force will produce either a change in the body's speed or a change in the direction of its motion (see acceleration). The motion may be either translational (straight-line) or rotational. With the principles of dynamics one can solve problems involving work and energy and explain the pressure and expansion of gases, the motion of planets, and the behavior of flowing liquids and gases. Solids are rigid, having a definite shape, but fluids (liquids and gases) are not, and special branches of dynamics have been developed that treat the particular effects of forces and motions in fluids. These include fluid mechanics, the study of liquids in motion, and aerodynamics, the study of gases in motion. The applications of liquids both at rest and in motion are studied under hydraulics, a branch of engineering closely related to dynamics. The principles of dynamics may also be combined with the study of other phenomena, as in electrodynamics, the study of charges in motion.

## Dynamics

That branch of mechanics which deals with the motion of a system of material particles under the influence of forces, especially those which originate outside of the system under consideration. From Newton's third law of motion, namely, to every action there is an equal and opposite reaction, the internal forces cancel in pairs and do not contribute to the motion of the system as a whole, although they determine the relative motion, if any, of the several parts.

Particle dynamics refers to the motion of a single particle under the influence of external forces, particularly electromagnetic and gravitational forces. The dynamics of a rigid body is the study of the motion, under given forces, of a system of particles, the distances between which are postulated to be constant throughout the motion.

In classical dynamics the basic relation that enables the motion to be determined once the force is known is Newton's second law of motion, which states that the resultant force on a particle is equal to the product of the mass of the particle times its acceleration. For a many-particle system it becomes impracticable to write and solve this equation for each individual particle and, in general, the motion may be computed only on a statistical basis (that is, by the methods of statistical mechanics) unless, as for a few particles or a rigid body, the number of degrees of freedom is sufficiently small. *See* Degree of freedom (mechanics), Kinematics, Kinetics (classical mechanics), Newton's laws of motion, Rigid-body dynamics, Statistical mechanics

## dynamics

The study of the motion of material bodies under the influence of forces. Newton's laws of motion form the basis of classical dynamics. When speeds approach the speed of light the special theory of relativity must be taken into account.*The Great Soviet Encyclopedia*(1979). It might be outdated or ideologically biased.

## Dynamics

a branch of mechanics devoted to study of the movement of material bodies under the influence of forces applied to them. Three laws of I. Newton form the basis of dynamics. All of the equations and theorems necessary for solving problems of dynamics are obtained as corollaries of these laws.

According to the first law (the law of inertia), a mass point not acted upon by any forces is at rest or in uniform rectilinear motion. Only the action of a force may alter this condition. The second law, the basic law of dynamics, establishes that when a force F acts on a mass point (or a body that is moving translationally) or mass *m*, the point or body receives an acceleration w, defined by the equality

(1) *m*w = F

The third law is the law of equality of action and reaction. When several forces are applied to a body, F in equation (1) designates their resultant force. This result follows from the law of independence of action of forces, according to which when several forces act upon a body, each of them imparts to the body the same acceleration which it would impart if it acted alone.

In dynamics two types of problems are analyzed; their solutions for a mass point or translationally moving body are found by means of equation (1). Problems of the first type consist in the determination of the forces acting on a body if the movement of the body is known. Newton’s discovery of the law of universal gravitation is a classic example of the solution of such a problem. Knowing the laws of motion of the planets, which had been established by J. Kepler on the basis of observations, Newton demonstrated that this motion occurs under the influence of a force inversely proportional to the square of the distance between the planet and the sun. Such problems arise in technology in determining the force with which bodies in motion act on constraints (that is, other bodies that limit their movement)—for example, in determining the force of pressure of wheels on a rail, as well as in finding the internal stresses in various parts of machines and mechanisms, when the laws of motion of these machines (mechanisms) are known.

Problems of the second type, which are basic in dynamics, consist in the determination of the law of motion of a body if the forces acting on it are known. In solving these problems it is also necessary to know the so-called initial conditions— that is, the position and velocity of the body at the moment of initiation of its motion under the influence of given forces. Examples of such problems are the following: Knowing the magnitude and direction of the velocity of a shell at the moment of its emergence from a gun barrel (muzzle velocity), and knowing the forces of gravity and air resistance acting on the shell during its motion, to find the law of motion of the shell, in particular its trajectory, horizontal range, and time to target; knowing the speed of a motor vehicle at the moment of the beginning of braking, and knowing the force of braking, to find the time and path of motion until stopping; and, knowing the force of elasticity of the springs and the weight of the body of a railroad car, to determine the law of the car’s oscillations, and specifically the frequency of these oscillations.

Problems of dynamics for a rigid body (upon nontranslational motion) and for various mechanical systems are solved by equations that also are obtained as corollaries of the second law of dynamics as applied to individual particles of a system or body. The equality of forces of interaction between these particles (the third law of dynamics) is also taken into consideration. Specifically, the formula for a rigid body rotating around a fixed axis *z* is obtained in such a way:

*I _{z}Є = M_{z}*

where *I _{z}* is the moment of inertia of a body relative to the axis of rotation, Є is the angular acceleration of the body, and

*M*is the torque, which is equal to the sum of the moments of the acting forces relative to the axis of rotation. This equation makes it possible to find the torque moment (a problem of the first type) if the law of rotation (the dependence of Є on time) is known or to find the law of rotation (a problem of the second type) if the torque moment and initial conditions (the initial position of the body and the initial angular velocity) are known.

^{z}General theorems of dynamics, which may also be obtained as corollaries of the second and third laws of dynamics, are frequently used in studying the movement of mechanical systems. These include theorems about the movement of the center of mass (or the center of inertia) and about changes in the momentum, moment of momentum, and kinetic energy of a system. Another means of solving problems of dynamics is associated with the use of other principles of mechanics than the second law of dynamics (d’Alembert’s principle, the d’Alembert-Lagrange principle, and variational principles of mechanics) and of equations of motion obtained through the use of such principles, in particular the Lagrange equations of mechanics.

Equation (1) and all its corollaries are valid only in studying motion relative to the so-called inertial frame of reference, which for movements within the solar system is a stellar system with a high degree of precision (a frame of reference with the origin at the center of the sun and axes directed toward remote stars); for solving the majority of engineering problems, a frame of reference associated with the earth is used. In studying motion relative to noninertial frames of reference, which are systems associated with bodies that are moving or rotating with acceleration, an equation of motion may also be formulated in the form of equation (1) if the force of following and the Coriolis force of inertia are simply added to the force F. Such problems arise in the study of the influence of the earth’s rotation on the movement of bodies relative to the earth’s surface, as well as in the study of the motion of various instruments and devices mounted on moving objects (ships, airplanes, and rockets).

In addition to general methods of studying the motion of bodies under the influence of forces, specialized problems are also analyzed in dynamics: gyroscope theory, the theory of mechanical oscillations, the theory of dynamic stability, the theory of impact, and the mechanics of a body of variable mass. The laws of dynamics are also used in studying the motion of a continuous medium—that is, elastically and plastically deformed bodies, liquids, and gases. Finally, as a result of the use of methods of dynamics in the study of concrete objects, a series of specialized disciplines has originated: celestial mechanics, exterior ballistics, and the dynamics of steam engines, motor vehicles, airplanes, and rockets.

In dynamics based on the laws of Newton, which is called classical dynamics, the methods used describe the movements of the most diverse objects (from molecules to celestial bodies), occurring with speeds from fractions of a millimeter per second to dozens of kilometers per second (the speeds of rockets and celestial bodies). These methods are of very great significance for contemporary natural science and technology. However, they cease to be valid for movement of objects of very small dimensions (elementary particles) and for motions with speeds close to the speed of light; these motions are subject to other laws.

S. M. TARG

## Dynamics

in music, the elements connected with the use of varying degrees of volume, or loudness.

The principal gradations of volume are piano (abbreviated p), meaning soft or weak, and forte (f), denoting loud or strong. Gradations softer than piano are pianissimo (pp), or very soft; piano-pianissimo (ppp), or extremely soft; and so forth (as far as ppppp). Gradations louder than forte are fortissimo (ff), or very loud; forte-fortissimo (fff), or extremely loud; and so forth (as far as fffff). Also employed are the indicators mezzo piano (mp), or moderately soft, and mezzo forte (mf), or moderately loud. All these indicators pertain to an extended musical segment in which a single and constant level of volume is, on the whole, maintained. Within such segments, individual sounds may be distinguished according to volume, and this is indicated by such terms as forzato and sforzato.

Also widely employed in music is a gradual increase or decrease in the volume: increase is indicated by the term crescendo (cresc., <) and decrease by the terms decrescendo or diminuendo (deer, or dim., >). An increase in the volume can lead to a new, higher degree of loudness, which is to be maintained for a certain time, or it can give way to a decrease in the volume, creating a dynamic “wave.” Dynamic indicators may be refined by the addition of such words as meno (less), quasi (as if, almost), molto (very), poco (a little), and poco a poco (little by little, gradually).

Dynamic gradations and their indicators have only a relative meaning in music; absolute magnitude of loudness depends on many factors, including the type of instrument and, in ensemble work, on the number of parts and the number of musicians in each part, as well as on the acoustical properties of the place of performance. Thus, in absolute terms, piano on a trumpet is much louder than the forte of a vocalist, and the volume of the piano of an entire choir is significantly higher than that of any one of its members. Absolute volume of sound is measured in acoustics and is expressed in phons.

## dynamics

[dī¦nam·iks]## dynamics

## dynamics

**1.**the branch of mechanics concerned with the forces that change or produce the motions of bodies

**2.**the branch of mechanics that includes statics and kinetics

**3.**the branch of any science concerned with forces

**4.**

*Music*

**a.**the various degrees of loudness called for in performance

**b.**directions and symbols used to indicate degrees of loudness