Proposition 27

Table of Contents

PROPOSITION 27. Problem.

If a body is always drawn downwards by some force, to find the curve AM (Fig. 32), upon which the body is thus moving, in order that the total compression force sustained by the curve has a given ratio to the compression arising from the normal force.

Solution

The body descends from A with a speed corresponding to the height b and on placing AP = x, PM = y and AM = s and let the force acting on the body at M be equal to P; [p. 106] the height corresponding to the speed, that the body has at M , is equal to b + Pdx , … now the total force of compresseion that the curve sustains at M following the direction of the normal MN, is equal to … on taking ds for the constant element.

This compression force is in the ratio m to 1 to the normal force ds ; hence …

which is the equation of the curve sought. Now this can be reduced by putting v in the place of … b + Pdx , to this form : …

which integrated gives :

…

From which there is obtained [as ds = sin θ = v m−1 , etc.]: a 2

which is the equation of the curve sought. Q.E.I.

Corollary 1

The speed of the body is zero there, where ds = 0 or where the tangent of the curve is vertical, if indeed m2−1 is a positive number or m is greater than one. Therefore in these cases we put the curve to be a tangent to the line AP at A and the initial speed or b = 0.

Corollary 2

Whereby if m > 1, or if the total compression force is greater than the normal force arising, the curve sought is given by this equation : in which ∫ Pdx thus must be taken so that it vanishing with x = 0.

Corollary 3

Whereby if m = 1, the centifugal force vanishes and therefore the line sought is straight. Moreover from the equation we have ddy = 0, which is the property of a straight line.

Corollary 4

If m = 0, then the total pressing force vanishes; whereby there is then produced the curve, that the body freely described, projected with a speed corresponding to its height b. Therefore for this curve this equation is found :

Corollary 5

If m is less than one, then the centrifugal force is in the opposite direction to the normal force and therefore the curve AM is concave downwards. Therefore we put the curve to be normal to AP at P ; and b = a. Therefore with b = a , this equation is found for the curve sought

Corollary 6

Therefore for the free motion, in which case m = 0, this curve described is found for the body, if it is projected horizontally from A with a speed corresponding to the height a, from this equation.

Example 1.

Let the uniform force acting or P = g; then we have Pdx = gx. Hence in the cases in which m > 1 and the body descends from being at rest at A, the equation for the curves sought with gc written in place of a is this :

But if m < 1 and the body is projected from A horizontally with a speed corresponding to the height a, the curve upon which the body must be moving, with gc written in place of a, is shown by this equation :

These curves are therefore algebraic, if either 23m−−m2 or 1−mm is a positive whole number. Now this comes about if m is a term of either this series 3, 53 , 75 , 97 , 11 etc. , or of this

series 0 , 12 , 23 , 34 , 54 , 56 etc. [e. g. on setting the expression squared equal to xm]

Corollary 7

Therefore if the total compression force is three times greater than the normal force, then the curve is a circle touching the line AP at A.

For it becomes the equation of the circle of radius c

Corollary 8

If the total compression force is twice as great as the normal force or the centrifugal force is equal to the normal force acting in the same way ; then the curve is a cycloid with the vertical cusp a tangent at A. For the equation is given by :

Example 2.

Whatever the force acting P should be, curves of the same kind are required, in order that the total compression that the curve sustains, is twice as large as the normal force or as the centrifugal force, which in this case that will be the equation.

Therefore putting m = 2 , and this equation is found for the curve sought :

Or by calling Pdx = X then

We have brought up this example, since in the following curves with this property are likewise lines of the quickest descent.

Corollary 9

Therefore it is evident that there are endless curves satisfying the question, on account of the quantity a being arbitrary. And all these boundless curves have a tangent to the line AP at A.

Scholium 1

It is apparent from the solution of this problem, how the inverse problem can be solved, in which the curve and the ratio between the total compression force and the normal force is given, and the magnitude of the force acting downwards is sought. Since it is or putting dy =pdx it becomes and hence by differentiation

Consequently it is found that …

Where it is to be noted that the initial speed must now be given ; for the formula gives b if we put x = 0.

Scholium 2

In a like manner, if the motion of the body or the speed of this is given at individual points and the relation between the total pressing force to the normal force is given, from the speed the force acting is found at once. For let v be the height corresponding to the speed at M; then as b + Pdx = v , P = dv and the equation ..

gives the nature of the curve sought. For since v is given, either x or s and constant quantities have to be given, clearly which are used in expressing the nature of the curve.

Moreover the same problems proposed under the hypothesis of centripetal forces or of many forces acting do not introduce more difficulties, even if more complex equations may be reached.

Since a simple example in a medium cannot be brought forwards as an illustration, this I rather abandon, and more towards that [study] which I am about to set out with great diligence in the following, where the nature of the brachistochrone is worked through, and curves of the same kind are produced.

Therefore I progress to that problem, in which a certain property of the motion is proposed, from which conjointly either the curve is sought with the force acting, or from the curve itself the force acting.

The exceedingly easy problems as when either the scale of the speeds or of the times is given, I omit, since from the expression of the speed or the force acting either the curve itself comes freely, and an expression of the time to the speed can be easily deduced. Because of these things, we bring forwards questions, in which neither the speeds nor the times are given, but certain relations depending on these.