« PreviousContinue »
respect to some other body at rest, it is said to be Absolute Motion. But when compared with others in motion, it is called Relative Motion.
12. Velocity, or Celerity, is an affection of motion, by which a body passes over a certain space in a certain time. Thus, if a body in motion pass uniformly over 40 feet in 4 seconds of time, it is said to move with the velocity of 10 feet per second; and so on.
13. Momentum, or Quantity of Motion, is the power or force in moving bodies, by which they continually tend from their present places, or with which they strike any obstacle that opposes their motion.
14. Forces are distinguished into Motive, and Accelerative or Retarding. A Motive or Moving Force, is the power of an agent to produce motion ; and it is equal or proportional to the momentum it will generate in any body, when acting, either by percussion, or for a certain time as a permanent force.
15. Accelerative, or Retardive Force, is commonly understood to be that which affects the velocity only : or it is that by which the velocity is accelerated or retarded; and it is equal or proportional to the motive force directly, and to the mass or body moved inversely. So, if a body of 2 pounds weight, be acted on by a motive force of 40; then the accelerating force is 20. But if the same force of 40 act on another body of 4 pounds weight ; then the accelerating force in this latter case is only 10; and so is but half the former, and will produce only half the velocity.
16. Gravity or Weight, is that force by which a body endeavours to fall downwards. It is called Absolute Gravity, when the body is in empty space; and Relative Gravity, when immersed in a fluid.
17. Specific Gravity is the relation of the weights of dif. ferent bodies of equal magnitude ; and so is proportional to the density of the body.
18. Evrry body naturally endeavours to continue in its present state, whether it be at rest, or moving uniformly in a right line.
19. The change or Alteration of Motion, by any external force, is always proportional to that force, and in the direction of the right line in which it acts.
20. Action and Re-action, between any two bodies, are equal and contrary. That is, by Action and l&e-action, equal
changes of motion are produced in bodies acting on each other; and these changes are directed towards opposite or contrary parts.
21. The relative magnitudes and directions of any two forces may be represented by two right lines, which shall bear to each other the relations of the forces, and which shall be inclined to each other in an angle equal to that made by the directions of the forces.
22, the name resultant is given to a force which is equivalent to two or more forces acting at once upon a point, or upon a body; these separate forces being named constituents or composants.
23. The operation by which the resultant of two or more forces applied to the same point, or line, or body, is determined, is called the composition of forces; the inverse problem is called the decomposition, or the resolution of forces.
24. The resultant of two or more forces which act upon the same line, in the same direction, is equal to their sum : and if some forces act in one direction, and others in a direction immediately opposite, the resultant will be equal to the excess of the sum of the forces which act in one direction above the sum of those which act in the opposite direction.
Composition and Resolution of Parallel Forces.
25. Prop. If to the extremities of an inflexible right line AB, are applied two forces, P and Q, whose directions are parallel and whose actions concur:—1st, The direction of the resultant, R, of those two forces is parallel to the right lines AP, Bø, and is equal to their sum. 2dly, That resultant divides the line Ab into two parts reciprocally proportional to the two forces.
1. It is manifest that if two new forces, p and F. M. C. m. G
make no change in the S
* will be the same as that P B. Q
of the resultant of the two original forces P, Q. Suppose, now, that s is the resultant of the two sorces p, P, while T is that of the two forces q, Q. These resultants, lying in the same place, will, if prolonged, necessarily meet in some point c; to which, therefore, we may suppose the forces s and T applied. Through this point let fe be drawn parallel to AB, and suppose each of the forces s and T resolved into two forces directed respectively in FG and CR. The forces, according to FG, being equal to p and q respectively, and applied in opposite directions, destroy each other's effects: the remaining forces, therefore, lying the same way on CR, must be added together for the resultant, which thus is equal to P + q ; being the first part of the proposition. 2. In order to establish the second part of the proposition, let Mc, cM, be lines in proportion to each other as the forces p, P.; and in c, cm, respectively proportional as q, Q: and draw Nr, nv., parallel to AB. Then, by the sim. triangles & P : p :: cn : Nr :: co: oA CNr, CoA ; cnv, cob ; q : Q :: nv : no :: OB : oc. Consequently, P. q : p. Q :: co. ob : co.o.A, or, since p = q, it is P : Q :: ob : oA. Q. E. D.
Corol. 2. When a single force R is applied to a point o, of an inflexible straight line AB, we may always resolve it, or conceive it resolved, into two others, which being applied to the two points A and B, in directions parallel to R, shall produce the same effect.
26. PROP. Any number of parallel forces, P, Q, R, s, &c. acting in the same sense, and their points of application being connected in an invariable manner; to determine their resultant.
Determining first, by the preceding prop. the resultant T of two of the forces P and Q, we shall have T = P + q ;
P+Q. : Q :: AB : AE. Thus, we may substitute for the forces p and q, the single force T whose value and point of application are known. Draw Ec from that point of application to the point c, at which another force, R, is applied. Compounding the forces T and R, their resultant v will be = T-HR = P-HQ+n; and its point of application, F, such that P + 9 + R : R :: Ec : EF.
A similar method may, obviously, be pursued for any number of parallel forces.
27. If parallel forces act in opposite directions; some, for example, upwards, others downwards; find the resultants of the first and of the second class separately, the general resultant will be expressed by the difference of the two forInter.
28. The point through which the resultant of parallel forces passes, is called the centre of parallel forces. If the forces, without ceasing to be respectively parallel, and without changing either their magnitudes or their points of application, assume another general direction, the centre of those forces will still be the same, because the magnitudes and relations, on which its position depends, remain the same.
29. Prop. The resultant of two forces p and q acting in ore plane, will be represented in direction and in magni. tude, by the diagonal of the parallelogram constructed on the directions of those forces.
1. In direction. Take, on the direc- A. tions AP, AQ, of the forces, P, Q, dis- g tances AB, Ac, proportional to those H forces, respectively. Suppose that the B /\ C force a is applied at the point c, and S that at the same point two other forces p, q, equal to each other, act in opposite directions, each of those forces being, PT 2D Q also, equal to Q. The effect of the four forces P, Q, p, q, will evidently be the same as that of the primitive forces P, Q ; since the other two annihilate each other’s effects. The forces a, q, will have a resultant s, whose direction, cs, will bisect the angle acq, made by the direction of the other two : since no reason can be assigned why it should lean to one rather than toward the other. The forces P, p, acting in parallel directions, would have a resultant, T, whose direction TH (art. 25.) would be parallel to them, and pass through a point, H, such as that P : p : : Ho ; HA. Now, the point K, where the directions cs, Th, of these two resultants intersect, will evidently be a point in the direction of the resultant of the four forces P, p, q, q; and, consequently, of the original forces P, Q. But the triangle chk is isosceles: for, since HT, cr; are
parallel, the alternate angles pok, HKc, are equal, and Dck, *ck, are equal, because sc bisects the angle acq: hence hck = HKc, and ink = Ho.
But, from what has precedeed, P : q : : HC : HA; and therefore p : p or Q : : HK : HA.
From B drawing bD parallel to Ac, we shall have
P : Q : : AB : AC : : CD : AC, whence cd : Ac: : HK : HA;
a proportion which indicates that the three points, A, K, D, all fall on the diagonal of a parallelogram Abcd.
2. In magnitude. For, with regard to the forces P, Q, represented in magnitude and direction by AB and AF, let T be opposed to those two forces so as to keep the whole system in equilibrio : then it will, of necessity, be equal and opposite to their resultant, R, whose direction is AG. Now, if we suppose that the sorce q is in equilibrio with the two forces p and T (which is consistent with our first hypothesis) the resultant of these latter will fall in the prolongation of QA, and will be represented by AH = AF. Also, is HD be drawn parallel to AB, and HB be joined, it will be equal and parallel to Ag; and we shall have
P : T : : AB : Ald.
Consequently, since AB represents, or measures, the force P, AD will represent or measure the force T ; and as that force is in equilibrio with the two forces P and Q, or with their resultant, R, this latter will be represented or measured by AG = AD ; that is, by the diagonal of the parallelogram ABGF. Q. E. D.
30. Corol. 1. If three forces, as A, B, c, acting simul. taneously in the same plane, keep one another in equilibrio, they will be respectively proportional to the three sides, DE, EC, CD, of a triangle which are drawn parallel to the directions of the forces AD, DB, CD.
For, producing AD, BD, and drawing cF, cE, parallel to them, then the force in cD is cquivalent to the two AD, BD, by the supposition; but the force cD is also equivalent to the two ED and CE or FD; therefore, if cD represent the force c, then ED will represent its opposite force A, and CE, or FD,