The Absolute and Relative Force

Definition 12: Absolute Force

111. The absolute force is the force that acts equally on a body either at rest or moving.

An absolute force of this kind is the force of gravity. which acts equally downwards on a body which is either at rest or moving.

Corollary.

112. If therefore the absolute effect of a force acting on a body at rest is known, then the effect of the force on the body is also known for any kind of motion.

Definition 13: Relative Force

113. A force is relative, which acts in one way on a body at rest, in another on a body in motion.

A force of this kind is the force acting on a body dragged away by a river ; where indeed the faster the body moves in the river, the smaller the force shall be : and that therefore the force vanishes when the speed of the body is the same as that of the river.

Corollary 1.

114. If therefore the speed of the body is given together with the law of the relative force, it is possible to find the strength, the magnitude of the force is exerted on the body.

Hence from this as the absolute force has to be considered, as long as the body has the same speed, the effect of this from the action of the absolute action can be determined. For the strength of the relative force is to be determined from the given motion of the body.

Corollary 2

115. Therefore these relative and absolute forces in turn are different from each other; for the magnitude and direction of the absolute force acting on the body may only depend on the location of the body ; while truly the magnitude and direction of the relative force acting on the body depends in addition on the speed of the body. [p. 45]

Scholium 1

116. Relative forces are chiefly to be considered in the relative motion of bodies in fluids.

For the action of these forces on bodies depends on their relative speed; for when that is greater, it is apparent that the force of the fluid acting on the body is also greater.

Moreover as well as the other causes of motion in fluids, which require a greater understanding of fluids, there are two which are easier to handle; the one when the fluid is at rest, and the other when the fluid is moving uniformly in a given direction.

Truly it is always possible for the one to be substituted for the other, with the relative motion always reduced to the absolute ; likewise, clearly for the state of a fluid considered to be at rest the proper forces will remain [i. e. those of a relative nature have vanished].

Therefore in what follows, concerning the relative forces which may be proposed, those properties which also pertain to fluids at rest will be the chief concern.

The action of fluids on the motion of bodies consists wholly in the diminution of their speed and on this account it is called the resistance, which is also greater when bodies move faster, and which always disappears when the bodies are at rest.

On this account we can put the true motions in place of the relative motion in what follows, and these are only affected by the absolute forces, and can be placed in a vacuum [in general].

Scholium 2.

117. The motion in mediums with resistance, if we wish the greatest order to be followed in the following chapters, must be referred to the last chapter [of vol. 1], in which the motion of the fluid is to be determined, since also it is not now agreed upon by which law the fluids resist the motions of bodies.

This matter is usually considered from many points of view, in order that the nature of fluids can be examined in a straight forward manner, these have been revoked and instead a purely mathematical hypothesis to be used : I have decided to retain this method as many elegant problems are passed over, which otherwise are not to be found in discussions on fluids. However I will apply this method only to the motion of points in fluids, as the calculations associated with bodies of finite magnitudes become insurmountable.

Moreover when the shape of a body taken as a finite number of points is considered, there is a convenient outcome from this, which is that the direction of the resistive force is in agreement with the direction of the motion, since indeed that arises from a fluid at rest.

Moreover, on account of this we agree that the motion of points in relative motion in fluids are always to be considered with relative forces in the same direction as the point itself, and that always as we will consider the motion to be decreasing.

PROPOSITION 14. PROBLEM.

118. For the given effect of an absolute force on a point at rest, to find the effect of the same absolute force on the same point in some kind of motion.

SOLUTION.

Let the point be placed at A (Fig. 13), from where it can be moved with speed c following the direction AB, and indeed the direction of the force acting on it is in the direction AC.

Some element of time is taken dt, and in this short length of time the point A is pulled forwards, if it were at rest at A, through the small distance AC, [p. 47] that may be called dz, so that after the time dt the point is no longer at A, but at C.

This motion of the point along AC is the effect of the force acting on the point at rest. The effect of the same force, which is put as absolute, should be the same on a moving point as on one at rest (111).

The direction of the point is taken so that it travels the distance AB along AB, since it travels the distance AB = cdt (30), with its speed c acting for the short time dt, if it is not influenced by any force.

Truly with the force acting after the short element of time dt, the point will no longer be found at B, but to be elsewhere at D, thus so that the effect, which is to be measured by the deviation from the point B, which is the distance BD, which is equal to the effect of the same force on the resting point (111), i. e. AC. Hence BD = AC.

Besides indeed BD is parallel to AC itself, since BD has been the effect of the force, and thus should be acting in the same direction, which does not change during the indefinitely short time dt.

On account of this the point A having the speed c along the direction AB and influenced by the absolute force, for the elapse of the short time, is to be found not from B but from D, with BD equal and parallel to AC itself.

Truly the distances traversed in the infinitely short time can be considered to be straight lines; on account of which the distance AD traversed in the very short time dt is agreed upon. Q. E. I.

Corollary 1

119. Since the motions in infinitely short distance traversed can be uniform, the speed with which the element AD (33) is traversed, is equal to AD (30)

dt

Corollary 2.

120. The speed along AD is put equal to c + dc, which preceding was c (35), will be given by c  dc  AD ; but before AB = cdt, from which c  AB . Hence there is dt

produced: dc  ADdt AB . Therefore on cutting off the portion Ab = AB from AD, there is left the equation dc  Db . dt

Scholium 1.

121. Moreover, it is to be understood that AC or BD is infinitely smaller than AB, since AB is the distance traveled with a finite speed in the time dt, but the absolute small distance AC is traversed in the same time element with an infinitely small speed.

It is not possible to infer a finite speed for a body at rest in an infinitely small increment of time.

Corollary 3

122. Hence on account of this, the angle BAD is indefinitely small, and with the points B and b joined, the line increment Bb is perpendicular to AD.

The sine of the angle BAC, which is surely to be given, is called k, with the total sine taken as 1, then the sine of the angle BDb is k also, and since BD = AC = dz, then Db  dz (1  kk ) and Bb = kdz.

Corollary 4.

123. Therefore, the increment of the speed dc, that we found before to be equal to Db , dt will be dz (1 kk )

It is understood that the distance dz is infinitely smaller than dt; dt for dz is infinitely small with respect to AB, i. e. cdt, and likewise with respect to dt, since c is put of finite magnitude.

Thus, we see that even at this time, 1733, the problems involved with an inadequate notation for orders of increments had not been resolved, and much was left to the intuitive powers of the practitioner. It does appear in general that Euler considers his infinitesimal elements set out in the diagrams as dx, dy, etc, as initially being small but finite linear quantities, from which relations are established, before undertaking a limiting process where dx, dy, etc are made infinitely small, while their ratio can remain finite. ]

Corollary 5.

124. With the increment of the speed dc found due to the force, the angle BAD of the change of direction of the point from the original direction represented by AB should be considered also, which likewise is found from the force. Truly the sine of the angle is Bb  kdz .

equal to AB cdt

Corollary 6.

125. Therefore there is a two-fold effect of the force affecting the motion of the point. One way is in agreement with the change in the speed, and the other with the change in the direction of this point. The first gives a change of the speed dc  second gives the declination of the sine of the angle kdz . cdt dz (1 kk ) , the dt

Corollary 7.

126. If the angle BAC is right then likewise k = 1, and dc = 0. Therefore in this case the speed remains unchanged by the force. Truly the sine of the angle of declination BAD dz . will be cdt

Corollary 8.

127. If the angle BAC is obtuse or greater than a right angle, then the cosine of this angle (1 kk ) is negative, and therefore the increment of the speed dc will be made negative, dz (1 kk ) . Which shows that the speed is diminished by the force. dt Declination kdz remains the same as it was before. cdt and equal to 

Corollary 9.

128. If the direction of the force AC agrees with the direction AB of the point A, this makes k = 0. Therefore in this case [p. 50] the direction of the motion is not changed by , if the the direction of the force. Truly the increment of the speed dc becomes equal to dz dt direction of the force agrees with the direction of the motion. But if moreover it should be in the opposite direction, then it becomes dc   dz . dt

Scholium 2.

129. Thus it is apparent from the solution to this proposition, how the absolute effect on a point in any kind of motion ought to be found, if the effect of the same force on the point at rest were known. Hence on account of this for the following propositions of this chapter, it will be sufficient for the force acting on a point to be placed either at rest or to be moving in the same direction as the force.

For if the point A (Fig. 14) has the speed c and is moving along the direction AB; meanwhile truly it is acted upon by a force in the same direction AB, thus so that in the passing of a small interval of time dt it will not be at B that the body will be found, for only with the speed c will the width be traversed, but at b, and the effect of the force will be the small distance Bb.

By the same small distance ao the point A, if it were at rest at a, would have been treated in the same small increment of time dt. Therefore from the motion of the point A by the force the effect of the same force acting on the point at rest will be known, and again hence the effect of the force on any moton of the point.