Matter and Motion
Matter and Motion was written 1876 by James Clerk Maxwell. It Is described in Thomas K. Simpson's Maxwell on the Elecromagnetic Field as "intended to introduce nonmathematical working people to the concepts of abstract mechanics."
Chapter I, Introduction
- The name of physical science... is often applied in a more or less restricted manner to those branches of science in which the phenomena considered are of the simplest and most abstract kind, excluding the consideration of the more complex phenomena, such as those observed in living beings.
- The first part of physical science relates to the relative position and motion of bodies.
- In all scientific procedure we begin by marking out a certain region or subject as the field of our investigations. To this we must confine our attention, leaving the rest of the universe out of account till we have completed the investigation in which we are engaged.
- A knowledge of the configuration of the system at a given instant implies a knowledge of the position of every point of the system with respect to every other point at that instant.
- The model or diagram is supposed to resemble the material system only in form, not necessarily in any other respect.
- A body so small that, for the purposes of our investigation, the distances between its different parts may be neglected is called a material particle. ...But we cannot treat them as material particles when we investigate their rotation. Even an atom when, we consider it as capable of rotation, must be regarded as consisting of many material particles. The diagram of a material particle is of course a mathematical point, which has no configuration.
- As indicating an operation is called a Vector, and the operation is completely defined by the direction and distance of the transference. ...All vectors ...are regarded as equal which are parallel (and drawn towards the same parts) and of the same magnitude.
- It appears then that the distance between one thing and another does not depend on any material thing between them, as Descartes seems to assert when he says that if that which is in a hollow vessel were taken out of it without anything entering to fill its place, the sides of the vessel, having nothing between them, would be in contact. This assertion is grounded on the dogma of Descartes, that the extension in length, breadth, and depth which constitute space is the sole essential property of matter. "The nature of matter," he tells us, "or of body considered generally, does not consist in a thing being hard, or heavy, or colored, but only in its being extended in length, breadth, and depth." By thus confounding the properties of matter with those of space he arrives at the logical conclusion, that if the matter within a vessel could be entirely removed the space within the vessel would no longer exist. In fact he assumes that all space must be always full of matter.
- The primary property of matter was indeed distinctly announced by Descartes in what he calls the "First Law of Nature": "That every individual thing, so far as in it lies, perseveres in the same state, whether of motion or of rest."
- Descartes... never attained to a full understanding of his own words (quantum in se est), and so fell back on his original confusion of matter with space—space being, according to him, the only form of substance, and all existing things but affections of space. This error... forms one of the ultimate foundations of the system of Spinoza.
- I would advise those who study any system of metaphysics to examine carefully that part of it which deals with physical ideas.
- We shall find it more conducive to scientific progress to recognise, with Newton, the ideas of time and space as distinct, at least in thought, from that of the material system whose relations these ideas serve to co-ordinate.
- The idea of Time in its most primitive form is probably the recognition of an order of sequence in our states of consciousness.
- Absolute, true, and mathematical Time is conceived by Newton as flowing at a constant rate, unaffected by the speed or slowness of the motions of material things. It is also called duration.
- Relative, apparent, and common time is duration as estimated by the motion of bodies, as by days months and years.
- There is nothing to distinguish one portion of time from another except by the different events which occur in them, so there is nothing to distinguish one part of space from another except its relation to the place of material bodies. We cannot describe the time of an event except by reference to some other event, or the place of a body except by reference to some other body. All our knowledge both of time and place is essentially relative.
- When a man has acquired the habit of putting words together, without troubling himself to form the thoughts which ought to correspond to them, it is easy for him to frame an antithesis between this relative knowledge and a so-called absolute knowlege, and to point out our ignorance of the absolute position of a point as an instance of the limitation of our faculties. Any one, however, who will try to imagine the state of a mind conscious of knowing the absolute position of a point will ever after be content with our relative knowledge.
- There is a maxim which is often quoted, that "The same causes will always produce the same effects." To make this maxim intelligible we must define what we mean by the same causes and the same effects, since it is manifest that no event ever happens more than once, so that the causes and effects cannot be the same in all respects. What is really meant is that if the causes differ only as regards the absolute time or the absolute place at which the event occurs, so likewise will the effects.
- The following statement... appears... and more capable of application to particular cases: "The difference between one event and another does not depend on the mere difference of the times or the places at which they occur, but only on differences in the nature configuration or motion of the bodies concerned." It follows from this, that if an event has occurred at a given time and place it is possible for an event exactly similar to occur at any other time and place.
- There is another maxim... which asserts "That like causes produce like effects." This is only true when small variations in the initial circumstances produce only small variations in the final state of the system. ...there are ...cases in which a small initial variation may produce a very great change in the final state of the system...
Chapter II, On Motion
- The motion of a material particle which has continuous existence in time and space is the type and exemplar of every form of continuity.
- If the motion of a particle is such that in equal intervals of time, however short, the displacements of the particles are equal and in the same direction, the particle is said to move with constant velocity. ...the path of the body will be a straight line, and the length of any part of the path will be proportional to the time of describing it.
- When the velocity of a particle is not constant, its value at any given instant is measured by the distance which would be described in unit of time by a body having the same velocity as that which the particle has at that instant.
- The phrase absolute velocity has as little meaning as absolute position. It is better, therefore, not to distinguish any point in the diagram of velocity as the origin, but to regard the diagram as expressing the relations of all the velocities without defining the absolute value of any one of them.
- The phrase "at rest" means in ordinary language "having no velocity with respect to that on which the body stands," as, for instance, the surface of the earth or the deck of a ship. It cannot be made to mean more than this.
- It is... unscientific to distinguish between rest and motion, as between two different states of a body in itself, since it is impossible to speak of a body being at rest or in motion except with reference, expressed or implied, to some other body.
- The word Acceleration is here used to denote any change in the velocity, whether that change be an increase, a diminution, or a change of direction.
- The process of constructing the diagram of total accelerations, by a comparison of the initial and final diagrams of velocities, is the same as that by which the diagram of displacements was constructed by a comparison of the initial and final diagrams of configuration.
- If the rate of acceleration is constant, it is measured by the total acceleration in a unit of time. If the rate of acceleration is variable, its value at a given instant is measured by the total acceleration in unit of time of a point whose acceleration is constant and equal to that of the particle at the given instant.
- The method of deducing the rate of acceleration from a knowledge of the total acceleration in any given time is precisely analogous to that by which the velocity at any instant is deduced from a knowledge of the displacement in any given time.
- In future... when we use the word acceleration without qualification, we mean... the rate of acceleration.
- In the diagram of configuration we use the capital letters A, B, C, &c., to indicate the relative position of the bodies of the system; in the diagram of velocities we use the small letters, a, b, c, to indicate the relative velocities of these bodies; and in the diagram of accelerations we use the Greek letters α, β, γ, to indicate their relative accelerations.
- Acceleration, like position and velocity, is a relative term and cannot be interpreted absolutely.
Chapter III, On Force
- If we take into account the whole phenomenon of the action between... two portions of matter, we call it Stress. This stress, according to the mode in which it acts, may be described as Attraction, Repulsion, Tension, Pressure, Shearing stress, Torsion, &c.
- External or "impressed" force considered with reference to its effect—namely, the alteration of the motions of bodies—is completely defined and described in Newton's three laws of motion. The first law tells us under what conditions there is no external force. The second shows us how to measure the force when it exists. The third compares the two aspects of the action between two bodies as it affects the one body or the other.
- It may thus be shown that the denial of Newton's [first] law is in contradiction to the only system of consistent doctrine about space and time which the human mind has been able to form.
- If a carriage in a railway train moves with constant velocity in a straight line, the external forces which act on it—such as the traction of the carriage in front of it pulling it forwards, the drag of that behind it, the friction of the rails, the resistance of the air acting backwards, the weight of the carriage acting downwards, and the pressure of the rails acting upwards—must exactly balance each other.
- We know that a thread of caoutchoucz` when stretched beyond a certain length exerts a tension which increases the more the thread is elongated. On account of this property the thread is said to be elastic. When the same thread is drawn out to the same length it will, if its properties remain constant, exert the same tension. Now let one end of the thread be fastened to a body, M, not acted on by any other force than the tension of the thread, and let the other end be held in the hand and pulled in a constant direction with a force just sufficient to elongate the thread to a given length; the force acting on the body will then be of a given intensity, F. The body will acquire velocity, and at the end of a unit of time this velocity will have a certain value, V. If the same string be fastened to another body, N, and pulled as in the former case, so that the elongation is the same as before, the force acting on the body will be the same, and if the velocity communicated to N in a unit of time is also the same, namely, V, then we say of the two bodies M and N that they consist of equal quantities of matter, or, in modern language, they are equal in mass.
- If equal quantities of the substance produce equal effects of any kind, we may employ these effects as measures of the quantity of the substance. For instance, if we are dealing with sulphuric acid of uniform strength, we may estimate the quantity of a given portion of it in several different ways We may weigh it, we may pour it into a graduated vessel, and so measure its volume, or we may ascertain how much of a standard solution of potash it will neutralize.
- In abstract dynamics, however, matter is considered under no other aspect than as that which can have its motion changed by the application of force. Hence any two bodies are of equal mass if equal forces applied to these bodies produce, in equal times, equal changes of velocity. This is the only definition of equal masses which can be admitted in dynamics, and it is applicable to all material bodies whatever they may be.
- It is an observed fact that bodies of equal mass, placed in the same position relative to the earth, are attracted equally towards the earth whatever they are made of; but this is not a doctrine of abstract dynamics founded on axiomatic principles, but a fact discovered by observation, and verified by the careful experiments of Newton on the times of oscillation of hollow wooden balls suspended by strings of the same length, and containing gold, silver, lead, glass, sand, common salt, wood, water, and wheat. ...measuring the length of a pendulum which swings seconds.
- The weight of a gramme—that is to say, the force which causes it to fall—may be asserted by letting it fall freely. At the end of one second its velocity will be about 981 centimeters per second if the experiment be in Britain.
- A pound or a gramme is greater in high latitudes than near the equator, and therefore a measurement of force in gravitation measure is of no scientific value unless it is stated in what part of the world the measurement was made.
- Let a unit of force act for unit of time upon unit of mass. The velocity of the mass will be changed, and the total acceleration will be unity in the direction of the force. The magnitude and direction of this total acceleration will be the same whether the body is originally at rest or in motion. For the expression "at rest" has no scientific meaning, and the expression "in motion," if it refers to relative motion, may mean anything, and if it refers to absolute motion can only refer to some medium fixed in space.
- To discover the existence of a medium, and to determine our velocity with respect to it by observation on the motion of bodies, is a legitimate scientific inquiry, but supposing all this done we should have discovered, not an error in the laws of motion, but a new fact in science.
- The effect of a given force on a body does not depend on the motion of that body. Neither is it affected by the simultaneous action of other forces on the body.
- We arrive at the following form of the [Newton's second] law. When any number of forces act on a body, the acceleration due to each force is the same in direction and magnitude as if the others had not been in action.
- When a force, constant in direction and magnitude, acts on a body, the total acceleration is proportional to the interval of time during which the force acts. For if the force produces a certain total acceleration in a given interval of time, it will produce an equal total acceleration in the next, because the effect of the force does not depend upon the velocity which the body has when the force acts on it.
- The total acceleration in a given time is proportional to the force. For if several equal forces act in the same direction on the same body in the same direction, each produces its effect independently of the others.
- The total effect of a force in communicating velocity to a body is therefore proportional to the force and to the time during which it acts conjointly.
- The product of the time of action of a force into its intensity if it is constant, or its mean intensity if it is variable ,is called the Impulse of the force.
- There are certain cases in which a force acts for so short a time that it is difficult to estimate either its intensity or the time during which it acts. But it is comparatively easy to measure the effect of the force in altering the motion of the body on which it acts...
- The word impulse was originally used to denote the effect of a force of short duration, such as that of a hammer striking a nail. There is no essential difference, however, between this case and any other case of the action of force.
- If a number of equal forces act in the same direction, each on a unit of mass, the different masses will all move in the same manner, and may be joined together into one body without altering the phenomenon. The velocity of the whole body is equal to that produced by one of the forces acting on a unit of mass. Hence the force required to produce a given change of velocity in a given time is proportional to the number of units of mass of which the body consists.
- The momentum of any body is... measured in terms of the momentum of unit of mass moving with unit of velocity which is taken as the unit of momentum.
- The direction of the momentum is the same as that of the velocity, and as the velocity can only be estimated with respect to some point of reference, so the particular value of the momentum depends on the point of reference which we assume. The momentum of the moon, for example, will be very different according as we take the earth or the sun for the point of reference.
- Statement of the Second Law of Motion in Terms of Impulse and Momentum—The change of momentum of a body is numerically equal to the impulse which produces it and is in the same direction.
- If any number of forces act simultaneously on a body, each force produces an acceleration proportional to its own magnitude. Hence if in the diagram of accelerations we draw from any origin a line representing in direction and magnitude the acceleration due to one of the forces, and from the end of this line another representing the acceleration due to another force, and so on, drawing lines for each of the forces taken in any order, then the line drawn from the origin to the extremity of the last of the lines will represent the acceleration due to the combined action of all the forces. Since in this diagram lines which represent the accelerations are in the same proportion as the forces to which these accelerations are due, we may consider the lines as representing these forces themselves. The diagram, thus understood, may be called a Diagram of Forces, and the line from the origin to the extremity of the series represents the Resultant Force.
- An important case is that in which the set of lines representing the forces terminate at the origin so as to form a closed figure. In this case there is no resultant force, and no acceleration. The effects of the forces are exactly balanced and the case is one of equilibrium. The discussion of cases of equilibrium forms the subject of the science of statics. It is manifest that since the system of forces is exactly balanced, and is equivalent to no force at all, the forces will also be balanced if they act in the same way on any other material system, whatever be the mass of that system. This is the reason why the consideration of mass does not enter into statical investigations.
- We have... used the word stress to denote the mutual action between two portions of matter. This word was borrowed from common language, and invested with a precise scientific meaning by the late Professor Rankine to whom we are indebted for several other valuable scientific terms.
- As soon as we have formed for ourselves the idea of a stress, such as the Tension of a rope or the Pressure between two bodies, and have recognized its double aspect as it affects the two portions of matter between which it acts, the third law of motion is seen to be equivalent to the statement that all force is of the nature of stress, that stress exists only between two portions of matter, and that its effects on these portions of matter (measured by the momentum generated in a given time) are equal and opposite. The stress is measured numerically by the force exerted on either of the two portions of matter.
- If... we neglect the weight of the string, each portion of the string is in equilibrium under the action of the tensions at its extremities, so that the tensions at any two transverse interfaces of the string must be the same. For this reason we often speak of the tension of the string as a whole, without specifying any particular section of it, and also the tension between the two bodies, without considering the nature of the string through which the tension is exerted.
- The fact that a magnet draws iron towards it was noticed by the ancients, but no attention was paid to the force with which the iron attracts the magnet. Newton, however, by placing the magnet in one vessel and the iron in another, and floating both vessels in water so as to touch each other, showed experimentally that as neither vessel was able to propel the other along with itself through the water, the attraction of the iron on the magnet must be equal and opposite to that of the magnet on the iron, both being equal to the pressure between the two vessels.
- Newton goes on to point out the consequence of denying the truth of this [third] law. For instance, if the attraction of any part of the earth, say a mountain, upon the remainder of the earth were greater or less than that of the remainder of the earth upon the mountain, there would be a residual force acting upon the system of the earth and the mountain as a whole, which would cause it to move off, with an ever increasing velocity, through infinite space.
- To prove the laws of motion by the law of gravitation would be an inversion of scientific order. We might as well prove the law of addition of numbers by the differential calculus.
- We cannot... regard Newton's statement as an appeal to experience and observation, but rather as a deduction of the third law of motion from the first.
Matter and Motion (1878)