Proposition 10

PROPOSITION 10. Theorem

74. The force that a body moving on a surface under the action of no external forces exercises on the surface is made normally to this surface towards the convex side and has the ratio to the force of gravity as the height corresponding to the speed of the body is to half the radius of osculation of the curve described by the body.

Demonstration

Let DMm (Fig. 7) be the curve on the surface ABC described by the body, the height corresponding to the speed of the body is equal to v and the radius of osculation of the curve MO is equal to r. Because the body is able to move freely from M, it can progress along the element Mn, now the surface brings it about that the body advances along the element Mm, with the distance nm being perpendicular to the surface, and the surface is required to exert a force of such a size along mn that the body follows the direction of Mm along the surface, departing from the direction Mn. [p. 33] Now this is performed by the force 2rv acting normally to the surface along the direction of the radius of osculation MO. On this account the force of the body is normal to the surface , clearly acting along mn, and is equal to 2rv with the force of gravity acting on the body taken as equal to 1. Q.E.D.

Corollary 1

75. This is therefore the centrifugal force that the body exerts on a surface in a similar way to that when it is forced to move on a given line.

Scholium 1

76. The force acting on the surface must necessarily be normal. For unless it is normal, it is possible to resolve it into two components, of which one is normal and the other is placed along the surface. Now of these only the normal devotes itself to pressing on the surface, while the other changes the motion of the body.

Corollary 2

77. We find that the length of the line of the radius of osculation r, that the body describes with no forces acting on a proposed surface (73). With this assumed, the centrifugal force is equal to :

Scholion 2

78. This centrifugal force acting on the surface and the above centrifugal force acting on a given curve that has been discussed above, have the same place in the equation; see Prop. 2 (20) with the adjoining corollaries and scholium. For the shortest line that the body can describe on the surface can be considered to resemble the channels along which a body can move, and then all that has been said concerning the motion in these channels prevails, which have been produced above for the motion upon a given line with no external forces acting.

PROPOSITION 11. Problem.

79. To determine the effect of any kind of force that acts upon a body on a given surface either in a vacuum or in a resisting medium.

Solution.

For any body, the direction of the external force acting can be resolved into three parts : the first of which that we call M, the normal direction to the surface; secondly, we designate by N that direction normal to the motion of the body as well as being normal to M, and the direction of this is in the tangent plane of the surface; and the direction of the third force called T agrees with the direction of the motion, which is therefore the tangential force ; surely the first two are the normal forces. Now since the directions of these three forces are mutually normal to each other, neither is able to disturb the others. [p. 35] Whereby, we investigate the effect that each can produce.

The first external force M, the direction of which is normal to the surface, has no effect in changing the motion of the body, as the whole is expended in pressing upon the surface. Therefore M either diminishes or increases the force arising from the centrifugal force, as the direction of this falls on convex or concave parts of the curve. For that force

acting towards an inner part of the curve ; the total force acting on the surface towards the outside is equal to (77). For the force arising from the centrifugal force is diminished in this case by the force M.

The second force N, since the direction of this is put both normal to the direction of the surface and to the direction in which the body moves, can only change the direction of the body and neither increase nor decrease the speed. Therefore this force makes the body move along the shortest line deduced, so that it no longer moves in a plane normal to the surface; therefore it is necessary to find the inclination of this plane of the shortest line in which the body moves, to the normal to the surface.

This angle of inclination is equal to the angle that the radius of osculation of the line described makes with the normal to the curve, and which we have determined previously in general (71).

After the body describes the element Mm with a speed corresponding to the height v [Italic v], is progressing, unless acted upon by the force N , along the element mv (Fig. 11) to v [Italic Greek ’nu’ : Microsoft seems to think these two letters are the same.], thus as Mm and mv are two elements of the shortest line and placed in a plane normal to the surface; the direction of the normal force N in the plane; the direction of the normal force in the plane of the paper is reduced to that above, if indeed we put this force N above to be put in this position of the elements, as represented in the figure. Therefore this force has the effect, that the body is moving along the mμ and by the angle vmμ deflected by the direction mv. For this angle corresponding to the radius of osculation is . Whereby when the force N generates this angle and the speed of the equal to = mv …

curve corresponds to the height v, from the effect of the normal force it becomes

Where the inclination of the plane Nmμ , in which the body actually moves, to the plane Mmv, which with the normal in the surface is found, the perpendicular vn is sent from v to the element Mm produced ; μn is also in the perpendicular mn and thus the angle μnv is the angle of inclination of the plane μmM to the plane vmM; and since μv is μv normal to vn, the tangent of this angle is equal to nv = N2v.mv . But nv is determined from ..

the inclination of the elements Mm and mv or the radius of osculation of the shortest line, = r and of which Mm and mv are elements. Here the radius of osculation is r, it is mv

..

thus the tangent of the angle μnv is equal to with the value found substituted in place of r (73). [p. 37] For now this angle is equal to the angle, which the radius of osculation of the elements Mm and mμ actually described by the body agrees with the radius of osculation of the elements Mm and mv or with the normals on the surface.

Moreover we have found the tangent of the above angle (71). Whereby with the equation made we have from which equation the effect of the force N is determined. Or since it is the case that this equation is found The third force T, since it is placed in the direction of the body, either only increases or decreases the speed. We can put this force to be an acceleration, the effect of this is expressed by this equation :

…

If the motion is made in a medium with resistance and the resistance is equal to R, only the tangential force T is to be diminished by the resistance R. On which account we have : Q.E.I.

Corollary

80. Therefore from the two equations, from one of which v is determined, and from the other dv, the one containing v solved with the position of the surface dz = Pdx + Qdy [p. 38] determines the curve that the body describes on the above proposed surface.

Scholium 1

81. The force N needs to be attended to well, in which place it acts, that is whether it inclines either to the right or to the left hand region of the motion of the body. For this indeed the different tangent of the angle μnv either positive or negative has to be taken. Concerning which we will not now be concerned, but defer further inquiry of this to the last chapter of this book.

Scholium 2

82. Therefore we progress to the following chapter, in which we examine the motion of the body upon a given line in a vacuum. In the third chapter we investigate the motion of a body on a given line with a resisting medium. Finally in the fourth chapter we carefully examine motion on a given surface both for the vacuum and resistive medium cases.