Proposition 26

PROPOSITION 26. Problem.

231. With the curve AM (Fig. 32) given, and with the initial speed at A corresponding to the height b, to find the size of the force always acting downwards, which arises in order that a body descending along the curve AM exerts the same force everywhere on the curve.

Solution

Let the force acting sought be equal to P, and with these lengths named : AP = x, PM = y et AM = s and the force that the curve sustains is equal to k , this equation is put in place : (224), in which ds is made a constant element.

Therefore from this equation the quantity P has to be elicited. [p. 102] Moreover, the equation multiplied by dy and integrated gives : from which there arises : which differentiated gives : But the integral kds dy 2 ∫ dxdy thus has to be taken, in order that with x = 0 it becomes ∫ dxdy = b. Moreover , so that this integration can advance easier, putting dy = pdx ; then we have and thus, From which equation there is produced :EULER’S MECHANICA VOL. 2. Chapter 2c. Translated and annotated by Ian Bruce. page 149 Q.E.I. Corollary 1. 232. From this equation the speed of the body at the individual points is also found at once; for the height corresponding to the speed at M is Now the time, in which the arc AM is completed, is equal to Corollary 2. 233. It is evident from the equation found that the magnitude of the force P is therefore to be greater where k is greater from the other terms; for the value of this variable has been multiplied by the compression k. Corollary 3. 234. Although now the force P is not seen to depend on the initial speed b, [p. 103] because b is not present in the expression, yet P depends on b on account of the integral ∫ ( 1+ pp ) , which must thus be taken, in order that with x = 0 it becomes k ( 1+ pp ) pdx ∫ ( 1+ pp ) = b. Hence with the initial speed varying other forces acting are p pdx 2 produced, even if the the proposed curve remains the same. Exemplum 1. 235. Let the curve AM be a parabola having the vertex at A and the axis horizontal, thus so that is is given by ay = x 2 . Therefore we have dy = 2 xdx and hence p = 2ax and a Whereby

Since this quantity must be equal to b, if x = 0, then C = −2a or

From which it is found : Therefore at the point A the force P is indefinitely small; as the numerator as well as the denominator vanish the value of this expression is equal to zero. Now the speed at A cannot be made arbitrary, also if the the constant C is seen to be determined from b. For C has only such a value which the expression returns of finite magnitude. Therefore b depends on a and the value of this can be found, if we put x = 0 in the expression Moreover, there is produced then b = ka . [p. 104] Therefore the descent must begin with 4 this speed, in order that the compression that arises is everywhere equal to that found from the force P. Example 2. 236. Let the curve AM be a circle of radius a touching the line AP at A; then it is given by y = a − ( a 2 − x 2 ) and p = x , and also ( a − x2 ) 2 ( 1 + pp ) = a . ( a − x2 ) 2 Therefore the integral 1+ pp 2 to which it is not required to add a constant, because pp = a 2 becomes infinite with x x vanishing. Therefore the equation becomes : whereby the speed of the bvody is uniform and thus the force acting vanishes. It is evident that the body is progressing uniformly on the circumference of the circle with no force acting and the centrifugal force everywhere is of the same magnitude.EULER’S MECHANICA VOL. 2. Chapter 2c. Translated and annotated by Ian Bruce. page 151 Example 3. 237. Let the curve AM be a cycloid having the base horizontal and with the cusp a tangent to the vertical AP at A, thus so that the equation becomes : Therefore we have : Whereby the equation becomes : and Therefore with the constant C taken of finite magnidute making b =∝ ; whereby making C = 0 ;then and b = 0. Therefore there is produced : Thus if the body descends on the cycloid AM from A from rest and is acted on by a downwards force [p. 105], which varies as the square root of the abscissa AP, the body is everywhere acted on by a constant force. Scholium. 238. Therefore cases are given, in which the speed b cannot be assumed at will, as has 1+ pp come about in these examples. For as often as pp is made infinitely large by making x = 0, a constant in the integraton of pdx has to be added, and generally this is itself ( 1+ pp ) determined because the initial speed cannot be infinitely large. But always, if the curve is 1+ pp a tangent to the line AP at A, then pp becomes infinite with x = 0, since that is also the reason in other examples considered why the initial speed cannot be made arbitrary.EULER’S MECHANICA VOL. 2. Chapter 2c. Translated and annotated by Ian Bruce. page 152