Dynamics

Dynamics is the science which treats of matter and motion. The term Mechanics (q.v.) has been, and still is, much employed to denote this science, but its use in this way is not justifiable. Kinematics, the science of motion—i.e. of space and time, does not take account of what moves, nor of what cause produces the motion. In dynamics, the nature of the moving body and the cause of its motion are both considered. The whole science is based upon Newton's Laws of Motion (q.v.), which are as follows: (1) Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it is compelled by force to change that state; (2) change of momentum is proportional to force, and takes place in the straight line in which the force acts; (3) to every action there is always an equal and contrary reaction. Definitions of the principal terms used in the science are given below.

The mass of a body—i.e. the quantity of matter which it contains, is proportional to its volume and density conjointly, the density being the quantity of matter contained in unit volume. If $V$ be the velocity of a moving body, the mass of which is $M$, the quantity $MV$ is termed its momentum: and similarly, if $A$ be the acceleration of velocity, $MA$ is called the acceleration of momentum. The quantity $\backslash frac\{1\}\{2\}MV^2$ (which represents the work done on a body of mass $M$, originally at rest, in order to produce in it the speed $V$) is called the kinetic energy of the body.

Force is defined as any cause which alters a body's state of rest or of uniform motion in a straight line. A force is measured (in accordance with Newton's second law) by the momentum which it produces in unit time—i.e. by the quantity $MA$. It is completely specified when its place of application, its direction, and its magnitude are given. Hence (and since every force produces its own change of momentum in a body quite independently of the action of other forces) forces are compounded and resolved in the same way as accelerations and velocities (see VELOCITY). A force does work when it moves a body in the direction in which it acts, and the work done is measured by the product of the force into the distance through which it moves the body; or, as has been already remarked, by the kinetic energy produced. A pair of equal and oppositely directed forces acting so as to rotate a body about an axis is termed a couple, and the product of either of the forces into the distance between their lines of action is called the moment of the couple.

In many cases of motion the moving body, though of finite dimensions, may be treated as if it were a mere material particle. Thus there is the dynamics of a particle. This subject is further subdivided into statics and kinetics of a particle according as the particle is or is not in equilibrium under the forces. The condition for equilibrium is that the sum of the resolved parts of the forces in any direction is zero; but, because of the tri-dimensional character of space, it is sufficient to show that the sums of the resolved parts in any three non-coplanar directions are zero. When motion occurs, three cases arise according as the motion of the particle (1) is limited to a given curve, (2) is limited to a given surface, or (3) is unlimited. Simple examples are those of particles (1) falling under the action of gravity, or sliding under gravity on a smooth or rough surface; (2) projected at any inclination under gravity, or revolving around an attracting centre (in both these cases the surface to which the motion is confined is a plane). In all these cases, in accordance with the second law of motion, the resultant of the forces acting on the particle is equal to the acceleration of momentum; and whenever two or more particles mutually influence each other, the third law is required, in addition, to completely determine the motion. When two smooth spheres impinge upon one another, and remain in contact, their common speed is that of their Centre of Inertia (q.v.) before impact. If they separate again, the centre of inertia retains its previous motion, while the relative speed of separation is always a definite fraction (less than unity) of the relative speed of approach. Thus the motions are determinate.

A moving body, though it cannot always be considered to be a mere particle, may often be regarded as rigid. We have thus the statics and kinetics of a rigid solid. The three necessary conditions for equilibrium of a particle are here insufficient, as the body may rotate. The other conditions are that the sums of the moments of the forces about any three non-coplanar axes shall vanish. When the rigid body moves under the action of forces, it is sufficient to know the motion of the centre of mass (which is a case of kinetics of a particle), and the moments of inertia of the body about three non-coplanar axes through the centre of mass. The moment of inertia about any axis is the sum of the products of the mass of each particle of the body into the square of its least distance from the axis. When a body rotates about an axis, it is always possible to find a distance such that, if the whole mass of the body were condensed at that distance from the axis, its moment of inertia would be the same as that of the actual body. This distance is called the radius of gyration. The quantity $MAR^2$, where $R$ is this radius, $A$ is the angular acceleration, and $M$ is the mass, is the rate of increase of moment of momentum; and, by the second law, this is equal to the moment of the resultant couple about the axis of rotation.

The case of equilibrium of a flexible cord or chain is readily treated by means of the consideration that the difference of the horizontal parts of the tension at each end of any link is zero, while the difference of the vertical parts is equal to the weight of the link; and at least one case of motion of a flexible cord can be treated by an elementary statical method (see WAVE). The subject of dynamics of an elastic solid is of great complexity. For a slight discussion of the more elementary parts, see ELASTICITY and RIGIDITY.