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OF THE COMPOSITION AND RESOLUTION OF FORCES.
45. Composition of Forces, is the uniting of iwo or more forces into one, which shall have the same effect; or, it is the finding of one force that shall be equal to several others, taken together in different directions.
Resolution of Forces, is the finding of two or more forces which, acting in any different directions, shall have the same effect as any given single force.
46. Two lines which represent the momenta communicated to the same or to equal bodies, will represent the spaces uniformly described by them in equal times ; and conversely, the lines which represent the spaces uniformly described by them in equal times, will also represent the momen.ta.
The momenta of bodies may be represented by numbers, thus, if the quantities of matter in two bodies be represented by 6 and 7, and their velocities by 9 and 8, their momenta will be represented by 6x9 and 7x 8. But in many cases it will be much more convenient to represent momenta by lines, because lines will not only express the quantities of the momenta, but also the directions in which they are communicated.
Any line, drawn in the proper direction, may be taken to represent one momentum, but to represent a second, a line in the direction of the latter motion must be taken in the same proportion to the former,' that the second momentum has to the first.
Let two lines thus taken, represent the momenta communicated to the same, or equal bodies; then, since (22) malv, and 6 is here given, mav; therefore, the lines which represent the momenta, will
bs ulso represent the velocities ; and since mai (27), and in this case, b and t are constant a ms, and these lines will also represent the spaces uniformly described in equal times. Again, if the lines represent the spaces uniformly described in equal times, they represent the velocities, because saut and ( is constant ; and since mobv, and b is constant mav, but we have shown that sav, therefore also som, and these lines also represent the momenta.
47. Two uniform motions, which, when communicated separately to a body, would cause it to describe the adjacent sides of a parallelogram in a given time, will, when they are communicated at the same instant, cause it to describe the diagonal in that time ; and the motion in the diagonal will be uniform.
Let a motion be communicated to a body at A, fig. 3, which would cause it to move uniformly from A to B in t", and at the same instant another motion, which alone would cause it to move uniformly from A to C also in t"; complete the parallelogram BC, and draw dhe diagonal AD; then the body will arrive at the point D in t", having described AD with an uniform motion.
For the motion in the direction AC can neither accelerate nor retard the approach of the body to the line BD, which is parallel to AC, hence the body will arrive at BD in the same time that it would
have done, had no motion been communicated to it in the direction AC, that is, in t". In the same manner, the motion in the direction AB can neither make the body approach to, nor recede from CD; therefore, in consequence of the motion in the direction AC, it will arrive at CD at the same time that it would have done, had no motion been communicated in the direction AB, that is, in t". Hence it follows, that in consequence of the two motions, the body will be found both in BD and CD at the end of t", and it will therefore be found in D, the point of their intersection.
And since a body in motion continues to move uniformly forwara in a right line, till it is acted upon by some external force (18), the body A must have described the right line AD with an uditurin motion.
48. Cor. 1. If two sides of a triangle AB, BD, fig. 3, taken in order, represent the spaces over which two vniform motions would separately carry a body in a given time; when tiese motions are communicated at the same instant to the body at A, it will describe the third side AD uniforinly in that time.
For if the parallelogram BC be completed, the same motion, which would carry a body uniformly from B to D, would, if communicated at A, carry it in the same manner from A to C; and, in consequence of this motion, and et the motion in the direction AB, the body would uniformiy describe the diagonal AD, which is the third side of the triangle ABD.
49. Cor. 2. In the same manner, if the lines AB, BC, CD, DE, fig. 4, taken in order, represent the spaces over which any uniform motions would, separately, carry a body, in a given time, these motions, when communicated at the sane iustant, will cause the body to describe the line AE which completes the figure in that time ; and the inotion in this line will be uniform.
50. Cor. 3. If AD, fig. 5, represent the uniform velocity of a body, and any paralielogram ABDC be described about it, the velocity AD may be supposed to arise from the two siniform velocities AB, AC, or AB, BD; and if by any means one of them, as AB, be taken auay, the velucity remaining will be represented by AC or BD.
51. If the adjacent sides of a parallelogram represent the quanti ies and directions of two forces acting at the same time upon a lody, the diagonal will represent one equivalent to them both.
Let AB, AC, fig. 6, represent two forces acting upon a body at A, then they represent the momenta communicated to it in those directions (21), and, consequently, the spaces which it would unitormly describe in equal times (46). Complete the parallelogram CB, and draw the diagonal AD ; then, by the last article, AD is the space uniformly described in the same time when the two motions are communicated to the body at the same instant ; and since AB, AC, and AD, represent the spaces uniformly described by the same body in equal times, they also represent the momenta, and therefore the forces acting in those directions ; that is, the forces AB, AC, acting at the same time, produce a force which is represented in quantity and direction by AD. The force represented by AD is said to be compounded of the two, AB, AC.
52. Cor. 1. If two sides of a triangle, taken in order, represent the quantities and directions of tuo forcrs, the third side will represent a force equirulent to their both.
For a force represented by BD, acting at A, will produce the same effect that the force AC will produre, which is equal to it, and in the same direc
tion; and AB, AC, are equivalent to AD; therefore AB, BD, are also equivalent to AD.
53. Cor. 2. If any lines AB, AC, CD, DE, fig. 4, taken in order, represent the quantities and directions of forces conmunicated at the same time to a body at A, the line AE, which completes the figure, will represent a force equivalent to them all.
For the two AB, BC, are eqnivalent to AC; also AC, CD, that is, AB, BC, CD, are equivalent to AD; in the same manner AD, DE, that is, AB, BC, CD, and DE, are equivalent to AE.
54. Cor. 3. Let AB and AC, fig. 6, represent the quantities and directions of two forces, as before, theu the resulting force may be expressed analytically in terms of AB, AC, and the included angle BAC.
Because of the parallels AB, DC, and BD, AC, the BDA= DAC, and, consequently, < BAC = LBAD+ ZBDA. Again, the angle ABD=180°— (ZBAD + <BDA)= 180' BAC = supplement of BAC, by what was just shown.
But, by Trigonometry (art. 43), it appears that
and therefore by transposition, AD
2AB•BD AB*+BD? — AB BD cos. ABD; but cos. ABD = cos. (180°-BAC) = cos. BAC, and BD?ZAC', therefore AD=NAB? + AC + 2AB.AC cos.BAC.
If we call AB, a; AC, 6; and the angle BAC, A, then the resultant NO+6+ 2ab cos. A.
The sine of the avgle which this diagonal makes with AB, may be found by the proportion AD: BD :: sin. B : siu. BAD; or
b sin. A „(+62 + 2ab cos. A):6:: sin. A :
(a* +62 +zub cos. A) .54. Cor. 4. Two given forces produce the greatest effect when they act in the same direction, and the least when they act in opposite directions ; for, in the forner case, the diagonal AD becomes equal to the sum of the sides AB, BC; and, in the latter, to their difference.
55. Cor. 5. Two forces cannot keep a body at rest unless they are equal and in opposite directions.
For this is the only case in which the diagonal, representing the compound forces, vanishes.
56. Cor. 6. In the composition of forces, force ie lost; for the forces repre. sented by the two sides AB, BD, by composition, produce the force represented by AD; and the two sides AB, BD, of a triangle, are, together, greater than the third side.
57. If a body at rest, be acted upon at the same time by three forces, which are represented in quantity and direction by the three sides of a triangle, laken in order, it will remain at rest.
Let AB, BC, and CA, fig. 7, represent the quantities and directions of three forces acting at the same time upon a body at A ; then, since AB and BC are equivalent to AC (53); AB, BC, and CA, are equivalent to AC and CA ; but AC and CA, wbich are egual and in opposite directions, keep the body at rest; therefore AB, BC, and CĂ, will also keep the body at rest.
58. If a body be kept at rest by three forces, and two of them be represented in quantity and direction by two sides of a triangle AB, BC; the third side, taken in order, will represent the quantity and direction of the other force.
Since AB, AC, represent the quantities and directions of two of the forces, and AB, BC, are equivalent to AC, the third force must be sustained by AC ; therefore CA must represent the quantity and direction of the third force.
59. If a body be kept at rest ly three forces, acting upon it at the same time, any three lines which are in the directions of these forces, and which form a triangle, will represent them.
Let three forces acting in the directions AB, AC, AD, fig. 8, keep the body A at rest. In AB take any point B, and through B draw BI parallel to AC, meeting DA produced in I; then will AB, BI, and IA, represent the three forces.
For AB being taken to represent the force in that direction, if BI do not represent the force in the direction AC, or BI, let BF be taken to represent it; join AF; then since three forces keep the body at rest, and AB, BF, represent the quantities and directions of two of them, FA will represent the third (58); that is, FA is in the direction AD, which is impossible ; therefore, BI represents the force in the direction AC; and, consequently, IA represent the third force (58).
Any three lines, respectively parallel to AB, BI, IA, and forming a triangle, will be proportional to the sides of the triangle ABI, and therefore proportional to the three forces.
60. Cor. 1. If three forces keep a body at rest, they act in the same plane; because the three sides of a triangle are in the same plane. (Euclid %, II.)
61. Cor. 2. If a body be kept at rest by three forces, any two of them are to each other, inversely, as the sines of the angles which the lines of their direction make with the direction of the third force.
Let ABI be a triangle, whose sides are in the directions of the forces ; then these sides represent the forces; and AB : BI :; sin. BIA: sin. BAI :: sin. IAC:sin. BAI :: sin. CAD:sin. BAD.
62. Cor. 3. If a body at rest be acted upon at the same time by thrce forees, in the directions of the sides of a triangle, taken in order, and any two of them be to each other, inversely, as the sines of the angles which their direc. tions make with the direction of the third, the body will remain at rest.
For, in this case, the forces will be proportional to the three sides of the triangle; and, consequently, they will sustain each other.
63. If any number of forces, represented in quantity and direction by the sides of a polygon, taken in order, act at the same time upon a body at rest, they will keep A at rest.
Lct AB, BC, CD, DE, and EA, fig. 4, represent the forces; then, since AB, BC, CD, and DE, are equivalent to AE (53) ; AB, BC, CD, DE, and EA, are equivalent to AE and EA ; that is, they will keep the body at rest.
64. if any number of lines, taken in order, represent the quantities and directions of forces which keep a body at rest, these lines will form a polygon.
Let AB, BC, and DE, in the preceding figure, represent the forces which keep a body at rest ; then the point E coincides with A; if not, join AE, then AB, BC, CD, and DE, are equivalent to AE; and the body would be put in motion by a single force AE, which is co trary to the supposition ; therefore, the point E coincides with A, and the lines form a polygon.
The propositions in the two last articles are true, when the forces act in different planes.
65. A single force may be resolved into any numler of forces.
Since the single force AD, fig. 5, is equivalent to the two AB, BD, it may be conceived to be made up of, or resolved into, the two AB, BD. The force at may, therefore, be resolved into as many pairs of forces as there can be triangles described upon AD, or parallelograms, about it. Also AB, or BD, may be resolved into two; and, by proceeding in the same manner, the original force may be resolved into any number of others.
66. Cor. 1. If two forces are together equivalent to AD, and AB be one of them, BD is the other.
67. Cor. 2. If the force AD be resolved into the two AB, BD, and AB be wholly lost, or destroyed, the effective part of AD is represented in quantity and direction by BD.
68. Cor. 3. In the resolution of forces, the whole quantity of force is increased. For the force represented by AD is resolved into the two AB, BD, which are together greater than AD.
69. The effects of forces, when estimated in given directions, are not altered by composition or resolution.
Let two forces AB, BC, fig. 9, and the force AC, which is equivalent to them both, be estimated in the directions AP, PQ. Draw BD, CP, parallel to AQ, and CE parallel to AP. Then the force AB is equivalent to the two AD, DB; of which AD is in the direction AP, and DB in the direction AQ; in the same manner, BC is equivalent to the two BE, EC; the former of which is in the direction BD, or QA, and the latter in the direction EC or AP; therefore the forces AB, BC, when estimated in the directions AP, AQ, are equivalent to AD, EC, DB, and BE; or AD, DP, DB, and BE, because EC is equal to DP; and, since DB and BE are in opposite directions, the part EB of the force DB is destroyed by BE; consequently, the forces are equivalent to AP, DE, or AP, PC. Also AC, when estimated in the proposed directions, is equivalent to AP, PC; therefore, the effective forces in the directions AP, AQ, are the same, whether we estimate AB and BC, in those directions, or AC, which is equivalent to them.
70. Cor. When AP coincides with AC, EC also coincides with it, and D coincides with E. In this case the forces DB, BE, wholly destroy each
and thus, in the composition of forces, force is lost.
ON THE COLLISION OF BODIES.
Definitions, 71. The collision of two or more bodies, is the shock by which, when they come in contact, they alter each other's motion : it is synonymous with percussion.
72. The force of percussion, or collision, is the same as the momentum, or quantity of motion, and is measured by the product arising from the mass moved, multiplied by the relocity.
73. Bodies are either hard, soft, or elastic.