I8-17.J 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



129 



ON THE MEASURES OF FORCE AND LAWS OF MOTION. 



If a body be distiirbeii from a stite of rest, or if tlie rate of a 

 moving body be acce^lerated or retarded, tlie cause of the motion in 

 tlie first instance, and of tlie acceleration or retardation in the second 

 instance, is called Force. When a material particle, acted on by only 

 two forces in opposite directions, is kept at rest, the two forces are 

 said to be in equilibrinm and statically eqnal. The material particle, 

 last considered, is said to be kept at rest by the pressures of the two 

 forces. The notion of pressure seems to arise from the peculiar sen- 

 s itiun experienced in the muscles of the human frame, when the 

 limbs are supporting a heavy weight or thrusting against an opposed 

 obstacle. 



By pressure, as manifested in the sense of touch, we are acquainted 

 v\ith the forms of all objects within our reach and grasp. If we had 

 no other means of communicating with the outer world than by con- 

 tact, our knowledge of it would be extremely limited ; we could have 

 no conception of colour, and but very little of distance; the extent of 

 a hundred miles would be as difBcult to imagine as a million with the 

 aid of vision. These deficiencies in the sense of touch are compen- 

 sated by the sense of sight — that is, by the consciousness of the 

 presence and relations in space to each other of external objects, — as 

 evidenced by vibrations in ether, which are communicated through 

 the optic nerve to the brain. Hearing is excited by vibrations trans- 

 mitted through the air or any other elastic matter, and which, in many 

 instances, are so iatense, as to be sensibly felt. Windows are fre- 

 queatly broken by the report of artillery, — and thunder, when close, 

 shakes the walls of the stoutest buildings. We observe, then, that 

 all our experience of the phenomena of the universe is derived from 

 force. 



Force acquaints us with the existence of matter; — nay, more, we 

 might,with perfect propriety, consider matter as composed of geo- 

 metrical points, the loci of radiating forces. In by far the greater 

 number, however, of investigitions which require the aid of me- 

 chanical science, it is sufficient to consider the properties of matter, 

 without any reference to its ultimate constitution. Thu*, having 

 previously by experiment determined how far elasticicity, rigidity, 

 flexibility, &c, influence the circumstances of statical or dynamical 

 phenomena, we are enabled to solve problems involving these con- 

 siderations, without any furtlier enquiry into the nature of internal or 

 molecular forces. 



Before, however, we can apply mathematical reasoning to deter- 

 mine or predict what happens when any number of forces act upon a 

 Imdy, it is necessary that some of the effects of force should be sus- 

 ceptible of numerical comparison. In order to render our meaning 

 clearer, let us, by way of analogy, consider the method usually adopted 

 to measure heat. Heat is evidenced by many effects ; among others, 

 by the sensation of warmth, — by the impetus which it gives, when 

 developed within certain limits, to the growth of plants, — and by its 

 interference with the laws of chemical affinity. Yet none of these 

 effects are sufficiently definite for the purposes of measurement. We 

 cannot be certain that the same source of heat will always, under the 

 same circumstances, excite the same sensations; — nay, we cannot be 

 certain at any two times that the sensations of hot or cold we experi- 

 ence are the same. Still less can we avail ourselves of the effects of 

 heat on vegetable life. While, as to the changes occasioned by a 

 high temperature in the chemical constitution of bodies, they are 

 involved with so many accompanying phenomena — so complex and 

 discontinuous — that they could scarcely be compelled to furnish a 

 scale of measurement. 



There is another efFect, however, of heat, which we have not yet 

 noticed, and that is — its power of expandmg the volume of bodies. 

 This effect is rendered the more valuable by the fact, that whatever 

 'pheuomena of heat are due, at any one time, to a particular tempera- 

 ture — that is, to a particular amount of expansion of the liquid of 

 the thermometer — are likewise due to the same temperature at any 

 ^o. UO.— Vol. X.— May, lS-17. 



other time. Here we have a class of effects which are always the 

 same fur the s.iine causes, and are susceptible of arithmetical com- 

 parison — the two qualities necessary for a measure. Consequently, 

 temperature is universally adopted as the measure of heat; and in 

 thcrmotic?, all the symbols and numerals have reference, not to heat, 

 but temperature. 



To return now to the effects of ordinary forces: among these, 

 weight — or the statical effect of the force of gravity — suggests itself 

 as an appropriate measure, not only of the gravity of diff-rent bodies, 

 but of the pressures occasioned by any kind of forces whatsoever. 

 By comparing the weight of bodies with the force of a spring- 

 balance, it is ffmnd that the weight of the same b-idy, at the same 

 pl.ice on the earth's surface, is always the same — and independent of 

 the position of the body in space. 



Agai'.-/, if we take a prismatic body, homogeneous throughout — say 

 a cylinder of lead— and divide it into two equal parts, we shall find 

 the weights of the two halves equal. Also, il we divide the cylinder 

 into any number of equal parts, we shall find the weights of all these 

 equal parts equal each to each, and the sums of their weights equal 

 to the weight of the undivided cylinder. Let the weight of the 

 cylinder be represented by the number ro; then the weight of an nth 



part would be represented by -, and the weight of p (equal parts) by 



— : but, as we have shown the weight of a body is not altered by 



dividing it into parts — consequently, the weight of a portion of lead, 

 of which the volume is equal to the volume of ihe p parts, would be 



represented by - — ; and its volume would be - X volume of the un- 

 )i n 



divided cylinder. Hence we infer, that the weights of homogeneous 

 substances vary as their volumes. If now we take the weight of a 

 specified volume of a given homogeneous substance as the unit of 

 measurement, — a force which would mike equilibrium with a weight 

 r times the specified weight is denoted by the number r: anl all for- 

 mulae in statics concerning the relations of forces in equilibrio, repre- 

 sent each force by the number of times the unit of force must be 

 multiplied in order to make equilibrium with it. 



When we have to consider the motion of bodies, it is more conve- 

 nient to employ another measure of force, the nature of which we 

 now proceed to explain. We must first, however, define velocity. 

 The velocity of a moving body, at any time /, is the space which the 

 body would pass through in an unit of time, supposing the rate of 

 the body uniform and the same as at the time t. As for example, — 

 if 1 foot be taken as the unit of space, and 1 second for the unit of 

 time, a body moving uniformly at the rate of 3 feet a second is said 

 to have a velocity expressed by the number 3. 



Now, it is found by experim-'nt— First, " that if a body be at rest, 

 it will continue at rest until acted on by some force; and if it be in 

 motion, and acted on by no extraneous forces, it will continue in mo- 

 tion with an uniform velocity, and in a straight Um."* Secondly, if 

 when a body is in motion, it be acted upon by an invariable force, in 

 the direction of its motion, the quantity by which the velocity of the 

 body will be increased or diminished (according as the force is acce- 

 lerating or retarding,) will always be the same in the same time ; and 

 is quite independent of the initial velocity which the body possessed 

 before it was subject to the influence of the force. 



This latter fact at once furnishes us with a convenient dynamical 

 measure of force, known by the name of the measure of accelerating 

 force. Professor Whewell well observes that the measure of the ac- 

 celerativity of force would be a much better term for it. This mea- 

 sure of accelerating force, which, for the sake of brevity, is frequently 

 simply designated '■ acceleratiug force," is the velocity generated in 

 amoving body, during an unit of time, by an invariable impressed 

 force. If the force vary with the time, the measure adopted for any 

 time /, is the velocity which would be generated in an unit of time bv 

 the force if invariable, aud the same as at the given time /: thus 



* The paragraph between iaverted commas eouociales the first low of motioi]. 



18 



