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with which it descends, or endeavours to descend, is as the sine of the angle A of inclination.
122. Corol. 3. Hence, if there be two planes of the same height, and two bodies be laid on them which are proportional to the lengths of the planes; they will have an equal tendency to descend downtheplanes. And consequently they will mutually sustain each other if they be connected by a string acting parallel to the planes.
123. Corol. 4. In like manner, when the power P acts in any other direction whatever, we; by drawing cDE perpendicular to the direction we, the three forces in equilibrio, namely, the weight w, the power P, and the pressure on the plane, will still be respectively as AC, CD, AD, drawn perpendicular to the direction of those forces.
124. If a Weight w on an Inclined Plane AB, be in Equilibrio with another Weight P hanging freely; then if they be set a-moving, their Perpendicular Velocities, in that Place, will be Reciprocally as those Wrights.
LET the weight w descend a very small space, from w to A, along the plane, by which the string PFw will come into the position PFA. Draw wh perpendicular to the horizon Ac, and we perpendicular to AF: then wh will be the space perpendicularly descended by the weight w; and AG, or the difference between FA and Fw, will be the space perpendicularly ascended by the weight P; and their perpendicular velocities are as those spa and AG passed over in those directions, in the same time. Draw CDE perpendicular to AF, and DI perpendicular to Ac.
Then, in the sim. figs. AGwh and AED1, AG : wh: : AE : DI;
and in the sim. tri. AEC, DIC, AC : CD :: AE : D1 ; but, by cor. 4, prop. 23, AC : CD :: w : P ;
therefore, by equality, AG : WH :: W : P ; - - That
That is, their perpendicular spaces, or velocities, are reciprocally as their weights or masses.
125. Corol. 1. Hence it follows, that if any two bodies be in equilibrio on two inclined planes, and if they be set amoving, their perpendicular velocity will be reciprocally as their weights. Because the perpendicular weight which sustains the one, would also sustain the other.
126. Corol. 2. And hence also, if two bodies sustain each other in equilibrio, on any planes, and they be put in motion; then each body multiplied by its perpendicular velocity, will give equal products.
127. The Velocity acquired by a Body descending freely down an
* Inclined Plane AB, is to the Velocity acquired by a Body failing
Perpendicularly, in the same Time; as the Height of the Plane BC, is to its Length AB,
For the force of gravity, both perpendicularly and on the plane, is constant; and these two, by corol. 2, prop. \ 23, are to each other as AB to BC. But, by art. 28, the velocities generated by A. - C any constant forces, in the same time, are as those forces. Therefore the velocity down BA is to the velocity down Bc, in the same time, as the force on BA to the force on BC : that is, as Bc to B.A.
128. Corol. 1. Hence, as the motion down an inclined plane is produced by a constant force, it will be a motion uniformly accelerated; and therefore the laws before laid down for accelerated motions in general, hold good for motions on inclined planes; such, for instance, as the following: That the velocities are as the times of descending from rest; that the spaces descended are as the squares of the velocities, or squares of the times; and that if a body be thrown up an inclined plane, with the velocity it acquired in descending, it will lose all its motion, and ascend to the same height, in the same time, and will repass any point of the plane with the same velocity as it passed it in descending.
129. Corol. 2. Hence also, the space descended down an inclined plane, is to the space descended perpendicularly, in the same time, as the height of the plane CB, to its length AB, or as the sine of inclination to radius. For the space described
described by any forces, in the same time, are as the forces, or as the velocities.
130. Corol. 3. Consequently the velocities and spaces descended by bodies down different inclined planes, are as the sines of elevation of the planes. -
131. Corol. 4. If CD be drawn perpendicular to Ab; then, while a body falls freely through the perpendicular space Bc, another body will, in the same time, descend down the part of the plane BD. For by similar triangles, - BC : BD :: BA : BC, that is, as the space descended, by corol. 2, -
Or, in any right-angled triangle BDC, having its hypothenuse Bc perpendicular to the horizon, a body will descend down any of its three sides BD, BC, DC, in the same time. And therefore, if on the diameter Bc a circle be described, the time of descending down any chords BD, Be, BF, DC, EC, FC, &c, will be all equal, and each equal to the time of falling freely through the perpendicular diameter Bc.
132. The Time of descending down the Inclined Plane BA, is to the Time of falling through the Height of the Plane Bc, as the Length BA is to the Height Bc.
DRAw cd perpendicular to AB. - B Then the times of describing BD and Pé-E Bc are equal, by the last corol. Call that time t, and the time of describing BA call T. A C
Now, because the spaces described by constant forces, are as the squares of the times; therefore t” : To :: BD : B.A.
But the three BD, BC, BA, are in continual proportion; therefore BD : BA :: Eco :: BA*; - . hence, by equality, to : To :: Bc : BA”, Or • - t : T : : BC : B.A.
133. Corol. Hence the times of descending down different planes, of the same heights are to one another as the lengths of the planes.
- PROPOSITION PROPOSITION xxWII.
134. A Body acquires the Same Velocity in descending down any Inclined Plane BA, at 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, f the force on the plane AB, - R t the time of describing Bc, and T the time of descending down AB; then by art. I 19, F : f :: BA : Bc ; and by art. 132, t : T :: Bc : BA.; A C theref. by comp. Ft : for:: 1 : 1.
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 planes, 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 altitudes A, C.
137. Corol. 3. Hence also, the velocities acquired by descending down any planes, are as the square roots of the
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 changingfrom one Plane to another.
PRODUCE the planes Dc, cB, to meet the horizontal line EA produced in F and G. Then, by art. 135, the velocity at B is the same, whether the body descend through AB or FB. And therefore the velocity at C will be the same, whether the body descend through ABc 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 S2IIle.
PRO POSITION XXIx.
142. The Times in which Bodies descend through Similar Parts
of Similar Curvey, 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 WAc to Vac.
For, as the curves are similar, they may |2–4 A be considered as made up of an equal number of corresponding parts, which are C every where, each to each, proportional to | 2^13
--~~ 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,