Kinematics of Three Dimensions

Newton's "Third Law..

I8. As we have stated above, force is to be regarded merely as one aspect of a mutual action between two bodies. If a body A experiences a force P in consequence of interaction with another body B, then B will also be subjected to force. This principle is propounded in Newton's third law of motion: "Action and reaction are equal and opposite;" i.e., we postulate that the force experienced by B will be a force equal in magnitude but opposite in direction to the force P, and having the same line of action.

On the basis of these three laws, with a further postulate regard ing the mutual attraction of two bodies at a distance (i.e., the "law of gravitation"), Newton erected the whole structure of his dynam ical scheme. That structure, as we have already inferred, is purely mathematical, concerned to work out, according to the laws of kinematics, the consequences of accelerations which are postu lated in the so-called "laws of motion." Force and mass are secon dary notions, not really essential to the scheme, but they simplify its presentation, and for this reason it is convenient to retain them, and even to introduce further dynamical concepts which may be based upon the fundamental relation (22). These concepts we now proceed to develop. We shall show that the laws of motion, applied to a single mass, lead to relations which hold in respect of any system of masses, and form the basis of general equations (see DYNAMICS).

Units of Measurement.-19.

Before we can employ equation (22) to deduce exact numerical results, we must define the units in which force, mass and acceleration are to be measured. The unit of acceleration will be that which involves a unit increase of velocity in a unit of time ; and the unit of velocity will be the velocity of a point which moves through a unit of distance in a unit of time. Thus, in the first place, we have to fix the funda mental units of length and time: if, for example, we choose the foot and the second, the unit of velocity will be I ft. per sec. and the unit of acceleration will be 1 ft. per sec. per sec. Inertia being defined (§ 17) as a constant quantity associated with a given body, independent of its velocity or acceleration, the unit of mass will be another fundamental (and therefore arbitrary) unit. It must be defined as the mass of some particular piece of matter ; e.g., the standard pound or kilogram.

When we have specified the fundamental units of length, mass and time, we may deduce, according to (22), the corresponding unit of force. If, for example, we adopt the "C.G.S. system," in which the fundamental units are the centimetre, gram and second, the unit force will be that force which produces an acceleration of I cm. per sec. per sec. when it acts on a mass of I gram: this

unit is termed the dyne. If we adopt as units the foot, pound and second, the unit force will be that which produces an acceleration of I ft. per sec. per sec. when it acts on a mass of I pound: this unit is termed the poundal.

A unit of force derived in this way from (22) is termed an absolute unit, since it is the same in all places and at all times. Now consider the case of a body of unit mass falling freely under the influence of the earth's attraction. In (22) if g is the measured acceleration, we have hence the attractive force, i.e., the "weight" of the body, consists of g absolute units. Measured in centimetres per second per sec ond, g is 981, nearly; so the weight of I gram is a force of 981 dynes, and conversely, the unit of force in the C.G.S. system (i.e., I dyne) is about of a gram weight. Measured in feet per second per second, g is 32.2 nearly; so the weight of I pound is a force of 32.2 poundals, and conversely, the poundal is a force of about lb. weight, i.e., roughly equal to the weight of half an ounce.

For scientific purposes, great advantages are possessed by an absolute system of measurement, and the C.G.S. system is now almost universally employed (see UNITS, PHYSICAL), but in practical applications of mechanics (e.g., engineering) it is cus tomary to take as the unit of force the weight of one pound ; i.e., a force of g poundals. This change of units will evidently involve a change in the form of (22). A force which in poundals is meas ured by MF will be measured in pounds weight by MF/g, where g=32.2 approximately: hence we have the expression for the accelerating force measured in pounds weight, when M, the mass accelerated, is measured in pounds, and F, the acceleration, is measured in feet per second per second.

The same expression will hold for the accelerating force meas ured in grams weight, when M is the mass in grams and F the acceleration in centimetres per second per second, provided that g is given the appropriate value 981. From a scientific standpoint, the use of gravitational units of force, such as is contemplated here, is open to the objection that the value of g varies to a slight extent with position on the earth's surface, and hence the weight of one pound or gram is not, strictly speaking, a constant quantity. In what follows, we shall assume that absolute units are em ployed, so that the relation between force, mass and acceleration is expressed by (22).