An Introduction to the Theory and Practice of Mechanics

On Central Forces.

418. Defs. The point towards which any body is attracted, or solicited, is called the centre of attraction, or the centre of force.

419. Centripetal force is that force by which a body is solicited towards another, or to a fixed point as a centre.

420. Centrifugal force is a force acting directly opposite to the centripetal force, and would carry the body away from the centre were it not prevented by the centripetal force.

When a body in free space describes a curve returning into itself, the centripetal force is equal to the centrifugal force; and they are called by one common name, central forces.

421. The force which tends to bend or alter the course of a body at every instant, is called a deflecting force.

422. A body describing a curve, is acted upon by two forces at the same time, viz. a projectile and a centripetal force;-by the projectile force it would run out in a tangent to its orbit, but it is prevented from so doing by the centripetal force.

423. With respect to a single body revolving round its own axis, the centrifugal force of any of its particles may have to the centripetal force of those particles, any proportions of inequality.

424. The path described by a body acted upon by a central force, is frequently called its trajectory, or orbit.

425. The time employed by a body after passing a certain point in its orbit, before it returns to that point again, is called the periodic time.

426. Angular-velocity is the velocity with which a body describes the arc contained by the angle which is formed by two radii drawn from the centre of force to the orbit in which the body moves; and the angular velocity is greater or less, according as the angle passed over in the same time is greater or less.

PROPOSITION CII.

427. The central force varies with the quantity of matter in the central or attracting body.

For let m denote the quantity of matter in the attracted body, and M that in the attracting body; then, since the force with which m is attracted is as the number of particles in M, a body equal to 2M would attract the body m with a double force, and a body equal 3M with a triple force, and so on: therefore, the central force or force of attraction, is as the quantity of matter in the attracting body.

428. Cor. When m and M are equal, their forces of attraction are equal.

PROPOSITION CIII.

429. If a body revolve in an orbit by the joint effects of a projectile and centripetal force; it will describe equal areas in equal times, and, in unequal times, areas proportional to the times.

Let F (fig. 149.) be the centre of force, and ABCDE part of the orbit in which the body moves; when the body is at A, let the projectile force be such as would carry it from A to B in an indefinitely small space of time; then, because the motion is uniform, the body would describe BT in the same time in which AB was described, were it not drawn from the direction of the tangent AT,, by the centripetal force TC, acting in a direction parallel to BF; therefore, by art. 52, the body, by the joint action of the two forces BT, TC, will describe the diagonal BC, and in the same time that the projectile force alone would have caused it to describe BT.

In the same manner, after equal successive intervals, similar effects will take place at the points C, D, E, &c.

Now the triangular areas AFB, BFC, CFD, DFE, &c. described in equal times, are equal.

For join FT, then the triangles AFB, BFT, (having equal bases AB, BT, and terminating in the same point F, have equal altitudes) are equal. Also the triangles BFT, BFC, having the same base BF, and being between the same parallels BF, TC, are equal; therefore, the triangle AFB is equal to the triangle BFC.

In the same manner the other triangles CFD, DFE, may be proved to be each equal to the triangle AFB; and by composition, any sums of those areas are to each other as the times in which they are described, that is, the areas are universally as the times.

Let now the number of these triangles be augmented, and their breadths diminished indefinitely; then, the ultimate perimeter ABCD, &c. will be a curve line, which is always concave towards the centre of force F; and the above reasoning being still applicable to those triangles whose breadths are evanescent, the areas will be as the times.

430. Cor. 1. The velocity of the body in any point of its orbit, is reciprocally as the perpendicular from the centre F upon the tangent to that point.

For the area of any one of the triangles AFB, BFC, &c. being constant, the base, which represents the velocity, is reciprocally as the perpendicular demitted upon it from F.

431. Cor. 2. The times in which equal parts of the orbit are described, are directly as the perpendiculars from F upon the tangents AB, BC, produced if necessary.

For triangles upon equal bases are as their perpendicular altitudes.

432. Cor. 3. The central forces with which the body is drawn towards the centre of force F, in any part of its orbit ABC, are as the versed sines of the indefinitely small arcs described in equal times.

For in the parallelogram TCNB, TC=BN; and Bm=1BN, is the versed sine of the indefinitely small arc ABC; and this is the case every where; therefore, in every part of the orbit, the centripetal force is as the versed sine Bm of the indefinitely small arc ABC.

433. Scholium. From what has been said, it follows, that every body which moves in a curve, about a centre of force F, and by a radius drawn from the body to that point, describes equal areas in equal times, is urged by a centripetal force tending to that point.

But if the areas described are not proportional to the times in which they are described, the direction of the central force is not to the fixed point F, but to some other variable point, on this or that side of the point F, according as the areas described in equal times increase or decrease.

It does not necessarily follow that the centripetal force should cause the body always to approach the centre of force, it may continue to recede from it, notwithstanding its being drawn by a force residing there; but this property must always belong to its motion, viz. the curve which it describes, must always be concave towards the centre of force.

PROPOSITION CIV.

434. To determine the ratio of the forces by which bodies tending to the centres of given circles are made to revolve in their peripheries.

In fig. 150, let AH represent the space which a body A would describe in a constant particle of time, by a uniform projectile force, and HE the distance it is drawn from the tangent AH by the centripetal force in the same time,

Then, since AE is but a nascent arc, the arc, its chord and AH are nearly equal; therefore, AE may represent the projectile force; but (Eu. 8. 6. cor.) AD : AE :: AE: Aa=HE≈ AE2 the central force at E; and in the same manner the

AD
central force may be found in any other circle.

435. Cor. 1. Let HE=ƒ, AE=v, and AD=2r; then by

the proposition, fa; and if F, V, and R denote similar

2R'

quantities in any other circle, then Fx; therefore, F:ƒ:: V2 22

That is, the central forces are as the squares of the

velocities directly, and as the radii inversely.

In the same circle, that is when R and r are given, we have F: ƒ : : V2 : v2; or the forces are directly as the squares of

the velocities.

When the velocities are the same, then F:ƒ::

1 1 R

R; that is, the forces are inversely as the radii or distances.