Chapter 5b

Proposition 71?

Motion is the translation of a body from the place it occupies to another place. True rest is a body remaining at the same place.

Euler
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PROPOSITION 71. PROBLEM

  1. If the body, while it traverses the element Mm (Fig. 47), is acted on by some oblique force in the direction MC, then it is required to determine the effect of this force in changing the motion of the body.

SOLUTION

Let this oblique force be in the ratio to the force of gravity, if on the surface of the earth, as P to 1, the element Mm = ds and the speed at M corresponds to the height v. Since the obliqueness of the force MC has been given, the angle CMT is given, and on account of this the triangles CMT and Mmt, which come from the perpendiculars dropped from C to MT and from m to MC, are as shown.

We can therefore put Mr = dy and mr = dx, then ds 2 = dx 2 + dy 2 , and the ratio between both ds, dx, and dy is given. They are now brought together with these equations (161, 163) that have been presented before (208). For these that we wish to find here are completely similar to these previous ones,; only in this respect do they differ, as for us here the ratio is P:1, while there it was p : A. On this account, we have : dv = Pdy ,

and with the radius of osculation MR at M put equal to r this equation is found : …

(208), I write P in place of A . Of these equations : dv = Pdy defines the increment of the speed, as the body travels through the element vds shows the lines of the curvatures Mm. [p. 230]

Truly the other, Pr dx = 2vds or r = 2Pdx at M described by the body. Hence the whole effect of the oblique force on the motion of the body can become known. Q. E. I.

[The component of P along the curve at M is Pcosθ = Pdy / ds : hence at .ds = dv , where at is the tangential acceleration, as in the previous chapters ; the component of P normal to the curve, or the normal acceleration an , is given by an = P sinθ = Pdx / ds = V 2 / r = 2v / r , where P is taken as the acceleration of a body of unit mass along MC. ]

Corollary 1

  1. If the oblique force is put in place at an obtuse angle with the element Mm (Fig. 48), everything remains as before, except that the line element mr =dy must be taken ( 1 − λ2 ) = μ . Hence dv = − Pdy , and the other equation Pr dx = 2dvds remains as before.

Corollary 2

  1. Therefore if the direction of the force MC falls between the normal MR and the element Mm as in Fig. 47, then the motion of the body is one of acceleration. But if CM falls outside each, as in Fig. 48, the motion is one of retardation.

Corollary 3.

  1. If the direction of the force MC falls on the tangent MT, then the angle mMr, Mr is made equal to Mm or dx = 0 and dy = ds. Therefore we have dv = Pds and r =∝ , which indicates that the direction of the body is not affected by this tangential force.

Corollary 4.

  1. If the direction of the force MC (Fig. 48) on the other part Mt, in which case Mr = dx = 0 and dy = ds , giving dv = − Pds and r =∝ as before. Therefore only the tangential speed of the body is affected, and the direction of the motion is clearly not changed.(544). [p. 231] Corollary 5.

  2. The direction of the force MC falls on the normal MR, where the effect of the normal force is known. Hence in this case dy = 0 and dx = ds . And consequently dv = 0 and Pr = 2v is produced. Hence the normal force does not affect the speed, but only the direction of the motion. (548). Scholium 1.

  3. Therefore both the effect of the normal force and of the tangential force on the motion of the body is known. On which account, when all the forces, as many as act on the body, are put in a plane in the same way in two’s, they can be resolved with one part normal and the other tangential, and the effect of any forces on the motion of the body can become known.

Scholium 2

  1. Hence it will be most convenient for motion in the same plane to be subdivided, as in the first place the directions of all the forces acting are parallel to each other, as in our region, the directions of the weights are observed to be parallel to each other. Then we will consider the case, in which the directions of all the forces converge to a single point, to which also the body is always attracted, and which is the case of the centripetal force, such as the singular discoveries made by Newton Part I of the Princ. Phil. Nat. Truly in the third place we introduce and investigate any forces acting on a body [p. 232], which the motion shall always be about to follow. Moreover we will turn from these individual cases, as at first we propose direct questions, then indeed also inverse questions, as far as we have done at this stage, that we may be able to resolve. Always finally, as much as is permitted, we will progress from the more simple to the more complicated and difficult cases. [Euler now solves simple projectile motion using his general equations.

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