The Absolute and Relative Force

PROPOSITION 21. PROBLEM

160. To determine the effect of any oblique forces acting on a moving point.

SOLUTION.

Let the point A (Fig. 13 repeated, [in which BAb is isosceles and the angle BAb is incremental.]) have the speed c in the direction AB. [A also refers to the mass of the body, while np/A is the acceleration in some set of units with constant of proportionality n.] Indeed it is acted on by a force p, the direction of which AC makes an angle with AB, the sine of which is k. It is evident that the point A left to itself unless acted on by a force progresses along the line AB and in an element of time dt travels through the distance AB = cdt (30).

Truly with the force p acting the point A will be deflect from the line AB and meanwhile travel along the element of distance AD, as has been shown in Prop. 14 (118). Moreover we have put AC or BD = dz there in the diagram, which is the element of distance through which the point A, if it should be at rest would be drawn forwards by the force p in the time dt. Hence it follows that dz  npdt 2 (159) [Recall that the final speed in the increment is A taken as the speed throughout the increment]. Therefore the sine of the angle BAD which has been found, is equal to kdz (124) [as sinθ = k and dz / sin BAD  cdt / k ], which is cdt nkpdt . And the increment of the speed dc [= Db/dt] which was equal Ac npdt (1 kk ) dz (1 kk ) to (123), is now equal to . Q. E. I. [p. 66] dt A equal toEULER’S MECHANICA VOL. 1. Chapter two. page 66 Translated and annotated by Ian Bruce. Corollary 1. 161. The distance AD is called ds (Fig. 18), and the element of time by dt  ds , then with ds in place of dt above there is c c produced dc  npds (1 kk ) . The perpendicular DF is drawn Ac from D to the direction of the force AE; let AF = dy and DF = dx, then [the element of the distance squared] ds 2  dx 2  dy 2 and k  dx and ds dy (1  kk )  ds . Hence it npdy comes about that dc  Ac or Acdc = npdy. Corollary 2. 162. [A circle] is drawn to the curve described by the small body in this way, at the point A with radius of osculation AO, and Bb : AB = AD : AO. [Essentially the law of the . AD . Since Bb is the sine of the angle centripetal force at that point.] Whereby AO  ABBb AB BAD, which was found to equal nkpdt Bb  npdxdt , and on account of AD = ds . Hence AB Ac Acds 2 Acds . it comes about that : AO  npdxdt Corollary 3. 2 ds . The radius of the osculating circle 163. Since it is the case that dt  ds , then AO  Ac c npdx is taken as AO = r, and hence we have nprdx  Ac 2ds . Corollary 4. 164. If the direction AE of the force p is incident along the normal AO, then there arises AF = dy = 0 and DF = dx = AD = ds. On account of which it follows that cdc = 0, and therefore the force does not change the speed.

Corollary 5. 2

165. Again in this case it follows that npr  Ac 2 on account of dx = ds, and r  Ac . np Therefore this force, the direction of which is normal to the direction of the body [p. 67] results in the arc of the [equivalent circular] curve, as the body is not able to complete its 2 rectilinear motion. [The centripetal force is np  Acr . This situation arises in projectile motion at the highest point, where dy is zero. We should perhaps recall that the force always acts downwards, while the initial speed is at any angle we choose, in Euler’s derivation of the equations governing motion in two dimensions that he has presented here step by step, in a very careful manner, that I have tried to reproduce in this translation.]

Corollary 6.

166. If the direction of the force p is incident along the tangent AB, then dx = 0 and dy = ds. In this case we have Acdc  npds . Therefore the force acting in this direction will give the body the greatest increase in speed. [This corresponds to motion vertical downwards under gravity.]

Corollary 7.

167. If the direction of the force p is incident in the opposite direction to AB , thus in order that it is contrary to the direction of the motion of the body, the quantity p becomes negative, and we have Acdc  npds . Therefore in this case the speed is decreased by the same amount as it was increased before. [Motion vertically upwards under gravity.]

Corollary 8.

168. Moreover in each case in which the direction of the force p is incident along the 2 ds , on account of dx = 0. Therefore in that case the direction of the tangent, and r  Ac np.0 body will not then change, and it will accelerate in a straight line.

Corollary 9.

169. Therefore any case the value of the constant letter n determined by experiment will be put in place for all cases. Therefore than everything which could be wished for in the motion will be given absolute values.

Corollary 10. npdy 170. From the first corollary there arises A  cdc . With this value substituted in the third corollary, we have nprdx  npcdyds or rdxdc  cdyds . [p. 68] dc In which equation neither n, A, nor p is present, and this prevails for any motion of the body whatsoever, and for any force to be acting.

Corollary 11

171. Nevertheless however, although the force p itself is not to be found in this equation, yet the direction of this, upon which the relation of the elements dx et dy depends, still remains. Therefore from the given direction of the force acting at any point on the curve and from the curve itself, along which the point may be moving, from these alone the speed at any point can be determined. For indeed it will be given by : dyds dc  dyds or c  e  rdx c rdx is 1. where e specifies the number the hyperbolic logarithm of which

Corollary 12.

172. Again since dt  ds , then t   e c dyds   rdx ds . Hence therefore it should be noted that likewise for the time in which any part of the motion is to be described, only the curve itself and the direction of the force needs to be given.

Corollary 13.

173. If from O the perpendicular OE is sent to the direction of the force AE, then ds : dx  AO : AE. [See Fig. 18.] Hence with the position AE = q, which line is called the co-radius by certain people, will be rdx  q . Hence the speed becomes ds  dy  dy c  e q and t   e q ds . [Joh. Bernoulli, Concerning the motion of heavy bodies, pendula and projectiles. Acta erud. Lips. 1713; Opera Omnia, Book I, Lausanne et Geneve 1742, p. 531.] [p. 69] [Thus, if dx = ds as in (165) then r = q and dy = 0 and we have circular motion with c given by a constant. Again, for motion vertically downwards (166), dx = 0 and dy = ds, which is a degenerate case.]

Scholium.

174. From the solution of this problem it appears to be possible to determine the motion of a small body, acted on by any kind of forces. For the motion is defined by two equations : both the speed of the body anywhere, and the curvature or the radius of osculation of the curve traversed.

With these known, likewise the time can be found, in which the motion along some part of the curve is completed, which is sufficient for the motion to be determined. [As no examples are provided here, we are to imagine that Euler has just come across this result, and has not yet followed through with any consequences. He now moves on to another ‘pet theory’.]

Definition 13.

175. The force of restitution is that imaginary infinite force, which restores the separate parts of the body again to their previous state. We considered a force of this kind to be present in the solution of Prop. 18 (146), by which the two parts of the small body, which momentarily we considered to be free, recombined again.

Corollary 1

176. If a point is considered to be divided into two points and these were separated by forces, the restoring force draws these into the middle line, as was shown (146) with sufficient explanation at the start.

Corollary 2

177. Since the effect of the restoring force must be produced instantaneously, the restoring force must be considered as provided by an infinite elastic force, by which the separated parts are again joined together.

Scholium.

178. The use of this force of restitution now is clear in a certain way from proposition 18, yet the use of this will be most fully understood when we are to begin investigating the motion of bodies of finite magnitudes in what follows. Truly we will investigate the effect of this here with many separate parts of a point joined together, which will be of great use in what follows. Hence a certain principle of restitution is embraced, with the help of which many questions are easily resolved, that we call the principle of restitution.