=

Wherefore the friction fg sin. i; and since it counterbalances the force with which the body endeavours to descend, we have

Farther, the angle whose tangent is - f + 1 + f is half the angle whose

Let, therefore, BF be the slope which loose earth would,

of itself, naturally

assume then, the line BE which determines the triangle of earth that exerts the greatest horizontal stress against the vertical wall bisects the angle CBF.

132. SCHOLIUM.

Sandy and loose earth takes a natural declivity of 60° from the vertical; stronger earth will take a declivity of 53°. Therefore, for a terrace of loose earth we have i = 30° ; for another of strong and close earth i = 2610.

Hence, for the former kind, where tan. 30° = } 3, the value of the stress is la's, and that of the momentum of the stress a3s.

For the latter kind, where tan. 2610 = } nearly, the stress

mentum = as.

=

la's, its mo

132. The horizontal stress and momentum being thus known, it is easy to proportion to them the resistance of the wall ABCD.

Let b = AB, while BC = a, and let s be the spec. grav. of the wall. For brick, s = 2000, for strong earth, s = 1428. Then the momentum of the resistance referred to the point AB, being ab'S; we shall have

Thus, if a 39.37 feet, s and S as above, we shall find b = 9·6033 feet. EXAM. 2. Supposing the earth of the same kind as in the above example, s to S, as 4 to 5, and the height of the wall and bank each 12 feet; required the thickness of the wall, being rectangular.

Ans. 2.986 feet.

Note. The preceding investigation proceeds upon the principles assumed by Coulomb and Prony. They who wish to go thoroughly into this subject, and have not opportunity to make experiments, may advantageously consult Traité Expérimental, Analytique et Pratique de la Poussée des Terres, &c. par M. Mayniel.

DYNAMICS.

134. THIS branch of Mechanics (as already defined in art. 5.) relates to the circumstances of bodies in actual motion: that is, of bodies acted upon forces, constant or variable, which do not at every instant of time neutralise each other. In this science then, we have to consider the forces as they vary in their in

tensity with the time during which they act; the law of their action according to their mode of transmission to the body acted upon; the spaces which bodies so acted upon describe (either linear or angular); the velocities of those bodies at any given period of their motion; and the paths themselves along which the bodies, under these circumstances, pass.

135. DEFINITIONS AND PRINCIPLES.

A body is said to be in uniform motion when the spaces which it passes over in equal successive portions of time are equal. And in this case the space is, obviously, as the time and the velocity conjointly.

A force is said to be constant when it generates equal increments of velocity in equal successive times.

The unit of time is generally taken one second; and the unit of space one foot. Hence in Dynamics, time is usually estimated in seconds and space in feet.

The measure of force is the velocity which it would communicate, if it continued constant, to a given body in a second of time.

When there is only a certain quantity of force acting on a body it will generate a less velocity in given time than it would if the body were less; or, in other words, the velocities produced by equal forces are inversely as the bodies on which they act.

In these researches, the several quantities are generally denoted by their initial letters: viz. the force by f, time by t, space by s, velocity by v, and the mass of the body by b.

136. PROP. I. If a body be acted on by a constant force during any given time, the relation between the force, time, and velocity, is expressed by

For the velocity generated at each second is the same and denoted by ƒ; hence in t seconds it is ft.

137. PROP. II. The space described in t seconds by a body subjected to a constant force is denoted by

S = {vt = {ft2.

For, the whole space being described by velocities which are in arithmetical progression, it is the same as would be described by their mean velocity in the same time. Hence it is

Here, also, S varies as v2 (4), because v varies as t.

These equations are sufficient for resolving all problems relative to constant forces.

138. PROP. III. If there be a second body b' subjected to the same force as in the preceding propositions we suppose b was, then

For the velocity generated by the given force is inversely as the body moved, and hence the conclusion follows.

312

ON THE COLLISION OF SPHERICAL BODIES.

139. PROP. I. If a spherical body strike or act obliquely on a plain surface, the force or energy of the stroke, or action, is as the sine of the angle of incidence.

Or, the force on the surface is to the same if it had acted perpendicularly, as the sine of incidence is to radius.

A

C

Let AB express the direction and the absolute quantity of the oblique force on the plane DE; or let a given body A, moving with a certain velocity, impinge on the plane at B; then its force will be to the action on the plane, as radius to the sine of the angle ABD, or as AB to AD or BC, drawing AD and BC pendicular, and AC parallel to DE.

D

BE

per

For, by art 34, the force AB is equivalent to the two forces AC, CB; of which the former AC does not act on the plane, because it is parallel to it. The plane is therefore only acted on by the direct force CB, which is to AB, as sin. BAC, or sin. ABD, to radius.

Corol. 1. If a body act on another, in any direction, and by any kind of force, the action of that force on the second body, is made only in a direction perpendicular to the surface on which it acts. For the force in AB acts on DE only by the force CB, and in that direction.

Corol. 2. If the plane DE be not absolutely fixed, it will move, after the stroke, in the direction perpendicular to its surface. For it is in that direction that the force is exerted.

140. PROP. If one body A, strike another body B, which is either at rest or moving towards the body A, or moving from it, but with a less velocity than that of A; then the momenta, or quantities of motion, of the two bodies, estimated in any one direction, will be the very same after the stroke that they were before it.

For, because action and reaction are always equal, and in contrary directions, art. 24, whatever momentum the one body gains one way by the stroke, the other must just lose as much in the same direction; and therefore the quantity of motion in that direction, resulting from the motions of both the bodies, remains still the same as it was before the stroke *.

* If bodies were exposed to no resistance, the result of a single impulsion or of a simple impact, would be uniform motion. The result of a mere continuance of action must be an acceleration of the motion, or a retardation, according as the action is in the same or the opposite direction to that of the first motion. It is equally obvious that the effect of the continued action of a transverse force must be a continual deflection, that is, a curvilinear motion. In inquiries relative to collision, we assume that action and reaction are equal, because otherwise the surplus action or reaction would be action or reaction against nothing, which is absurd: and we assume b v as the true measure of moving force, or momentum, because that measure alone consists with the well known universal fact now stated: namely, that the relative motions of bodies, resulting from their mutual actions, are not affected by any common motion, or the action of any equal and parallel force on both bodies: for this universal fact imports, that when two bodies are moving with equal velocities in the same direction, a force applied to one of them, so as to increase its velocity, gives it the same motion relative to the other, as if both bodies had been at rest. Here it is plain, that the space described by the body in consequence

B

C

141. Thus, for illustration, if A with a momentum of 10, strike B at rest, and communicate to it a momentum of 4, in the direction AB. Then A will have only a momentum of 6 in that direction; which, together with the momentum of B, viz. 4, make up still the same momentum between them as before, namely 10.

142. If B were in motion before the stroke, with a momentum of 5, in the same direction, and receive from A an additional momentum of 2. Then the motion of A after the stroke will be 8, and that of B, 7; which between them make 15, the same as 10 and 5, the motions before the stroke.

143. Lastly, if the bodies move in opposite directions, and meet one another, namely, A with a motion of 10, and B, of 5; and A communicate to B a motion of 6 in the direction AB of its motion. Then, before the stroke, the whole motion from both, in the direction of AB, is 10 - 5 or 5. But, after the stroke, the motion of A is 4 in the direction AB, and the motion of B is 6 the same direction AB; therefore the sum 4 + 1, or 5, is still the same motion from both, as it was before.

5 or 1 in

144. PROP. The motion of bodies included in a given space, is the same with regard to each other, whether that space be at rest, or move uniformly in a right line.

For, if any force be equally impressed both on the body and the line on which it moves, this will cause no change in the motion of the body along the right line. For the same reason, the motions of all the other bodies, in their several directions, will still remain the same. Consequently their motions among themselves will continue the same, whether the including space be at rest, or be moved uniformly forward. And therefore their mutual actions on one another, must also remain the same in both cases*.

145. PROP. If a hard and fixed plane be struck perpendicularly by either a soft or a hard unelastic body, the body will remain upon it. But if the plane be struck by a perfectly elastic body, it will rebound from it again with the same velocity with which it struck the plane.

For, since the parts which are struck, of the elastic body, suddenly yield and give way by the force of the blow, and as suddenly restore themselves again with a force equal to the force which impressed them, by the definition of elastic bodies; the intensity of the action of that restoring force on the plane, will be equal to the force or momentum with which the body struck the plane. And, as action and re-action are equal and contrary, the plane will act with the same force on the body, and so cause it to rebound or move back again with the same velocity as it had before the stroke.

But hard or soft bodies, being devoid of elasticity, by the definition, having no restoring force to throw them off again, they must necessarily remain upon the plane struck.

146. Corol. 1. The effect of the blow of the elastic body, on the plane, is double to that of the unelastic one, the velocity and mass being equal in each.

of the primitive force, and of the force now added, is the sum of the spaces which each of them would generate in a body at rest. Therefore the forces are proportional to the velocities or changes of motion which they produce, and not to the squares of those velocities, as was asserted by Leïbnitz. This measure of forces, or the fact that a force makes the same change on any velocity whatever, and the independence of the relative motions on any motion that is the same on all the bodies of a system, are counterparts of each other, and the corresponding laws of the communication of motion and force may, therefore, be assumed without hesitation. *See also the preceding note.

For the force of the blow from the unelastic body, is as its mass and velocity, which is only destroyed by the resistance of the plane. But in the elastic body, that force is not only destroyed and sustained by the plane; but another also equal to it is sustained by the plane, in consequence of the restoring force, and by virtue of which the body is thrown back again with an equal velocity. And therefore the intensity of the blow is doubled.

147. Corol. 2. Hence unelastic bodies lose, by their collision, only half the motion lost by elastic bodies; their mass and velocities being equal. For the latter communicate double the motion of the former.

148. PROP. If an elastic body A impinge on a firm plane DE at the point B, it will rebound from it in an angle equal to that in which it struck it; or the angle of incidence will be equal to the angle of reflection; namely, the angle ABD equal to the angle FBE.

C

D

BYO

E

Let AB express the force of the body A in the direction AB; which let be resolved into the two AC, CB, parallel and perpendicular to the plane.—Take BE and CF equal to AC, and draw BF. Now action and re-action being equal, the plane will resist the direct force CB by another BC equal to it, and in a contrary direction; whereas the other AC, being parallel to the plane, is not acted on or diminished by it, but still continues as before. The body is therefore reflected from the plane by two forces BC, BE, perpendicular and parallel to the plane, and therefore moves in the diagonal BF, by composition. But, because AC is equal to BE or CF, and that BC is common, the two triangles BCA, BCF are mutually similar and equal; and consequently the angles at A and F are equal, as also their equal alternate angles ABD, FBE, which are the angles of incidence and reflection.

149. PROP. To determine the motion of non-elastic bodies, when they strike each other directly, or in the same line of direction.

B

Let the non-elastic body B, moving with the velocity V in the direction Bb, and the body b with the velocity v, strike each other. Then, because the momentum of any moving body is as the mass into the velocity, BVM is the momentum of the body B, and bv = m the momentum of the body b, which let be the less powerful of the two motions. Then, by art. 140, the bodies will both move together as one mass in the direction BC after the stroke, whether before the stroke the body b moved towards C or towards B. Now, according as that motion of b was from or towards B, that is, whether the motions were in the same or contrary ways, the momentum after the stroke, in direction BC, will be the sum or difference of the momentums before the stroke; namely, the momentum in direction BC will be

BV+bv, if the bodies moved the same way, or

BV - bv, if they moved contrary ways, and

BV only, if the body b were at rest.

Then divide each momentum by the common mass of matter B + b, and the quotient will be the common velocity after the stroke in the direction BC; namely, the common velocity will be, in the first case,

150. For example, if the bodies, or weights, B and b, be as 5 to 3, and their

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