Definition 16

Definition 16

253. That fixed point is called the centre of attraction, to which bodies are attracted by a force, which depends on the distance from this point, or which is as some fraction of this distance.

Corollary 1

254. Therefore given the distance from this centre of force in which the body placed is drawn to the centre by as large a force, as if it should be the force of gravity acting on this, and placed on the surface of the earth.

Corollary 2

255. Therefore with this distance and the law of the attraction known, clearly given by a function of the distance, to which the attraction is proportional, the ratio of this force is known, for any position of the body trying to fall towards the centre of force, to the force of gravity on the same body acted on if it should be on the surface of the earth. [p. 101]

Corollary 3

256. Thus in this manner, any variable forces are allowed to be compared with the force of gravity, since the effect of this force on the body is known, and thus also the effect of any force on the body can be determined.

Scholium 1

257. I put this attraction of the centres of the forces similar to the force of gravity, thus in order that likewise the forces of different bodies placed in the same position are as the masses themselves, and thus the magnitudes of the accelerations are all the same (212).

Therefore in handling these problems, it is not necessary to call upon the mass of the body in the computation of the motion, but only the magnitude of the acceleration, to which the force of attraction to the centre divided by the mass is in proportion.

It can be compared with the acceleration of the force of gravity, that we put equal to one, and we will compare all the accelerations due to the magnitudes of these forces with this acceleration of unity, clearly homogeneous quantities.

Corollary 4

258. Thus when we talk about the forces as being as the distances from the centre of force or proportional to a certain function of these, it is not only the forces that the bodies have to the centre, but also the accelerations associated with these force, i. e. it is the ratio of the force to the mass, that should be understood.

Corollary 5

259. Therefore since the direction of the force which presses upon the body, always pulls towards the centre of force it is evident, if the body is either at rest or it has motion, the direction of which passes through the centre of motion, then the body must be moving on this straight line perpetually crossing through the centre of force (189).

Definition 17

260. The force, which presses upon bodes to the centre of this kind of force, is called the centripetal force. And that, if it is negative, in order that the body is repelled from the centre, is called the centrifugal force.

Corollary 1

261. Since this will be the question about the motion, the centripetal force will be for us the magnitude of the acceleration, or the force pulling the body towards the centre divided by the mass of the body.

Corollary 2

262. Therefore the effort or the striving that the body has towards the centre of force, is expressed by the strength of the centripetal force [i. e. the acceleration] multiplied by the mass of the body.

On account of which it will be to the weight of the same body, if it were put in place on the surface of the earth, as the strength of the centripetal force or the strength of the acceleration to unity. (257).

Scholion

263. Newton has 3 ways in which the effects of centripetal force can be measured.

264. A measure of the absolute quantity of the effectiveness of the centre of the force, without regard to the [mass of the] body being attracted.

Thus he asserts, in the case of the larger loadstone the greater the absolute quantity of the centripetal force present, and in the case of the lesser lodestone, the corresponding centripetal force is smaller too

The absolute quantity is greater on the sun than it is on the earth. Moreover this comparison is to be understood from the similarity of the centres of force, i. e. according to the same function of the distances, and with attractions; indeed the comparison does not have differences of this kind at the same place.

Hence this absolute quantity of force is to be measured from a known effort, which the body has exerted on it at a given distance from the given centre of force.

Moreover in place of this consideration, I put in place the distance into which the body can be put with a force equal to its weight pushing towards the centre (254).

2. The strength of the centripetal force has the magnitude of an acceleration, which is perceived by the senses at the object itself, where the centripetal force itself is acting (261)

It is measured by the ratio of the effort applied to the mass.

3. The strength of the centripetal force leads us to the magnitude of the motive force, which is specified by the force the bodies experience on approaching the centre of force.

The motive force is the quantity of motion, and that is usually measured by the product of the speed by the mass, and which is produced in a given time proportional to this effort itself.

For this force p, and the mass A, the increment in the speed in a given element of time varies as A p (154), that multiplied by the mass A gives the increment in the quantity of motion, that is proportional to the force p.