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29. 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.
THAT is, s is as, or as 12 when ƒ and 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 v is the velocity att, 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 velocityv, 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 tv. But, by the last corol. the velocity v is as 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. Or s is as t2, the square of the time only, when b and ƒ are given.
30. Corol. 1. The space s is also as tv, or in the compound ratio of the time and velocity; b and ƒ being given. For, stv 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.
31. Corol. 2. The space s is also as v2, the square of the velocity; because the velocity is as the time t.
32. Propositions 3, 4, 5, 6, give theorems for resolving all questions relating to motions uniformly accelerated. Thus,
bany body or quantity of matter, f the force constantly acting on it, the time of its acting,
the velocity generated in the time t,
the momentum at the end of the time.
Then, from these fundamental relations, m œ bv, m × ft,
8 ∞ tv, and v∞ we obtain the following table of the ge
neral relations of uniformly, accelerated motions:
33. And from these proportions those quantities are to be left out which are given, or which are proportional to each other. Thus, if the body or quantity of matter be always the same, then the space described is as the force and square of the time. And if the body be proportional to the force, as all bodies are in respect to their gravity; then the space described is as the square of the time, or square of the velocity; and in this case, if x be put =, the accelerating force; then will
THE COMPOSITION AND RESOLUTIon of
34. COMPOSITION OF FORCES, is the uniting of two or more forces into one, which shall have the same effect; or the finding of one force that shall be equal to several others taken together, in any different directions. And the resolution of Forces, is the finding of two or more forces which, acting in any different directions, shall have the same effect at any given single force.
35. If a Body at a be urged in the Directions AB and Ac, by any two Similar Forces, such that they would separately cause the Body to pass over the Spaces AB, AC, in an Equal Time; then if both Forces act together, they will cause the Body to move in the same Time, through AD the Diagonal of the Parallelogram ABCD.
Draw cd parallel to AB, and bd parallel to AC. And while the body is carried over ab, or cd by the force in that direc- A tion, let it be carried over bd by the force. in that direction; by which means it will be found at d. Now, if the forces be impulsive or momentary, the motions will be uniform, and the spaces described will be as the times of description:
theref. Ab or cd: AB or CD time in ab; time in AB,
And as this is always the case in every point d,d, &c. there~ fore the path of the body is the straight line AdD, or the diagonal of the parallelogram.
But if the similar forces, by means of which the body is moved in the directions AB, AC, be uniformly accelerating ones, then the spaces will be as the squares of the times; in which case, call the time in bd or cd, t, and the time in AB or AC, T; then
it will be
Ab or cd: AB or CD : t2 : T2 "
theref, by equality, ab: bd :: AB : BD;
and so the body is always found in the diagonal, as before.
36. Corol. 1. If the forces be not similar, by which the body is urged in the directions AB, AC, it will move in some curved line, depending on the nature of the forces.
37. Corol. 2. Hence it appears that the body moves over the diagonal AD, by the compound motion, in the very same time that it would move over the side AB, by the single force impressed in that direction, or that it would move over the side Ac by the force impressed in that direction.
38. Corol. 3. The forces in the directions, AB, AC, AD, are respectively proportional to the lines AB, AC, AD, and in these directions.
39. Corol. 4. The two oblique forces AB, AC, are equivalent to the single direct force AD, which may be compounded of these two, by drawing the diagonal of the parallelogram. Or they are equivalent to the double of AE, drawn to the middle of the line Bc. And thus any force may be compounded of two or more
other forces; which is the meaning of the expression composition of forces.
40. Exam. Suppose it were required to compound the three forces AB, AC, AD; or to find the direction and quantity of one single force which shall be equivalent to, and have the same effect, as if a body A were acted
on by three forces in the directions AB, AC, AD, and proportional to these three lines. First reduce the two ac, ad, to one AE, by completing the parallelogram ADec. Then reduce the two AE, AB to one AF by the parallelogram AEFB. So shall the single force AF be the direction, and as the quantity, which shall of itself produce the same effect, as if all the three AB, AC, AD acted together.
41. Corol. 5. Hence also any single direct force AD, may be resolved into two oblique forces, whose quantities and directions are AB, AC, having the same effect, by describing any parallelogram whose diagonal may be AD: and this is called the resolution of forces. So the force AD may be resolved into the two AB, AC by the parallelogram
ABDC; or into the two AE, AF, by the parallelogram AEDF; and so on, for any other two. And each of these may be resolved again into as many others as we please.
42. Corol. 6. Hence too may be found the effect of any given force, in any other direction, besides that of the line in which it acts; as of the force AB in any other given direction CB. For draw AD perpendicular to then shall DB be the effect of the force AB in the direction CB. For the given force AB is equivalent to the two ad, DB, or AE; of which the former Ad, or db, EB, being perpendicular, does not alter the velocity in the direction cв; and therefore DB is the whole effect of AB in the direction CB. That is, a direct force expressed by the line DE acting in the direction DB, will produce the same effect or motion in a body B, in that direction, as the oblique force expressed by, and acting in the direction AB, produces in the same direction CB. And hence any given force AB, is to its effect in DB, as AB to DB. or as radius to the cosine of the angle ABD of inclination of those directions. For the same reason, the force or effect in the direction AB, is to the force or effect in the direction AD or EB, as AB to AD; or as radius to sine of the same angle ABD, or cosine of the angle DAB of those directions.
43. Corol. 7. Hence also, if the two given forces, to be compounded, act in the same line, either both the same way, or the one directly opposite to the other; then their joint or compounded force will act in the same line also, and will be equal to the sum of the two when they act the same way, or to the difference of them when they act in opposite directions; and the compound force, whether it be the sum or difference, will always act in the direction of the greater of the two.
44. If Three Forces A, B, C, acting all together in the same Plane, keep one another in Equilibrio; they will be proportional to the Three Sides DE, EC, CD, of a Triangle, which are drawn Parallel to the Directions of the Forces aD, DB, CD.
Produce ad, bd, and draw CF, CE, parallel to them.