The New Student's Reference Work/Dynamics
Dynam′ics is that science whose aim it is to describe most completely and most simply the motions which occur in nature. This is the definition given by Kirchhoff. But his illustrious colleague, Helmholtz, defines dynamics as the science of those phenomena in nature which may be reduced to the motion of ponderable masses. The difference between the two definitions is slight; and it vanishes completely if we assume that all the phenomena of nature can be explained, ultimately, in terms of matter in motion.
Perhaps the simplest method of getting a clear idea of the object and the method of dynamics is to consider the orderly sequence of steps by which this science, in its modern form, has been built up.
1. Before the motion of a particle can be described, we must be able to define the position of that particle: for motion is merely change of position. For this purpose dynamics makes use of the ordinary geometry of position. As is well known to everyone, when we wish to locate a point on the earth’s surface, we must tell three things about it, namely, its latitude, its longitude and its height above the sea-level. So in dynamics it requires always three specifications (called co-ordinates) to locate a particle. Knowing how to locate points, one can then easily locate any rigid body. For if any three points (not in the same straight line) in such a body are fixed, the entire body itself is fixed.
2. The second step in dynamics is to describe the change of position or displacement in bodies. But since change of position is the same kind of quantity as position itself, we still use simply the geometry of position for this purpose.
3. The third step is to describe the rate of change of position. This introduces a new quantity, time. To measure time we use a body in uniform motion, that is, we use a body which is set in motion and let alone. The most uniform motion that we know anything about is the rotation of the earth on its axis. Accordingly the average period of rotation of the earth with respect to the sun is called a day. And 1-864,00th part of a day is called a second. This is the unit of time most frequently employed in dynamics. The idea of time being clear, the rate of motion or the velocity of a particle is defined as “the ratio of the change of position to the time occupied in this change.” The numerical value of any velocity, without reference to its direction, is called its speed.
4. But since practically all velocities in nature are changing, the fourth step in dynamics is to define the rate of change of velocity. This is called acceleration, and is measured by the ratio of the change in velocity to the time occupied in the change.
Up to this point we have considered merely the motion of a body, but have not asked any questions concerning the causes of these motions. This science of motion alone is called kinematics, which is merely a Greek word for science of motion. Kinematics may be taken as a purely mathematical subject; but, as a matter of fact, it nearly always forms the first chapter of any treatise on dynamics.
5. Passing now to the thing in motion—that which we call matter—we find that the quantity of it can be measured just as it was in the case of time and space; but we cannot define it, any more than we could time or space. Time may be called duration and space may be called extension; but no one is any the wiser for all that. So the amount of matter in a body is called its mass; but this is not a definition of matter; it is merely a convenient name for the amount of substance in any body. The measure which is employed for matter is its inertia, that is, its tendency to remain at rest if it is at rest or to remain in motion if it be in motion. If one body is twice as hard to set in motion as another, it is said to have twice the inertia of the other, that is, twice the mass of the other. It was shown by Newton that all bodies, regardless of their chemical composition, are attracted by the earth with forces which are strictly proportional to their masses. And hence we use the balance when we wish to compare (i. e., to measure) two masses.
These three ideas, mass, space and time, are the fundamental concepts of dynamics. Proceeding from these, Galileo, Huygens, Newton and Lavoisier constructed what might be called the first modern system of dynamics. The experimental facts which lie at the foundation of this system are Newton’s laws of motion and Lavoisier’s discovery of the conservation of matter.
Newton’s Laws of Motion
Before these experimental facts can be stated in a clear and definite manner, it is essential to introduce two more fundamental quantities, viz., the linear momentum of any body, which is defined as the product of its mass by its velocity, and the force acting upon any body, which is defined as the product of its mass by its acceleration; or, what amounts to the same thing, the time-rate at which the linear momentum is changed. Newton’s laws may now be stated as follows:
I. If a body is in translation under no external force, its linear momentum remains constant.
II. The change in the linear momentum of a body is proportional to the force acting upon the body; and the direction of the change is the same as the direction of the force.
III. Action is always equal and opposite to reaction; by which is meant that the mutual forces of any two bodies or of any two parts of a body are always equal and oppositely directed.
Lavoisier showed that the amount of matter in any isolated system cannot be increased or diminished by any known means: and therefore, presumably, that “the amount of matter in the universe is a constant.”
6. The next great step in dynamics is the introduction of the idea of work and energy. Work, in dynamics, is employed to denote one definite quantity, viz.: the product of a force multiplied by the distance through which it is exerted, both distance and force being measured in the same direction; while energy is defined simply as the ability to do work, and is, therefore, measured in the same units as work.
By 1847 Helmholtz, together with a number of his contemporaries, had proved that though the energy which a system of bodies possesses may assume a great variety of forms, yet the amount of that energy is a constant quantity. The energy of the universe is continually undergoing transformation; but there is no evidence for thinking that the slightest bit of energy has ever been annihiliated or created by man. This summary is known as the principle of the conservation of energy. Nearly all the problems of dynamics are solved by applying to the particular case in question either Newton’s laws of motion or the law of the conservation of energy.
Dynamics of Rotation
7. A very great simplification in the treatment of dynamical problems is secured by the fact that all cases in rotation may be solved not only by the same principles, but by the same formulæ, as in the case of translation.
We have only to replace linear displacement, x, by angular displacement, θ; and linear inertia, m, by rotational inertia, I. The following table gives a summary of the principal quantities involved in translation and rotation, and shows that they are identical in form in the two cases:
|Inertia ( = Mass)||m||Rotational inertia ( = moment of inertia)||I|
|Linear Displacement ( = change of position)||x||Angular Displacement||θ|
|Linear Velocity||v = x/t||Angular Velocity||ω = θ/t|
|Linear Acceleration||α = v/t||Angular Acceleration||γ = ω/t|
|Linear Momentum||M = mv||Angular Momentum||G = Iω|
|Force||F = ma||Moment of Force||L = Iγ|
|Energy of Translation||E = ½mv²||Energy of Rotation||E = ½Iω²|
8. Up to this point we have no principle which will determine the direction in which any dynamical process occurs. The law of the conservation of energy would be equally well-satisfied whether a clock-weight ran down and delivered energy to the clock-train and the air, or whether the clock-weight ran up, deriving the necessary energy from the heat in the train and in the air. It has been found convenient to divide all kinds of energy into two groups—energy of position and energy of motion—called potential and kinetic energy respectively. And it has been discovered by Clausius, Kelvin, Helmholtz and others that, in general, “the potential energy of any system tends to become a minimum.” Armed with this theorem, which is, perhaps, the most general principle of dynamics, we are prepared for the solution of all the ordinary problems of mechanics. The method in general consists in writing, in the form of equations, the six conditions of equilibrium, namely, that all the forces shall be zero and that all the moments of force shall be zero, and then solving for the quantity desired.
Upon the eight general principles enunciated above are constructed the entire superstructures of astronomy, physics and mechanical engineering.
For the history of the subject see Mach’s Science of Mechanics, translated by McCormack; for the laws of motion see Tait’s Newton’s Laws of Motion; for a treatment of rotation see Worthington’s Dynamics of Rotation.