Proposition 98

Table of Contents

PROPOSITION 98. Problem

If a body is moving on the surface of a solid of revolution, of which the axis is the vertical line AL (Fig. 96), in vacuo with gravity g acting uniformly, to determine the motion of the body on a surface of this kind.

Solution

The solid of revolution is generated by the rotation of the curve AM about the vertical axis AL; all the sections of this are horizontal circles, of which the radii are applied lines of the curve AM. Therefore the equation expressing the nature of this surface is dz = xdx + ydy , Z with Z some function z; indeed it is given by : ∫ 2 Zdz = x 2 + y 2 = LM 2 . Therefore if the equation is given between AL = z and LM = 2 Zdz for the curve AM , then Z is also given. [p. 489] With these put in ∫ y place, therefore P = Zx and Q = Z ; from which values if substituted the two following equations are obtained, from which the described curve as well as the motion on that surface become known : v = b − gz and Whereby this becomes : Putting x 2 + y 2 = u 2 ; the function u is a certain function of z, clearly ∫ u 2 = 2 Zdz , and the above equation changes into this : Now the projection in the horizontal plane is obtained through the equation between x ∫ and y, if from the equation x 2 + y 2 = 2 Zdz the value of z is substituted in terms of x

and y ; and the arc of this projection is page 746 ∫ (dx + dy )

Moreover in the vertical plane the 2 2 projection is obtained by eliminating y; with which done the equation is produced : which equation, if it is divided by u, allows the construction. Now the pressing force, that the surface sustains towards the axis, is equal to : Q.E.I.

Corollary

If the increment of the time in which the element Mm is completed, is put as dt, then and the integral of this is : [From equation (876) v = c3( dx 2 + dy 2 + dz 2 ) it follows that the constant α = c ( xdy − ydx ) 2 c . Note in the O.O.] Therefore the time becomes as : ∫ with ydx denoting the area of the projection in the horizontal plane. Corollary 2. [p. 490]
If the body is considered to be moving in the projection in the horizontal plane, then the speed of this at Q corresponds to the height from which motion in the projection the motion on the surface itself can be found.

Corollary 3

Therefore let BQC (Fig. 97) be the projection of the curve in the horizontal plane, in which the body is moving, thus so that the motion of this corresponds to the motion of the body on the surface itself; the time in xy ∫ which the arc BQ is completed, is as 2 − ydx ,or with the negative as ∫ xy ydx − 2 , i. e. as the area BAQ drawn by the radius AQ. Corollary 4.
Moreover the element of the area BAQ is ydx− xdy . Therefore with the tangent QT 2 drawn, and on sending the perpendicular AT to that from A, then 3 3 Whereby the height corresponding to the speed at Q is equal to c 2 = c 2 on putting AT p AT = p.

Corollary 5

Therefore the body moving in the projection likewise is moving on that surface, and if it is moving freely attracted by some centripetal force to the centre A (Book I. (587)).

Corollary 6. [p. 491]

The point B corresponds to the start of the motion made on the surface, and since the direction for the first motion on the surface is given, the perpendicular to the tangent at B is given. Therefore let AB = f and the perpendicular to the tangent is equal to h ; then the 3 height corresponding to the speed at B is equal to c 2 . h

Corollary 7.

Therefore the centripetal force tending towards A, which acts, as the body in the projection BQC is moving freely, is equal to 2c 3 dp with u put in place fore p 3 du ( x2 + y2 ) . Now the equation between u and z expresses the nature of the curve, and the rotation of this curve thus has given rise to the proposed surface, and is therefore given.

Corollary 8

Again it is the case that These values, if they are substituted in the equation found, give or

Corollary 9

Therefore this quantity, with the differential of this divided by du, gives the centripetal force required at A, as the body in the projection BQC is moving freely, in a motion corresponding to the motion of the body on the surface.

Corollary 10

If c = 0, also making p = 0. Whereby in this case the projection is in the horizontal plane with a straight line passing through A, upon which the body thus approaching A is thus attracted, as it is the centripetal force

Corollary 11. [p. 492]

If the direction of the body is first horizontal, the tangent at B is normal to AB and thus h = f. Now in this case the speed on the surface is equal to the speed in the 3 projection ; whereby if i is the value of z, and if u = f, then b − gi = c 2 . f

Corollary 12.

If besides the centripetal force at B is equal to the centrifugal force, then the curve BQC is a circle and therefore so also the curve described on the surface, and both the motion on the surface as well as that in the projection are equal. Let it be the case, when u = f and z = i, that dz = mdu ; as p = f and u = f and z = i in the case in which a circle is described, then 2c3 = gmf 3 . [See (892)]

Corollary 13.

If there is put in place π : 1 as the ratio of the periphery to the diameter, then the perimeter of our circle is equal to 2πf , which divided by the speed gives the time of one period in the circle, that hence becomes equal to c 3 , i. e. f2 2π 2 f gm gmf , 2 . Hence the f pendulum completing whole isochronous oscillations with this period is equal to m in the same hypothesis of gravity. Corollary 14. [p. 493]
Therefore on the surface which is generated by the rotation of the curve AM (Fig. 96) about the vertical axis AL, a projected body can describe a circle of radius LM in the same time that a pendulum of length equal to the subnormal LS can complete a whole oscillation.

Corollary 15.

Therefore if the curve AM is a parabola, all the periods along horizontal circles in the parabolic conoid are completed in equal times; and pendulums in the same times by performing whole oscillations with a length equal to half the parameter.

Scholium 1.

Whatever curve AM is assumed, if the centripetal force motion in the projection towards A is defined from a given formula and that is put equal to the centrifugal force 3 and joined together with the equation b − gi = c 2 , there is produced 2c3 = gmf 3 ; for in f this case as the projection is a circle as is the curve described on the surface. Now so that this can be more apparent, there is put dz = qdu, and the (884) centripetal force comes about equal toEULER’S MECHANICA VOL. 2. Chapter 4b. Translated and annotated by Ian Bruce. page 750 Now putting u = f, z = i and bf 2 = gf 2i + c3 and q = m; the centripetal force is equal to : 3 which put equal to the centrifugal force 2c2 , gives 2c3 = gmf 3 . f Example. [p. 494]
Let the surface be a circular cone or AM (Fig. 96 and Fig. 97) a right line inclined at some angle to the axis AL; whereby then z = mu and dz = mdu. Therefore the centripetal force tending towards A which is produced, in order that the body projected in BQC is moving freely, is equal to Therefore this force is composed from a constant force and from a force that varies inversely as the cube of the distance from the centre A. If c = 0, then the projection is a right line passing through A and the centripetal force is equal to gm , or is 1+ m 2 constant. Therefore the body approaches A with a uniformly accelerated motion; now the motion on the surface of the cone agrees with the descent or the ascent on a right line inclined equally and the acceleration is the same. But if the projection is curvilinear and the tangent at B is normal to AB, then 3 i = mf and b − gmf = c 2 on putting AB = f. Therefore in this case we have f b= c + gmf f2 3 3 , and if the body is revolving in a horizontal circle, it becomes above: 2c3 = gmf 3 . Hence it comes about that the speed of the body corresponds to the height c 3 = gmf . And the periods completed in the same times in this circle, in which 2 f2 f pendulums with lengths m complete whole oscillations. But if it is not the case that

3 3 c gmf + 2c3 = gmf 3 , now still b = , the curve BQC has a certain tangent at B normal to f2 AB, but this projection is not a circle. But if 2c3 is almost equal to gmf 3 , then the curve does not depart much from a circle ; but it has various apses, at which the tangent is perpendicular to the radius. Now the positions of these apses has been determined by proposition 91 of the preceding book. For since the centripetal force is if this is compared with that centripetal force P3 , [p. 495] on account of y = u then y and thus on putting u = f , the line of the apses also is distant from the preceding apse by the angle But since 2c 3 is approximately equal to gmf 3 , the angle intercepted between the two 2 2 apses is equal to 180 m 3+1 degrees. Hence as m 3+1 is always greater than 13 , then the angle intercepted is greater than1030 55’. Scholium 2. 894. I do not include here the example of the surface of the sphere, but I resolve the motion on that boundary in the following proposition, since this matter is worthy of being handled most carefully. For if a pendulum is not set in motion following a vertical plane, then the body moves on the surface of a sphere and the body describes a circle or another not less elegant curves, as becomes known from any experiment set up. Indeed the case, in which a pendulum completes a circle, has now been published in the Acta Erud. Lips. A. 1715, [p.242] by the most celebrated Johann Bernoulli under the title De centro turbinationis inventa nova. [Opera omnia, Tom. 2, p. 187]. But if the curve is not circular, no one as far as I know, has consider either the motion of this pendulum nor determined it. [Isaac Newton had considered an analogous problem in the Principia, London, 1687, Book. I sect. X Prop. LV and LVI, in connection with the apses of the moon, where gravity follows the inverse square law and therefore is not constant; see Cohen’s translation and commentary.]

Definition 6. [p. 496]

The rotational [or whirling] motion of a pendulum is the name given to the motion imparted to a pendulum which is not in a vertical plane. Therefore in this case the pendulum is not moving in the same vertical plane, but describes some curve on the surface of the sphere of which the radius is the length of the pendulum in place. [This is now called the spherical pendulum.]