130 



THE CIVIL ENGINEER AND ARCHITECT'S JOURNAL. 



[Mat, 



gravity accelerates the velocity of a body falling in vacuo by 32^ feet 

 a second ; taking feet and seconds as units of space and time, the 

 accelerating force of gravity is represented by 32J. 



Our next object must be to endeavour to discover some law con- 

 necting tile statical and dynamical measures of motion. We are con- 

 scious, from every day experience, that the velocity we can communi- 

 cate to a large and heavy obstacle by thrusting against it with all our 

 strength, is much less than the velocity we could communicate in the 

 same time to a smaller and less ponderous obstacle. We know tliat 

 the same pressuie will not always communicate the same velocity to 

 different bodies in the same time. Let us now define all bodies to 

 have the same masses in which the same pressure would create the 

 same velocities in the same time. This definition of the word mass 

 will aave^ much unnecessary explanation in the following experi- 

 ment. 



Suppose n equal balls made of the same material, quite smootb, 

 and capable, by some mechanical contrivance, of being fastened to 

 each other at pleasure, and thus forming one or any number of solid 

 iKidies. 



Let H — 1 of the balls be| fastened together and placed on a 

 smooth horizontal table, let the remaining ball be tied to one end of a 

 tbiu inextensible string, and the other end of the string attached to 

 the n^ 1 balls. If now the e single ball be allovred to hang down 

 bryond the table and descend, dragging the other balls after it on the 

 table, and the velocity at a time / from the commencement of the mo- 

 tion be measured, and if the experiment be again tried with 2, 3, &c., 

 balls hanging down, and 7i — 2, 7! — 3 balls, &c. on the table, the ve- 

 locities at the end of the same time i will be found to be proportional 

 to the numbers 1, 2, 3, &c. ; but the pressures communicating motion 

 were the weights of the one, two, three, equal balls, &e., and the mass 

 moved is invariable — namely, the mass of all the balls ; consequently, 

 vpe learn that when the mass is constant, the velocity acquired at the 

 end of any time is proportional to the pressure causing it — the pres- 

 sure not varying with the time. Moreover, we infer that the velocity 

 gtiierated in a given lime, and therefore in the unit of time, is propor- 

 tional to the pressure when the mass is constant. 



Next suppose that the ?! balls are all united, and as one mass, com- 

 pelled to move by the gravity of k, other equal balls ; in this case, we 

 shall find that the velocity generated in an unit of time is the same, 

 whatever be the value of n ; consequently, when the velocity generated 

 in an unit of time is constant, the pressure varies with the mass ; and 

 we have already shown that when the mass is constant the velocity 

 generated in an unit of time varies as the pressure ;— therefore, when 

 both the mass and velocity vary, the pressure varies as the product of 

 the mass and velocity generated in an unit of time. It is not neces- 

 sary in these experiments that the balls should be made of the same 

 materials, provided they be of such a magnitude that anyone of them, 

 when attached in succession to each of the rest by the inextensible 

 string above alluded to, should generate in them all the same veloci- 

 ties at the same time. Since the dynamical measure of the force of 

 gravity is the same for all bodies, it follows that the weight of bodies 

 varies as their masses. It is sometimes assumed that the masses of 

 bodies varies as their weights, which of course leads to the same re- 

 sults. 



If m denote the mass of a body, g the accelerating force of gravity, 

 the unit of mass is so chosen that mg shall =;», where ro is the 

 weight of the body. The property of matter by which it apparently 

 resists a force tending to move it, in proportion to its mass, has some- 

 times been called the vis iiiertis, — an useless term, since it expresses 

 nothing more than is expressed by the word mass. If v be the velo- 

 city generated in a body in an unit of time, v is the measure of the 

 accelerating force acting upon the body: )» X o is called the measure 

 of the moving force, or more frequently the moving force, where the 

 word force is transferred from the cause to the measure of the effect. 



Consequently, when pressure, which does not vary with the time, 

 acts directly on a body, the moving force is proportional to the pres- 

 sure. In obtaining the above relation between the statical and dyna- 



mical measures of force, which is known by the name of the third law 

 of motion, we assumed that the same pressure would generate the 

 same, velocity in any material system, provided its mass were con- 

 stant, and its parts so connected that they must all have the same 

 velocity. We assumed, in fact, that the pressure of the hanging balls 

 produced the same velocity in the whole number of balls as it would 

 have done on a single ball of the same material and equal in bulk to 

 all of them. 



'Ibis, perhaps, ought previously to have been demonstrated by es- 

 periment ; although, in proving the third law of motion by means of 

 Attwood's machine, most writers take the same principle for grant- 

 ed — as we think, most uuwarrantably. Newton stated the third law of 

 motion thus — action and reaction are equal and opposite: on this 

 Professor Whewell observes, "since, in virtue of the equality of the 

 action and reaction bewteen two bodies, the momentum gained and 

 lost are always equal, the momentum gained and lost are sometimes 

 called action and reaction, and the third law of motion is then ex- 

 pressed by saying that in the communication of motion reaction is 

 equal and opposite to action." 



By momentum is signified the product of mass by velocity. If we 

 are to understand by action and reaction only the momentum lost by 

 one body in transferring motion, and gained by the body to which 

 motion is transferred, we do not think that there is any connection 

 between the proposition of Newton and the third law of motion, as it 

 is stated by modern philosophers. But in fact by the equality of ac- 

 tion and reaction, is meant that force, whether measured by the pres- 

 sure exerted or momentum lost in the body communicating motion, is 

 productive of momentum in the body to which the motion is com- 

 municated, equal to the momentum lost, and proportional to tlie pres- 

 sure exerted. 



The principle of the equality of action and reaction is of the greatest 

 importance : taking the statement in its most extended meaning, It 

 enunciates not merely that in the communication of motion, the mo- 

 mentum gained and lost are equal, but that the internal forces con- 

 necting the different parts of a material system — provided the con- 

 nection and relation of those parts continue the same — are likewise 

 equal and opposite. We have now briefly described the various mea- 

 sures of force and the first and third laws of motion; the aecond law 

 of motion is generally given in the following words: when a force 

 acts upon a body in motion, the change of motion in magnitude and 

 direction is the same as if the force acted on the body at rest. As an 

 example of this, — if a body in vacuo were projected horizontally, it 

 would arrive at the surface of the earth in the same time as though 

 it had been simply allowed to fall from a state of rest. All the laws 

 of motion are suggested by ordinary experiments; which indeed only 

 prove them a[)proximately, owing to the utter impossibility of exclud- 

 ing all forces but those the effects of which we are examining: never- 

 theless, in proportion as we remove disturbing causes, so do we find 

 the results of our inquiries tend to coincide with the limitingstatement 

 of these fundamental laws. A far more accurate test, however, is fur- 

 nished by astronomical observations: — the orbits of the heavenly bodies 

 calculated on the supposition of the truth of the laws of motion, are 

 found to coincide with their observed orbits so nearly, that any diffe- 

 rence may fairly be ascribed to errors of observation. 



The only planet that could not be made to keep to its tables, was 

 Uranus; the differences of its observed and predicted places were 

 alwavs, however, extremely small ; — yet, from such data as these, Mr. 

 Adams, previously, in England, and afterv^ardsM. Leverrier, in France, 

 computed the orbit of the new planet, long before its existence was 

 announced by the telescope of the observer. In conclusion, we beg to 

 state that we have not endeavoured to give any new definitions, or to 

 vary the slalemcuts and terms usually employed to express the rela- 

 tions of force, motion, and malter; our aim has been to explain, to 

 persons not accustomed to the terse style of mathematical works, the 

 fundamental principles of mechanical science. 



