PROPOSITION XXVII. 134. A Body acquires the Same Velocity in descending down any Inciined Plane BA, as by falling perpendicular through the Height of the Plane BC. FOR, the velocities generated by any constant forces, are in the compound ratio of the forces and times of acting. But if we put. F to denote the whole force of gravity in BC, fthe force on the plane AB, the time of describing BC, and B D T the time of descending down AB; then by art. 119, F: ƒ :: BA BC; and by art 132, T; BC: BA; theref, by comp. Ft:ƒT:: 1: 1. A That is the compound ratio of the forces and times, or the ratio of the velocities, is a ratio of equality. 135. Corol. 1. Hence the velocities acquired, by bodies descending down any places, from the same height, to the same horizontal line are equal. 136. Corol. 2. If the velocities be equal, at any two equal altitudes, D, E; they will be equal at all other equal alti tudes A, C. 137. Corel. 3. Fece also the velocities acquired by descending down any planes, are as the square roots of the heights. PROPOSITION XXVIII. 138. If a Body descend down any Number of Contiguous Planes, AB, BC, CD; it will at last acquire the Same Velocity, as a Body falling perpendicularly through the Same Height ED, supposing the Velocity not altered by changing from one Plane to another. E PRODUCE the planes DC, CB, to meet the horizontal line EA produced in F ald G. Then, by art. 135, the velocity at B is the same whether the body descend through AB OF FB And therefore the volocity at c will be the same, whether the body descend through ABC C F G or through FC, which is also again, by art. 135, the same as by descending through GC. Consequently it will have the same velocity at D, by descending through the planes AB, BC, CD, as by descending through the plane GD; supposing no obstruction to the motion by the body impinging on the planes at B and c: and this again, is the same velocity as by descending through the same perpendicular height ED. 139. Corol. 1. If the lines ABCD, &c, be supposed indefinitely small, they will form a curve line, which will be the path of the body; from which it appears that a body acquires also the same velocity in descending along any curve, as in falling perpendicularly through the same height. 140. Corol 2. Hence also, bodies acquire the same velocity by descending from the same height, whether they descend perpendicularly, or down any planes, or down any curve or curves. And if their velocities be equal, at any one height, they will be equal at all other equal heights. Therefore the velocity acquired by descending down any lines or curves, are as the square roots of the perpendicular heights. 141. Corol. 3. And a body, after its descent through any curve, will acquire a velocity which will carry it to the same height through an equal curve, or through any other curve, either by running up the smooth concave side, or by being retained in the curve by a string, and vibrating like a pendulum Also, the velocities will be equal, at all equal altitudes; and the ascent and descent will be performed in the same time, if the curves be the same. 142. PROPOSITION. XXIX. The Times in which Bodies descend through Similar Parts of Similar Curves ABC. abc, placed alike, are as the Square Roots of their Lengths. THAT is, the time in AC is to the time in ac, as √ Ac to ac. B A For, as the curves are similar, they may D a be considered as made up of an equal number of corresponding parts, which are every where, each to each, proportional to the whole. And as they are placed alike, the corresponding small similar parts will also be parallel to each other. But the time of describing each of these pairs of corresponding parallel parts, by art. 128, are as the square roots of their lengths, lengths, which by the supposition, are as roots of the whole curves. Ac to ac, the Therefore, the whole times are in the same ratio of✔ AC to ✔ac. 143. Corol 1. Because the axes DC, DC, of similar curves, are as the lengths of the similar parts Ac, ac; therefore the times of descent in the curves AC, ac, are as DC to DC, or the square roots of their axes. 144. Corol. 2. As it is the same thing, whether the bodies run down the sinooth concave side of the curves, or be made to describe those curves by vibrating like a pendulum, the lengths being DC, DC; therefore the times of the vibration of pendulums in similar arcs of any curves, are as the square roots of the lengths of the pendulums. SCHOLIUM. 145. Having in the last corollary, mentioned the pendulum, it may not be improper here to add some remarks concerning it. E B A pendulum consists of a ball, or any other heavy body B. hung by a fine string or thread, moveable about a centre A, and describing the arc CBD; by which vibration the same motions happen to this heavy body, as would happen to any body descending by its gravity along the spherical superficies CBD, if that superficies were perfectly hard and smooth. If the pendulum be carried to the situation AC, and then let fall, the ball in descending will describe the arc CB; and in the point B it will have that velocity which is acquired by decending through CB, or by a body falling freely through EB. This velocity will be sufficient to cause the ball to ascend through an equal arc BD, to the same height D from whence it fell at c; having there lost all its motion, it will again begin to descend by its own gravity; and in the lowest point B it will acquire the same velocity as before; which will cause it to re-ascend to c; and thus, by ascending and descending, it will perform continual vibrations on the circumference CBD. And if the motions of pendulums met with no resistance from the air, and if there were no friction at the centre of motion A, the vibrations of pendulums would never cease. But from these obstructions, though small, it happens, that the velocity of the ball in the point в is a little diminished in every vibration; and consequently it does not return precisely to the same points c or D, but the arcs described con tinually tinually become shorter and shorter, till at length they are insensible; unless the motion be assisted by a mechanical contrivance, as in clocks, called a maintaining power. again, making just one revolution, and thereby measuring out a straight line ABA equal to the circumference of the cir cle, while the point a in the circumference traces out a curve line ACAGA; then this curve is called a cycloid; and some of its properties are contained in the following lemma. LEMMA. 147. If the generating or revolving circle be placed in the middle of the cycloid, its diameter coinciding with the axis AB, and from any point there be drawn the tangent CF, the ordinate CDE perp. to the axis, and the chord of the circle AD Then the chief properties are these : The right line CD the circular arc AD; The semi-cycloid ACA = double the diameter AB, and PROPOSITION XXX. 148. When a Pendulum vibrates in a cycloid; the Time of one Vibration, is to the Time in which a Body falls through Half the Length of the Pendulum, as the Circumference of a Circle is to its Diameter. DB. On LB describe the semicircle LMB, whose centre is o; draw мp parallel to DB; also draw the chords BE, BH, EH, and the radius OM. Now the triangles BEH, F BHK, are equiangular; the re- G fore BK: BH BH: BE, or BH2 BK. BE, or BH = ✔ (BK. BF). And the equiangular triangles мmp, MON, give Mp мm:: MN: MO. Also, by the nature of the cycloid, His equal to Gg. If another body descend down the chord EB, it will have the same velocity as the ball in the cycloid has at the same height. So that Kk and Gg are passed over with the same velocity, and consequently the time in passing them will be as their lengths Gg, Kk, or as нh to кk, or BH, to BK by similar triangles, or(BK. BE) to BK, or ✓ BE to ▼ BK, or as BL to BN by similar triangles. That is, the time in Gg time in кk: : √ BL: √ BN. Again, the time of describing any space with a uniform motion, is directly as the space, and reciprocally as the velocity; also, the velocity in K or Kk, is to the velocity at B, as Ek to EB, or as LN to LB; and the uniform velocity for EB is equal to half that at the point B, therefore the INLB ✔LN kk time in kk time in EB :: : :: : LB (BL. LN.) (by sim. tri):: Nn or MP: 2√ (BL LN.) That is, the time in кk: time in EB:: MP: 2 But it was, time in Gg: time in Kk :: √ BL: √BN; theref. by comp time in Gg: time in EB :: MP: 2√(BN. NL) or 2nm. But, by sim tri. мm: 20м or BL:: мp: 2NM. Theref. time in co time in EB Min: BL. Consequently the sum of all the times in all the Gg's, is to the time in EB, or the time in DB, which is the same thing, as the sum of all the мm's, is to LB; that is, the time in and the time in or the time in That is, the time of one whole vibration, 149. Corol. |