Proposition 72

Table of Contents

PROPOSITION 72. Problem

If the curve AM (Fig.71) is given, upon which the body is moving in vacuo, to find the curve am, upon which the body moves in a medium with resistance thus descends, so that the speed at a is equal to the speed at A and with equal arcs AM and am taken, so that the speeds at individual points M and m are also equal.

Solution

With the vertical axes AP and ap drawn, and with the horizontal lines MP, mp let AM = am = s, AP = t and ap = x; the given curve AM is given by means of the equation between s et t. Now the speeds at the points A and a correspond to the height b and the speeds at M and m correspond to the height v. Let the absolute force acting downwards be g and the resistance as the 2mth power of the speeds. With these put in place, then for the motion in vacuo upon the curve AM: dv = – gdt or v = b – gt

and with the descent in the resisting medium upon the curve ma :

… dv = − gdx + v mds ; k

in which equation if in place of dv and v there are substituted values found from the previous equation, there is produced :

− gdt = − gdx + (b − gt ) m ds (b − gt ) m ds or dx = dt + . m k gk m Since moreover the equation is given between t and s, if in place of t the value of this in terms of s is substituted, the equation is obtained between x and s for the curve sought am. Q.E.I.

Corollary 1

If on the curve AM the point B is the start of the descent and thus the height of this above A = bg , there is also obtained the start of the descent b on the curve am, on taking the arc amb = AMB.

Corollary 2

From the solution it is apparent that dx > dt always; whereby the height ap is greater than the height AP; for in the medium with the resistance there is a need for more height to generate the same speed as in vacuo.

Corollary 3

Because on the curves AM and am taken with equal arcs the speeds at these places are the same, the times too are equal, in which the equal arcs AM et am are described. And thus the time of the descent in the medium with resistance along bma is equal to the time of the descent in vacuo along BMA.

Corollary 4

Lest the curve bma becomes imaginary, it is necessary that everywhere dx < ds. On this account it is necessary that : …

Moreover this is thus obtained, if it is the case that [p. 343] gk m dt < (gk m − b m )ds , which is so if t vanishes and [the inequality] pertains to the point a, unless t has a negative value somewhere. Concerning this it is only required to be considered that the point a is made real, which comes about if gkmdt is not greater than ( gk m − b m )ds .

Corollary 5

Therefore lest the curve am becomes imaginary, before everything it is necessary that b < k m g . At the point A, let ds = αdt ; α is a number greater than one, and thus

…

Therefore if the curve MA in A has a horizontal tangent, it must be the case that b < k m g on account of α =∝ .

Corollary 6

Moreover, if ds = αdt at the point A, let b m =g ( α −1 ) ( α −1 )ds= ds . … then at this point a

Therefore in this case the curve am has a vertical tangent at a.

Corollary 7

At the start of the motion at B let b = gt or t = bg . Therefore for the point b, dx = dt with the elements of the curves taken equal. Whereby the tangents at the points B and b are equally inclined.

Scholion 1

Since it is not possible to completely construct the curve am from the curve AM, for besides it is necessary to know the speed at the point A or at the start of the descent B, if another starting point of the descent is taken on the curve AM then another curve am is found. Therefore for this reason the curves BMA and bma are thus only in agreement for a single descent, in order that the speeds in which the intervals are traversed are equal to each other, and if the starts of the descents are placed at other points, then this agreement is no longer in place. Therefore two curves are not given, upon which all the descents as far as to a given point are in agreement with each other, the one in vacuo, and the other placed in a resisting medium.

Example 1

Let AMB be a straight line inclined at some inclination, thus so that s = αt , and the curve amb is sought, upon which the body in a like manner is progressing in a resisting medium upon as the above AMB in vacuo. Moreover in putting αs in place of t, the following equation is produced between x and s for the curve sought amb : and the integral of this is :

Lest the point a becomes imaginary, it is necessary that :

For if it happens that

…

then the curve has a vertical tangent at a nor therefore can b have a greater value. Therefore the body can put on the inclined line BMA to descend from such a height, as it becomes :

…

then the equation becomes :

which is the equation for the curve amb, at which the start of the descent is taken from b, where ds = αdx , or the arc amb is equal to: km α m−1 (α −1) g m−1 . If the resistance were proportional to the squares of the speeds, then m = 1 and thus

…

and on integrating : x = s − 2ss . αk Which is the equation for a cycloid described on a horizontal base, the diameter of the generating circle of which is α2k .

Corollary 8

Therefore let the cycloid AMB be described on the horizontal base CB (Fig. 72) by the generating circle ANC, and let the medium be resisting in the ratio of the square of the speeds, the exponent of which is equal to k.

If now in the circle ANC the chord AN = k2 is taken, and the horizontal PNM is drawn and from M the tangent MT and the two bodies are placed to descend, the one on MT in vacuo and the other on the curve MB in the resisting medium, both these bodies complete equal distances in equal times. [p. 346]

Example 2

Let AMB (Fig. 71) be a cycloid considered downwards, the diameter of the generating circle of which is α2 ; then we have ss =2at , t = 2ssa and dt = sds . Hence with a these substituted there is produced the following equation for the curve amb : or if the whole arc AMB, which in vacuo is put for the completed descent, is called c, then gcc b = 2a and thus for the curve amb this equation arises :

Therefore lest this curve becomes imaginary at the point a, it is necessary that g m−1 < 2m a m k m or the height of the arc AB must be less than kg m g ; for if the height of the arc AB = kg m g , then the tangent of the curve amb is vertical at b. Therefore if B is the cusp of the cycloid, then 2cca = a2 or c = a, and if in addition g m−1 a m = 2 m k m , in order that the tangent of the curve amb becomes vertical at a, this equation is obtained : which curve has vertical tangents at a and b.

Corollary 9

Therefore when the body thus descends on the cycloid, in order that the accelerations of this body are proportional to the distances traversed, then descents on the curve amb in the resisting medium have the same the same property in place, if the start of the descent is taken at the point b, that is determined through the equation to the curve amb; clearly it is amb = c.

Corollary 10

If another starting point is taken at B on the curve AMB, another whole curve amb is found, since the length of the arc AMB = c is contained in the equation of this. Whereby, even if the cycloid is a tautochronous curve in vacuo, yet such a curve amb is not [a tautochrone] in a medium with resistance, since for several descents on the same curve AMB as many different curves correspond in the resisting medium.

Scholium 2

In this example the curves have been elicited that the Cel. Hermann in Comm. Book II found for tautochrones in resisting mediums [Jacob Hermann, General theory of the motion that arises with forces acting constantly on bodies.

But likewise he has shown that it is not possible to give a satisfactory answer to the question. Furthermore from these is understood, in a like manner, that it is possible to find in a curve in a resisting medium, upon which a body on ascending is moving in the same way as that on a given curve in vacuo. [p. 348] For A and a are the initial points of the ascent, that one in vacuo, and this in a resisting medium, and let the initial speed correspond to the height b; this equation is obtained for the curve amb :

from which equation it is understood that the curve amb does not become imaginary, unless the curve itself AMB were such. For because, lest the curve amb is imaginary, it must be that dx < ds, here dx is less than dt, that per se must be less than ds. As if the line

AMB is a vertical line, the other amb is possible to be assigned; for on putting AB = c then b = gc and s = t; wherefore for the curve amb this equation is found : the integral of which is :

This equation can be adapted to resistance proportional to the square of the speed; let m = 1 and thus which is related to the cycloid in this way: the cycloid AMB (Fig.72) is described with the generating circle of diameter AC = k2 upon the horizontal base BC; then the arc AM = k – c is taken; then M is the start of the ascent, from which point, if the body ascends on the curve MA with a speed corresponding to the height gc, in the medium resisting as the square ratio of the speeds, the body is moving in the same way as in vacuo with the same initial speed, rising vertically up.