Proposition 3

PROPOSITION 3. Theorem.

37. If the body, which is moving in the channel AM (Fig. 3), is acted on at M by the force MN, the direction of which is normal to the curve AM, then the speed is neither increased or decreased and the whole force is taken up in pressing against the channel.

Demonstration

From the first book (164) it was shown that the force, the direction of which is normal to the direction of the motion, neither increases nor decreases the speed. Though indeed there for free motion it has a place for stiffness, [p. 15] since the normal force neither before or after pulls on the body. Truly in free motion the direction of the normal force does not change, as it is not able to have an effect in this situation. Therefore the body is pressed by this force on the channel and consequently only the force of the channel presses in the direction MN. Q.E.D

Corollary 1.

38. Therefore the direction of such a normal force is either incident in the direction of the centrifugal force or is in the direction contrary to that. In the first case the centrifugal force is increased, and in the other it is diminished.

Corollary 2.

39. Because the direction of the centrifugal force falls on the convex part of the curve, the effect of this is to increase the force, if the normal force falls in the same place; but if the normal force is directed to a concave part of the curve, the effect is diminished the force.

Corollary 3.

40. If the normal force is equal to N and the centrifugal force as before is equal to 2rv , the curve is pressed either by the force 2rv + N , if the forces act together, or by the force 2v − N , if they act in the opposite directions. r

Corollary 4.

41. If the normal force is equal and opposite to the centrifugal force, then the curve sustains no force, or the body does not try to escape from the curve. Therefore in this case the body is free to describe the same curve; [p. 16] it is also evident that the normal force is equal to 2rv ; for it is brought about here, in order that the body is free to move on any curve.

PROPOSITION 4.Theorem.

42. If the body, which is moving in the channel AM (Fig. 3), is acted on at M by a force, the direction of which is along the tangent MT, the effect of this is consistent with this, that the speed of the body is either increased or diminished in the same way as for free motion.

Demonstration.

Since the direction of this force is the tangent MT of the channel, the effect of the channel cannot impede the effect of this force ; nor can this force exercise any effect on the channel. On account of which the force either augments or diminishes the speed of the body, according as the direction of this either acts in the same direction as the body or in the opposite direction, and clearly if the body is moving freely. And with the height corresponding to the speed at M equal to v, the element Mm = ds and with the force MT = T, there is dv = Tds with the accelerating force T; but with retardation that becomes dv = −Tds . Q.E.D.

Corollary 1.

43. Therefore in the motion of bodies on given lines, the normal force only generates a force on these lines, and the tangential force truly only affects the speed.

Corollary 2.

44. Since the force of the retarding resistance may be greater than the tangential force, [p. 17] it acts in the same manner in the motion of bodies on given lines as in the case of free motion. If therefore as well as the accelerating tangential T there is the resistance R present, then with both joined together we have dv = Tds − Rds .

PROPOSITION 5. Problem.

45. If a body is moving on some given line AM (Fig. 4) in some medium with resistance and in addition it is acted on by some absolute force, the direction of which is MP, to determine the effect of the absolute force as well as of the resistance as well as the force supported by the curve AM.

Solution

Let the height corresponding to the speed at M be equal to v, the force of the resistance is equal to R and the absolute force MP = P, the direction of which is such that, as with the element Mm taken equal to ds , the perpendicular mn from m sent to MP is equal to dx and Mn = dy = ( ds 2 − dx 2 ) . The force P is resolved into these two forces along the normal MN to the curve and the force pulling along the tangent MT ; on this account the triangles MPT and Mmn are similar and the normal force MN or PT = Pdx and the tangential force ds Pdy MT = ds increased the speed. Since truly the force of resistance decreases the speed, the Pdy speed is only increased by the excess ds − R ; on this account there is (42) dv = Pdy − Rds. The normal force truly is effected by Pdx , as the curve is pressed just as much at M along ds the direction MN to the convex part of the curve in place. [p. 18] Whereby, since the centrifugal force acting at the same place is equal to 2rv , with the radius of curvature designated by r the radius of osculation at M, the total force by which the curve is pressed on normally at M, along MN, is equal to Pdx

• 2rv . Hence the motion of the body on the ds given curve as well as the force acting on the curve can be found at individual points. Q.E.I.

Corollary 1

46. Therefore from these two formulas both the acceleration and the force on the wall can be expressed can all be deduced from the expressions, which pertain to the motion on the given lines.

Scholium 1

47. Here indeed we have put in place a single absolute force; yet nevertheless from that it is understood how the effect of many forces can be understood. Of course as we made in free motion, thus also here the individual forces are to be resolved into two parts, truly the normal and the tangential, from which by gathering these together a single normal force and a single tangential force arises; the effect of which can be determined by Propositions 3 and 4.

Scholion 2.

48. Therefore up to the present we have set out the fundamentals, from which in the following it is permitted to determine the motion of bodies on given. But before we treat the similar principles of the motion on given surfaces, it is expedient that we consider a few cases in which the motion on a given line in effect can be deduced.[p. 19] In as much as with the help of channels, in which the body is contained, it is of minimal use to produce such motion on account of friction and other obstacles, which by no means are able to be removed. Moreover constrained motion of this kind is most conveniently brought into being with the aid of pendulums, as was first done by Huygens[Original reference presumably used by Euler: Chr. Huygens, Horologium oscillatorum sive de motu pendulorum ad horologia aptato demonstrationes geometricae. Paris 1673; Opera varia, Vol. 1, Lugduni Batavorum 1724, p. 89. See the English translation in this series.]; why we arrange matters to make use of pendulums we explain in the following proposition.