Proposition 17

PROPOSITION 17. Problem.

137. With the force present acting uniformly downwards, a body moves on some curve AM (Fig.21) with an given initial speed at A; to determine the motion of the body on this curve and the force pressing the body to the curve sustained at individual points. [p. 63]

Solution

With the force acting put as g and with the initial speed at A corresponding to the height b and as well, AP = x, PM = y, AM = s and with the speed at M corresponding to the height v, with these in place, there is dv = gdx (93), and thus v = b + gx . And again the time in which the arc AM is completed is equal to ∫ ( bds+ gx ) .

Furthermore, the total force experienced by the curve along the direction of the normal MN, is equal to (93) with the element dx present constant.

This solution only differs from that solution for proposition 13, because there it was v = gx , here it is v = b + gx . From these formulae therefore the motion as well as the pressing force are known. Q.E.I.

[We note that this is not the reaction force, as that force acts on the body, and which is the force we would now calculate. For some reason, Euler persists with the force the body exerts on the curve throughout this book.]

Corollary 1

138. If the line AM is straight, now from (88) it is understood that the time to traverse AM is to the time to traverse AP beginning with the same speed b , is as AM to AP.

Now on account of the centrifugal force vanishing, the force pressing the body on the .. curve is equal to ds or constant.

Corollary 2.

139. Also it appears in this case, since the motion does not start from rest, that the speed only depends on the height. Whereby, whatever the curve AM shall be, the speed of the body at any point of this is known, also with the kind of curve unknown. [p. 64] Example 1.

140. Let the curve AM be a parabola, having the vertex at A and the axis AP vertical; hence with the parameter of this is put equal to a , y 2 = ax and Hence the time obtained for the body to pass along AM is equal to Hence, as with dx kept constant, then ddy = − adx , and we have … Consequently the total pressing force is equal to

Corollary 3

141. Therefore if b = 4 , then the force on the curve vanishes. Thus in this case the body is free to move along this parabola, which is also the case treated in the preceding book

Corollary 4

142. Therefore with the present b = 14 ga , the time to traverse the arc AM is equal to Therefore this is equal to the time to descend beginning from rest along the abscissa AP .

Corollary 5

143. If b > 14 ga , then the force becomes negative; then the curve is therefore pressed in the direction away from the axis AP. But if b < 14 ga , the direction of the force is along MN. The size of the force pressing at the individual points of the curve varies inversely as the radius of osculation. [p. 65]

Example 2

144. If the curve AM is a circle, the radius of which is equal to a and the centre is placed on the vertical line AP, then it is given by y 2 = 2ax − x 2 , hence Hence the time in which the arc AM is traversed is equal to And since dxddy −1 = a , the force that the circle undergoes at a point is equal to ds 3

Corollary 6

145. The time can be expressed by logarithms, if b = 0; moreover it is equal to infinity , or the body perpetually remains at A. That is apparent from the above treatment (97). For since the normal to the curve at A is AP and since neither is the radius of osculation infinitely small, then the body cannot descend. Corollary 7.

146. If b = 2 or the initial speed is the same size as the body acquires in falling from a height of half the radius of the circle, the total force acting with the centrifugal force is … equal to a ; and thus it is in proportion to the height travelled through.

147. Oscillatory motion is reciprocal motion in which the body alternately approaches and recedes from the starting point of the motion M (Fig. 22). Thus if the body is moving on a given curve MAN [p. 66], first it descends on MA, then it ascends on AN, while it loses speed;

then it descends from N again and ascends on the arc AM, with which done it again descends and this periodic motion continues. Such motion is called oscillatory.

Corollary 1

148. Oscillatory motion hence consists of alternate descents and ascents on a given line; and in the descending motion the body moves with an acceleration, and in the ascents the body truly loses the speeds acquired.

Corollary 2

149. Hence whatever the previous ascent touched upon, the descent is made on the same part of the curve. Whereby, since the speed of the body depends only on the height in a vacuum, the body at the same point on the curve either in the ascent or in the descent has the same speed.

Corollary 3

150. From which it follows that the time for the descent along MA is equal to the time of the descent along AM and likewise in the same manner the time of the ascent along AN is equal to the time of the descent along NA.

Corollary 4

151. The body ascending on the arc AN until it reaches the point N, since the height is equal to that of the point M from which it fell. The one follows from the other, since the speed is determined only by the height. [p. 67]

Corollary 5

152. If the curve AN is similar and equal to the curve AM, then the motion along AN is equal to the motion along AM. Whereby all the ascents and descents are made in equal s times.

Corollary 6

153. If the curves MA and AN are dissimilar, at least the time along MAN is equal to the time along NAM, or the times of approaching and receding are equal to each other.

Corollary 7

154. Since the body always reaches the same height, clearly this oscillatory motion must last indefinitely.

Corollary 8

155. Hence any curve is suitable for the production of oscillatory motion if it has two arcs such as MAN ascending from the lowest point A.

Scholium 1

156. Here we have set out the properties of oscillatory motion, such as follow from the exposition of the hypothesis of uniform forces acting, and always pulling downwards. Indeed the same is true also, if the forces depend in some manner on the height, or even if they are directed towards a fixed point, that becomes more apparent in what follows. In a medium with resistance, truly the matter is otherwise, for neither is the ascent along a given curve similar to descent along the same, nor in the ascent does the body reaches an height equal to that from which in the descent it had fallen.

Scholium 2

157. It is usual to call the motion along MAN the going movement, following the returning movement along NAM ; hence oscillatory motion consists of alternate goings and comings [we do not have such handy words as the Latin itus and reditus used here in the English language to express this notion, though to and fro’ might be our equivalent]. Truly an oscillation is called by others constant to and fro’ motion, as the [term] oscillation is called by others, to and fro’. Here we accept the name oscillation in the basic sense, so that thus one oscillation is agreed to be one to and fro’ motion. The to motion and indeed the fro’ motion each consists of one descent and one ascent, and thus the whole oscillation includes two descents and two ascents. Therefore since the time for the to motion is the same as the time for the fro’ motion, the time of one oscillation is double the time for one to or one fro’ motion. [This may sound pedantic, but that is not the case, as previous writers incl. Newton and Huygens, had timed pendulum swings from one extreme to the other, and not back to the starting point, as is the more logical thing to do. Thus again we see the hand of Euler gently guiding humanity in the right direction.] Corollary 9.

158. Therefore in this chapter, in which motion is a vacuum is undertaken, if we wish to examine the motion of oscillations, we have a need to consider only either the ascent or the descent upon the two parts of the curve AM, AN.

Scholium 3

159. Nothing matters, provided the arcs AM and AN in succession make one curve, but if they are different curves, then they are connected at A thus so that they have a common tangent [p. 69]; for otherwise the motion is disturbed. Whereby there is only the need in an inquiry about oscillatory motion to define the motion on the curves AM and AN themselves. This then is sufficient in the determination of oscillations, as the relation between larger and smaller oscillations can then be found. Moreover these oscillations are called larger which are completed by larger arcs, and the smaller by lesser arcs. Scholium 4.

160. It is evident from Proposition 6 (49), how oscillations are able to be effected with the help of pendulums, clearly with the aid of the evolute of the curves AM and AN, around which the thread is taken. Also the use of pendulums was adapted to oscillations by Huygens, as is apparent from his habit of applying that motion to the perfection of clocks. Truly the same difficulties that we have mentioned in the place cited, they have here in this place. On account of which we only investigate the motion of points upon given lines, and we lead the mind away from all the circumstances of pendulums which are able to disturb our intention.