DYNAMICS.

124. THAT department of mechanics which relates to the circumstances and effects of bodies in motion (art. 5.) is of great extent, and of very comprehensive application. A selection of its most interesting topics will here be presented; but numerous other problems which, while they fall within its scope, require the aid of the fluxional analysis, will be solved in the collections in a subsequent part of this volume.

GENERAL LAWS OF MOTION, &c.

125. PROP. THE quantity of matter, in all bodies, is in the compound ratio of their magnitudes and densities.

That is, b is as md; where b denotes the body or quantity of matter, m its magnitude, and d its density.

For, by art. 10, in bodies of equal magnitude, the mass or quantity of matter is as the density. But, the densities remaining, the mass is as the magnitude; that is, a double magnitude contains a double quantity of matter, a triple magnitude a triple quantity, and so on. Therefore the mass is in the compound ratio of the magnitude and density.

Corol. 1. In similar bodies, the masses are as the densities and cubes of the diameters, or of any like linear dimensions. -For the magnitudes of bodies are as the cubes of the diameters, &c.

Corol. 2. The masses are as the magnitudes and specific gravities. For, by art. 10 and 17, the densities of bodies are as the specific gravities.

126. Scholium. Hence, if b denote any body, or the quantity of matter in it, m its magnitude, d its density, g its specific gravity, and a its diameter or other dimension; then,

(pronounced or named as) being the mark for general proportion, from this proposition and its corollaries we have these general proportions:

127. PROP. The momentum, or quantity of motion, generated by a single impulse, or any momentary force, is as the generating force.

That is, m is as f; where m denotes the momentum, and f the force.

So

For every effect is proportional to its adequate cause. that a double force will impress a double quantity of motion; a triple force, a triple motion; and so on. That is, the motion impressed, is as the motive force which produces it.

128. PROP. The momenta, or quantities of motion, in moving bodies, are in the compound ratio of the masses and velocities.

That is, m is as bv.

For, the motion of any body being made up of the mo tions of all its parts, if the velocities be equal, the momenta will be as the masses; for a double mass will strike with a double force; a triple mass with a triple force; and so on. Again, when the mass is the same, it will require a double force to move it with a double velocity, a triple force with a triple velocity, and so on; that is, the motive force is as the velocity; but the momentum impressed, is as the force which produces it, by art. 127; and therefore the momentum is as the velocity when the mass is the same. But the momentum was found to be as the mass when the velocity is the same. Consequently, when neither are the same, the momentum is in the compound ratio of both the mass and velocity.

Otherwise: M: m :: в :b, when v is constant :

and mμ

therefore, M

vv, when в is constant : :: BV: bv, when both vary.

129. PROP. In uniform motions, the spaces described are in the compound ratio of the velocities and the times of their description.

That is, s is as tv.

For, by the nature of uniform motion,

S: S :: T t, when v is constant : and s :: V v, when T is constant ; therefore sσ :: TV: tv, when both vary.

Corol. 1. In uniform motions, the time is as the space directly, and velocity reciprocally; or as the space divided by the velocity. And when the velocity is the same, the time is as the space. But when the space is the same, the time is reciprocally as the velocity.

Corol. 2. The velocity is as the space directly and the time reciprocally; or as the space divided by the time. And when the time is the same, the velocity is as the space. But when the space is the same, the velocity is reciprocally as the time.

Scholium.

130. In uniform motions generated by momentary impulse, any body or quantity of matter to be moved, force of impulse acting on the body b,

let b

f

บ

= the uniform velocity generated in b, m = the momentum generated in b,

s = the space described by the body b,

t = the time of describing the space s with the veloc. v. Then from the last three propositions and corollaries, we have these three general proportions, namely, fx m, m ∞ bv, and s ∞ tv; from which is derived the following table of the general relations of those six quantities, in uniform motions, and impulsive or percussive forces:

By means of which, may be resolved all questions relating to uniform motions, and the effects of momentary or impul. sive forces.

131. PROP. The momentum generated by a constant and uniform force, acting for any time, is in the compound ratio of the force and time of acting.

That is, m is as ft.

For, supposing the time divided into very small parts, by art. 127, the momentum in each particle of time is the same, and therefore the whole momentum will be as the whole time, or sum of all the small parts. But by the same prop, the momentum for each small time, is also as the motive force. Consequently the whole momentum generated, is in the compound ratio of the force and time of acting.

Corol. 1. The motion, or momentum, lost or destroyed in any time, is also in the compound ratio of the force and time. For whatever momentum any force generates in a given time; the same momentum will an equal force destroy in the same or equal time; acting in a contrary direction.

And the same is true of the increase or decrease of motion, by forces that conspire with, or oppose the motion of bodies. Corol. 2. The velocity generated, or destroyed, in any time, is directly as the force and time, and reciprocally as the body or mass of matter.-For, by this and art. 128, the compound ratio of the body and velocity, is as that of the force and time; and therefore the velocity is as the force and time divided by the body. And if the body and force be given, or constant, the velocity will be as the time.

132. PROP. The spaces passed over by bodies, urged by any constant and uniform forces, acting during any times, are in the compound ratio of the forces and squares of the. times directly, and the body or mass reciprocally.

Or, the spaces are as the squares of the times, when the force and body are given.

ft
b'

That is, s is as or as when fand b are given. For, let v denote the velocity acquired at the end of any time t by any given body b, when it has passed over the space s Then, because the velocity is as the time, by the last corol. therefore is the velocity at it, or at the middle point of the time; and as the increase of velocity is uniform, the same space s will be described in the same time t, by the velocity uniformly continued from beginning to end. But, in uniform motions, the space is in the compound ratio of the time and velocity; therefore s is as tv, or indeed s =

ft v. But, by the last corol. the velocity v is ast , or as b the force and time directly, and as the body reciprocally. Therefore s, or tv, is as ; that is, the space is as the force and square of the time directly, and as the body reciprocally. VOL. II.

26

Or s is as t, the square of the time only, when b and ƒ are given.

s=

Corol. 1. The space s is also as tv, or in the compound ratio of the time and velocity; b and f being given. For, tv is the space actually described. But to is the space which might be described in the same time t, with the last velocity v, if it were uniformly continued for the same or an equal time. Therefore the space s, or tv, which is actually described, is just half the space tv, which would be described with the last or greatest velocity, uniformly continued for an equal time't.

1

Corol. 2. The space s is also as v3, the square of the ve locity; because the velocity v is as the time t.

Scholium.

133. The last four propositions give theorems for resolv. ing all questions relating to motions uniformly accelerated. Thus, put b = any body or quantity of matter,

f

the force constantly acting on it,

t = the time of its acting,

v = the velocity generated in the time t,
the space described in that time,

S

m = the momentum at the end of the time. Then, from these fundamental relations, m ∞ bv, m ∞ ft, √, we obtain the following table of the ge

sx tv, and v α

ft
b

134. From the above relations those quantities are to be Jeft out which are given, or which are proportional to each other. Thus, if the body or quantity of matter be always