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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:sip. BAD.
62. Cor. 3. If a body at rest be acted apon at the same time by three forces, 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.
Let 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.
61. If any numler 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 D 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 in. creased. 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 other; 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 coine 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 measc red by the product arising from the mass moved, multiplied by the relocity. á 73. Bodies are either hard, soft, or elastic.
A body is said to be perfectly hard, when its component par. cannot be separated, or moved among themselves, by any finite force.
A soft body consists of particles which give way upon the applicae tion of the least force or impression. Such a body does not again, of itself, resume its figure, but remains altered.
An elastic body has a natural tendency to recover its former figure, after it has been altered by compression. By a perfectly elastic body, we mean one which recovers its figure with a force equal to that which was employed in compressing it. Almost all bodies, with which we are acquainted, are elastic in a greater or less degree, but none perfectly so. In steel balls, the force of elasticity is to the compressing force as 5 to 9, and, in glass balls, as 15 to 16.
74. The impact of two bodies is said io be direct, when their centres of gravity move in the right line which passes through the point of impact.
On the Impact of hard, or non-elastic Bodies. 75. If two bodies have no elasticity, when they come into contact by direct impact, there is no force whatever to separate then again; consequently they must either remain at rest, or move on uniformly together.
76. To determine the several particulars relating to the motion, velocities, and direction of two percutient bodies, let A and B, fig. 10, represent the bodies or quantities of matter, V and v their velocities, then will AV be the momentum of A, and Bv that of B.
77. Then, if the body A strikes the body B in motion, and both move the same way, or towards the same part, as from A towards B, then the sum of their motions, in that direction, will be AV + Bu, and the velocity of both bodies after the stroke, towards the same part,
AV + Bv will be =U. For the velocity is always as the momentum divided by the mass (27).
78. If one of the bodies, as B, has a contrary direction, in which case the bodies will meet, then the momentum of B will have a negative sign, viz. – Bv ; and the sum of the motion towards the same part, will be AV-Bv; and the velocity, after the collision, will be AV-Bv
79. The sum of the motions towards the same parts, is the sam. before and after the stroke. For let them both move the same way, and let A strike B ; then, by that impulse, the motion of B, viz. Bv, will be augmented, and become Bv + x after the stroke ; but, because action and re-action are equal (20). The body B will re-act upon A, and produce an equal effect by the impact; that is, it will diminish the motion of A by the same quantity f, so that its motion after the stroke will be AV-r; but the sum of the motions of both, after the stroke, is AV-+By+r=AV+Bv, the
U : V:; A : A+B, and V=4+B xU.
sum of the motions before the stroke. And the same may be shown if the bodies meet.
80. The magnitude of the stroke will be proportional to the quantity I, because that is the whole effect or mutation produced in the motion of each body. The greatness of the stroke is, therefore, measured by the loss x, which the body, having the greatest momentum, sustains in its motion.
81. In the above theurem (77), if the body B be supposed at rest, then v=0, and Bu vanishes ; the velocity, then, after the stroke,
AV is A+B= U; and, consequently, AV = (A + B)U. Whence
In the same theorem, if we suppose the bodies equal, viz. A=B ; then, if the bodies tend the same way, the velocity, after the stroke, will be V+v=U; or V-v=U, if they meet.
82. Cor. 1. If the bodies are equal, and one of them, as B, at rest, then AV
U; or the velocity, after the stroke, is equal to half that of the A+B striking body.
83. Cor. 2. If B, at rest, exceed A infinitely in magnitude, then because AV is infinitely sman respect of A+B, therefore so is U in respect of V (81); consequently, U will vanish, or the body A impinging against any firm, immoveable object, will, after the stroke, be at rest.
84. Cor. 3. If equal bodies, moving with equal velocities, meet, they will mutually destroy each other's motions ; for in this case, AV-Bv=0, and,
AV-Bo consequently, -A+B=V=0; or both bodies remain at rest after tầe
85. The momentum of the body A, after the stroke, is AU= A'V+ABv
A'V+ABv_ABV + ABv A+B (by art. 77); therefore AV.
A+B AB A+B*(VFv)= the loss of motion in the body A after the struke.
= AB But is a constant quantity; therefore the loss of motion in A is
A+B as VFv; and, consequently, the magnitude of the stroke in bodies, tending the same way, is as V-v; and as V+v, if they meet. And if B be at rest, then v=0, and the stroke will be as V, the velocity of the percutient body.
AB 86. Cor. The velocity lost by A-AAR BOV FO) AT BIV Fv); and
A(+ ) since B gains the same quantity of motion that A loses, the velocity gained
A by B A+BY
(VF o.) Hence the velocity lost by A, and that gained by B are reciprocally as the bodies themselves.
Er. Let the weights of two bodies, A and B, be 10 and 6, and Seir velocities 12 and 8 respectively : then, when they move the same
A way, the velocity gained hy B will be
When they move in opposite directions, we shall find that the rela city gained by B, in the direction of A's motion, equal 12; and the velocity lost by A equal 7; and the velocity of both bodies, in the same direction, will be found 4.
ON THE COLLISION OF ELASTIC BODIES. 87. If two perfectly elastic bodies infringe on one another, their relative velocity will be the same both before and after the impulse; or, in other words, they will recede from each other with the same velocity as that with which they approached or met. For the compressing force is as the intensity of the stroke, and this (85) is as the relative velocity with which they meet or strike. But perfectly elastic bodies restore themselves to their foriner figure, by the same force by which they were compressed ; that is, the restoring force is equal to the compressing force, or to the force with which the bodies approached each before the impulse. But the bodies are impelled from each other by this restoring force; and, therefore, this force acting on the same bodies, will produce a relative velocity equal to that which they had before; or, it will make the bodies recede from each other with the same velocity with which they before approached, or so as to be equally distant fron' one another, at equal times, before and after the impact.
We do not here mean to say, that each body will have the same velocity after the inpact as it had before, but that the velocity of the one will
, after the stroke, be so much increased, and that of the other so much decreased, as to have the same difference as before, in one and the same direction. So that if the elastic body A, fig. 10, move with a velocity V, and overtake the elastic body B, moving the same way with the velocity v; then their relative velocity, or that with which they strike, is V-v, and it is with this same velocity that they separate after the stroke. But if they meet each other, or the body B move contrary to the body A, then they meet and strike with the relocity V+v, and it is with this same velocity that they separate and recede from each other after the stroke.
It may further be observed, that the sums of the two velocities of each body, before and after the stroke, are equal to each other. Thus. if V and v be the velocities before impact, and I and y the correse ponding one after it, since V-v=y-x, we also have, by transposition, V+x=vty.
88. Let the elastic body A, fig. 11, move in the direction AC, with the velocity V; and let the velocity of the other body B, in the same line, be v; which latter velocity v will be positive, if B move the sanje way as A, but negative if B move in the opposite direction to A.