The Effect Of Forces Acting On A Free Point
Table of Contents
Definition 10: Forcec
- A force is an action on a free body that either: **- leads to the motion of the body at rest, or
- changes the motion of that body.**
The force of gravity is a force of this kind.
Through gravity, bodies fall freely downwards, and the descending motion is one of continuous acceleration.
Corollary
- Any body left to itself will continue in a state of rest or of uniform motion in a fixed direction.
Therefore to what extent it occurs that a free body at rest begins to move, or the uniform motion of a body becomes non-uniform, or the motion changes direction, the cause is to be ascribed to some external force, which we call the force acting on the body, according to whatever state of the body it produces.
Scholium 1.
- Concerning the principles of external forces, in as far as many external forces are applied to a body in equilibrium and they keep the body at rest, that has now been explained in Statics.
There the external force has also been defined, as that force denotes all the force that prevails to make the body move.
The motion of the body is not to be considered in Statics, but only these cases are to be investigated in which several forces cancel each other out [p. 40], and the body upon which they act remains in a state of rest.
Mechanics explains how forces acting on a body, which do not cancel each other out, can produce motion in a body at rest, and indeed for a body in motion, change that motion.
Scholium 2.
- Whether forces of this kind have their origin in the bodies themselves, or indeed through such forces that do exist in the world, these I will not consider here.
For here it is sufficient, in place of the forces that really arise in the world, only to consider the force of gravity, by which all bodies on the earth try to move downwards.
Besides truly forces of this nature are evident disturbing the motion of the planets, which unless they were influenced by a certain force, would be progressing in straight lines.
Similar forces arise from magnetic and electric bodies. These can attract bodies.
- The followers of Descartes think that all the forces arise from the motion of some subtle matter
- The followers of Newton think that the force is from the attraction or repulsion between the bodies themselves
Certainly, the forces of this kind can arise from vortices, and from elastic [energetic] bodies.
What is the effect of any forces on bodies?
Definition 11. Direction
- The direction of the force is the straight line along which the body is trying to move.
Thus the direction of the force of gravity is that vertical line, for a heavy body tries to fall along that line.
Scholium 1.
- In statics, everything is put in place to remain at rest.
All the forces have their directions set up, serving to keep the body at rest all the time.
But in mechanics [which we now call dynamics], when the body is always arriving at another place, the direction of the force acting will constantly be changing direction.
For different positions of the body the directions of the forces are either parallel to each other, or converging to a fixed point, or they act in response to some other law, from which so many different forms of the laws governing mechanics arise.
Scholium 2.
- The comparison and measurement of different forces, likewise from statics, should be recalled.
In which some force has been treated a that has the ratio to another force b as m to n, when the force a is applied n times in turn on the point A (Fig. 11) along AB, and the force b is applied m times along the opposite direction AC, and the point A continues in equilibrium.
Then indeed the force a, taken n times, is equivalent to the force b, taken m times, and will be related by na = mb or a : b = m : n.
Scholium 3.
- Now in this regard, the measurement of forces differs in mechanics from statics.
In statics, all the forces keep the same magnitude. In mechanics, as with the body arriving in another place, the directions of these are made changeable, thus the magnitude of these is able to be variable following some law.
PROPOSITION 13. THEOREM
- When a point is acted on by many forces, the same motion comes about from these, as if the point is acted on by a single force equivalent to all of these forces.
DEMONSTRATION
Let the point A (Fig. 12) be acted on by the forces AB, AC, AD, and AE, to which the force AM is equivalent.
The equal and opposite position to this, AN, is taken, and as it is known from statics, it will cancel the action of the forces AB, AC, AD, and AE.
Therefore [harking back to (104), ] in the first place only the force AN is impressed on the point A and the motion is along AN, the magnitude of the forces AB, AC, AD, and AE acting together impress on the point A the force along the line AM in the central direction.
Truly the force AM alone, since it is equal to the force AN, is of such a size that it moves the point A along AM, to the same extent as the force AN moves the point towards AN.
Whereby the force AM alone impresses on the point A a motion along AM, as much as the forces AB, AC, AD, AE acting together along the same direction [p. 43] AM. In each case therefore, the effect is the same. Q. E. D.
Corollary 1.
- If a point is therefore influenced by many forces, that can be considered to be influenced as it were by a single force, which is equivalent to all these forces.
Corollary 2.
- And in turn in place of a single force acting on a point, there can be considered to be many forces acting, to which that is equivalent ; that which, as has been shown from statics, can be made in a limitless number of ways.
Scholium.
- Because truly, as the body first has moved from its position, the forces acting on it change their directions and magnitudes or they are put in place to move with the body, there will be some other equivalent force for any moment.
Hence on account of this circumstance, for any time, the equivalent force of all the forces acting on the point ought to be investigated, and thus not the long term effect of the same force is to be put in place [i. e. the time average force,] but rather that from an infinitely small element of time.