The Tangential Force

DEFINITION 21.

543. A body describes the curved line AMB (Fig. 47) when acted upon by a force. The tangential force on the body is the component of the force along the direction of the tangent TMt to the curve at the point M.

Corollary 1.

544. Therefore the tangential force exerts no other effect on the body while the element Mm is traversed, except that the motion of the body is either accelerated or retarded, since clearly the body is pulled either following the direction of MT or Mt.

Corollary 2.

545. Therefore when a body is moving along a curved line, the tangential force continually changes its direction and exerts its influence at another place.

Scholium.

546. Indeed a tangential force is hardly ever able to arise of its own accord in the nature of things; truly nothing is known of this wider use. For a force acting has a direction that can always be resolved into two parts, one of which is placed along direction of the tangent. [p. 226]

DEFINITION 22

547. The normal force is the force acting on the body describing the line AMB (Fig. 47), the direction of which MR is normal to the element of the curve Mm or the tangent MT

Corollary 1

548. Therefore the normal force can neither increase nor decrease the speed of the body, since its direction MR is always at right angles to the direction of motion. (164).

Corollary 2.

549. The effect of this force is agreed upon, as we show in what follows, as only the direction of the body can be changed and affected, because by itself the body progresses along a straight line, and the action of the normal force makes it move along a curve

Scholium 1.

550. If a body moves in the same plane, and also the directions of the forces acting on it are in put in the same plane, then the individual forces can be resolved into two parts, one of which is the normal, and the other the tangent, as is apparent from the principles of statics. Whereby when we have determined the effect of the tangential and normal forces on the body, then likewise also, [p. 227] the effect of any oblique force is also known. Moreover we call all forces acting on the body oblique, which are neither along the normal nor the tangent.

Scholion 2.

551. Hence the first division of this chapter arises. For in the first part we consider these motions that have their paths in the same plane, and likewise all the forces are agreed to be acting in the same plane. Following this, we are to consider motions that follow paths that do not lie in the same plane; for which it is understood that it is not sufficient to resolve the individual forces into two parts, for these are required to be resolved into three parts, on account of the three dimensions in which the body is moving.

PROPOSITION 70.PROBLEM

552. If a body, as it traverses the element Mm (Fig. 47) in a plane, is acted on by two forces, the one normal and the other tangential, to determine the effect of each in altering the motion of the body.

SOLUTION

Let the speed of the body describing the element Mm correspond to the height v, the force pulling along the normal MR is equal to N, and the tangential force pulling along MT is equal to T, the element Mm = ds and the radius of osculation at M = r. [Which we now usually call the radius of curvature.]

To determine the effect of the force N, since the direction of this is along the normal at Mm, we use the formula (165), which is npr = Ac 2

This becomes pr = 2 Av (209). But here N has been put in place of A for us ; for we understand by N not only the impression of the force N on the body, but the strength of the acceleration or the absolute force divided by the mass of the body, [which is A] (213). p Whereby here in place of A must be substituted N, with which done we have : Nr = 2v .

Q. E.D. for the first part. [Thus in modern terms, we have the centripetal acceleration at the point M , ac = V 2 / r , where V is the tangential speed at this point.] Then to determine the effect of the tangential force T, I use this rule : Acdc = npds (166), or in place of this, it is modified to give : Adv = pds (209). And on p account of the reasons offered, here in place of A I substitute T, and it becomes dv = Tds . Q. E.D. for the second part. [We may wish to consider Tds as the increase in the kinetic energy of the body supplied by the force T acting along the tangent on a unit mass through the increment ds, while dv is the corresponding change in a hypothetical gravitational potential energy of a unit mass under unit gravitational acceleration. One needs to look at Euler’s work in Ch. 2 to see how relations equivalent to these can be found without recourse to work–energy relations.

Corollary 1

553. Because the acceleration of gravity is put equal to 1 [here called force, but in the sense force per unit mass], the normal acceleration is to the acceleration of gravity as the height corresponding to the speed to half the radius of osculation [curvature] ; which ratio follows from the equation Nr = 2v . [In modern terms, the centripetal acceleration ac = V 2 / r = 2 gh / r = 2.1.v / r = N . ]

Corollary 2

554. Therefore with the given acceleration to the normal, and the curve that the body describes, the speed of the body is known at once, for the speed is expressed by v , and v= Nr . 2

Corollary 3

555. Truly the increment of the height corresponding to the speed is always equal to the product of the tangential acceleration and the element of distance traversed by the body [p. 229]. Or the element of v is to the element of distance described ds as the tangential acceleration is to the acceleration of gravity.