Newton's Law

Newton's Laws of Motion
English physicist, mathematician, and natural philosopher Sir Isaac Newton is considered one of history’s most important scientists. Newton devised an analytic method more rigorous than that of any scientist before him, and his experimental method is still practiced today. His contributions to mathematics, physics, and the study of natural phenomena have proven extremely far-reaching and valuable. Published in 1687, Newton’s two-volume Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy) contains his important three laws of motion, also called Newton’s laws, excerpted here.
FROM Philosophiae Naturalis Principia Mathematica
By Isaac Newton


LAW I.
Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impress'd thereon.
Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impell'd downwards by the force of gravity. A top, whose parts by their cohesion are perpetually drawn aside from rectilinear motions, does not cease its rotation, otherwise than as it is retarded by the air. The greater bodies of the Planets and Comets, meeting with less resistance in more free spaces, preserve their motions both progressive and circular for a much longer time.
LAW II.
The alteration of motion is ever proportional to the motive force impress'd; and is made in the direction of the right line in which that force is impress'd.
If any force generates a motion, a double force will generate double the motion, a triple force triple the motion, whether that force be impress'd altogether and at once, or gradually and successively. And this motion (being always directed the same way with the generating force) if the body moved before, is added to or subducted from the former motion, according as they directly conspire with or are directly contrary to each other; or obliquely joyned, when they are oblique, so as to produce a new motion compounded from the determination of both.
LAW III.
To every Action there is always opposed an equal Reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Whatever draws or presses another is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone. If a horse draws a stone tyed to a rope, the horse (if I may so say) will be equally drawn back towards the stone: For the distended rope, by the same endeavour to relax or unbend it self, will draw the horse as much towards the stone, as it does the stone towards the horse, and will obstruct the progress of the one as much as it advances that of the other. If a body impinge upon another, and by its force change the motion of the other; that body also (because of the equality of the mutual pressure) will undergo an equal change, in its own motion, towards the contrary part. The changes made by these actions are equal, not in the velocities, but in the motions of bodies; that is to say, if the bodies are not hinder'd by any other impediments. For because the motions are equally changed, the changes of the velocities made towards contrary parts, are reciprocally proportional to the bodies. This Law takes place also in Attractions, as will be proved in the next Scholium.
COROLLARY I.
A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it would describe the sides, by those forces apart.
If a body in a given time, by the force M impress'd apart in the place A, should with an uniform motion be carried from A to B; and by the force N impress'd apart in the same place, should be carried from A to C: compleat the parallelogram A B C D, and by both forces acting together, it will in the same time be carried in the diagonal from A to D. For since the force N acts in the direction of the line A C, parallel to B D, this force (by the second law) will not at all alter the velocity generated by the other force M, by which the body is carried towards the line B D. The body therefore will arrive at the line B D in the same time, whether the sorce N be impress'd or not; and therefore at the end of that time, it will be found somewhere in the line B D. By the same argument, at the end of the same time it will be found somewhere in the line CD. Therefore it will be found in the point D, where both lines meet. But it will move in a right line from A to D by Law 1.
COROLLARY II.
And hence is explained the composition of any one direct force A D, out of any two oblique forces A B and B D; and, on the contrary the resolution of any one direct force A D into two oblique forces A B and B D: which composition and resolution are abundantly confirmed from Mechanics.
As if the unequal Radii O M and O N drawn from the centre O of any wheel, should sustain the weights A and P, by the cords M A and N P; and the forces of those weights to move the wheel were required. Through the centre O draw the right line K O L, meeting the cords perpendicularly in K and L; and from the centre O, with O L the greater of the distances O K and O L, describe a circle, meeting the cord M A in D: and drawing O D, make A C parallel and D C perpendicular thereto. Now, it being indifferent whether the points K, L, D, of the cords be fixed to the plane of the wheel or not, the weights will have the same effect whether they are suspended from the points K and L, or from D and L. Let the whole force of the weight A be represented by the Line A D, and let it be resolved into the forces A C and C D; of which the force AC, drawing the radius O D directly from the centre, will have no effect to move the wheel: but the other force D C, drawing the radius D O perpendicularly, will have the same effect as if it drew perpendicularly the radius O L equal to O D; that is, it will have the same effect as the weight P, if that weight is to the weight A, as the force D C is to the force D A; that is (because of the similar triangles A D C, D O K,) as O K to O D or O L. Therefore the weights A and P, which are reciprocally as the radii O K and O L that lye in the same right line, will be equipollent, and so remain in equilibrio: which is the well known property of the Ballance, the Lever, and the Wheel. If either weight is greater than in this ratio, its force to move the wheel will be so much the greater.
If the weight p, equal to the weight P, is partly suspended by the cord N p, partly sustained by the oblique plane p G; draw p H, N H, the former perpendicular to the horizon, the latter to the plane p G; and if the force of the weight p tending downwards is represented by the line p H, it may be resolved into the forces p N, H N. If there was any plane perpendicular to the cord p N, cutting the other plane p G in a line parallel to the horizon; and the weight p was supported only by those planes p Q, p G; it would press those planes perpendicularly with the forces p N, H N; to wit, the plane p Q with the force p N, and the plane p G with the force H N. And therefore if the plane p Q was taken away, so that the weight might stretch the cord, because the cord, now sustaining the weight, supplies the place of the plane that was removed, it will be strained by the same force p N which press'd upon the plane before. Therefore the tension of this oblique cord p N will be to that of the other perpendicular cord P N as p N to p H. And therefore if the weight p is to the weight A in a ratio compounded of the reciprocal ratio of the least distances of the cords p N, A M, from the centre of the wheel, and of the direct ratio of p H to p N; the weights will have the same effect towards moving the wheel, and will therefore sustain each other, as any one may find by experiment.
But the weight p pressing upon those two oblique planes, may be consider'd as a wedge between the two internal surfaces of a body split by it; and hence the forces of the Wedge and the Mallet may be determin'd; for because the force with which the weight p presses the plane p Q, is to the force with which the same, whether by its own gravity, or by the blow of a mallet, is impelled in the direction of the line p H towards both the planes, as p N to p H; and to the force with which it presses the other plane p G, as p N to N H. And thus the force of the Screw may be deduced from a like resolution of forces; it being no other than a Wedge impelled with the force of a Lever. Therefore the use of this Corollary spreads far and wide, and by that diffusive extent the truth thereof is farther confirmed. For on what has been said depends the whole doctrine of Mechanics variously demonstrated by different authors. For from hence are easily deduced the forces of Machines, which are compounded of Wheels, Pulleys, Leavers, Cords and Weights, ascending directly or obliquely, and other Mechanical Powers; as also the force of the Tendons to move the Bones of Animals.
COROLLARY III.
The Quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves.
For Action and its opposite Re-action are equal, by Law 3, and therefore, by Law 2, they produce in the motions equal changes towards opposite parts. Therefore if the motions are directed towards the same parts, whatever is added to the motion of the preceding body will be subducted from the motion of that which follows; so that the sum will be the same as before. If the bodies meet, with contrary motions, there will be an equal deduction from the motions of both; and therefore the difference of the motions directed towards opposite parts will remain the same.
Thus if a sphªerical body A with two parts of velocity is triple of a sphªerical body B which follows in the same right line with ten parts of velocity; the motion of A will be to that of B, as 6 to 10. Suppose then their motions to be of 6 parts and of 10 parts, and the sum will be 16 parts. Therefore upon the meeting of the bodies, if A acquire 3, 4 or 5 parts of motion, B will lose as many; and therefore after reflexion A will proceed with 9, 10 or 11 parts, and B with 7, 6 or 5 parts; the sum remaining always of 16 parts as before. If the body A acquire 9, 10, 11 or 12 parts of motion, and therefore after meeting proceed with 15, 16, 17 or 18 parts; the body B, losing so many parts as A has got, will either proceed with one part, having lost 9; or stop and remain at rest, as having lost its whole progressive motion of 10 parts; or it will go back with one part, having not only lost its whole motion, but (if I may so say) one part more; or it will go back with 2 parts, because a progressive motion of 12 parts is took off. And so the Sums of the conspiring motions 15+1, or 16+0, and the Differences of the contrary motions 17 - 1 and 18 - 2 will always be equal to 16 parts, as they were before the meeting and reflexion of the bodies. But, the motions being known with which the bodies proceed after reflexion, the velocity of either will be also known, by taking the velocity after to the velocity before reflexion, as the motion after is to the motion before. As in the last case, where the motion of the body A was of 6 parts before reflexion and of 18 parts after, and the velocity was of 2 parts before reflexion; the velocity thereof after reflexion will be found to be of 6 parts, by saying, as the 6 parts of motion before to 18 parts after, so are 2 parts of velocity before reflexion to 6 parts after.
But if the bodies are either not sphªerical, or moving in different right lines impinge obliquely one upon the other, and their motions after reflexion are required: in those cases we are first to determine the position of the plane that touches the concurring bodies in the point of concourse; then the motion of each body (by Corol. 2.) is to be resolved into two, one perpendicular to that plane, and the other parallel to it. This done, because the bodies act upon each other in the direction of a line perpendicular to this plane, the parallel motions are to be retained the same after reflexion as before; and to the perpendicular motions we are to assign equal changes towards the contrary parts; in such manner that the sum of the conspiring, and the difference of the contrary motions, may remain the same as before. From such kind of reflexions also sometimes arise the circular motions of bodies about their own centres. But these are cases which I don't consider in what follows; and it would be too tedious to demonstrate every particular that relates to this subject.
COROLLARY IV.
The common centre of gravity of two or more bodies, does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common centre of gravity of all bodies acting upon each other (excluding outward actions and impediments) is either at rest, or moves uniformly in a right line.
For if two points proceed with an uniform motion in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest, or proceed uniformly in a right line. This is demonstrated hereafter in Lem. 23. and its Corol. when the points are moved in the same plane; and by a like way of arguing, it may be demonstrated when the points are not moved in the same plane. Therefore if any number of bodies move uniformly in right lines, the common centre of gravity of any two of them is either at rest, or proceeds uniformly in a right line; because the line which connects the centres of those two bodies so moving is divided at that common centre in a given ratio. In like manner the common centre of those two and that of a third body will be either at rest or moving uniformly in a right line; because at that centre, the distance between the common centre of the two bodies, and the centre of this last, is divided in a given ratio. In like manner the common centre of these three, and of a fourth body, is either at rest, or moves uniformly in a right line; because the distance between the common centre of the three bodies, and the centre of the fourth is there also divided in a given ratio, and so on in infinitum. Therefore in a system of bodies, where there is neither any mutual action among themselves, nor any foreign force impress'd upon them from without, and which consequently move uniformly in right lines, the common centre of gravity of them all is either at rest, or moves uniformly forwards in a right line.
Moreover, in a system of two bodies mutually acting upon each other since the distances between their centres and the common centre of gravity of both, are reciprocally as the bodies; the relative motions of those bodies, whether of approaching to or of receding from that centre, will be equal among themselves. Therefore since the changes which happen to motions are equal and directed to contrary parts, the common centre of those bodies, by their mutual action between themselves, is neither promoted nor retarded, nor suffers any change as to its state of motion or rest. But in a system of several bodies, because the common centre of gravity of any two acting mutually upon each other suffers no change in its state by that action; and much less the common centre of gravity of the others with which that action does not intervene; but the distance between those two centres is divided by the common centre of gravity of all the bodies into parts reciprocally proportional to the total sums of those bodies whose centres they are; and therefore while those two centres retain their state of motion or rest, the common centre of all does also retain its state: It is manifest, that the common centre of all never suffers any change in the state of its motion or rest from the actions of any two bodies between themselves. But in such a system all the actions of the bodies among themselves, either happen between two bodies, or are composed of actions interchanged between some two bodies; and therefore they do never produce any alteration in the common centre of all as to its state of motion or rest. Wherefore since that centre when the bodies do not act mutually one upon another, either is at rest or moves uniformly forward in some right line; it will, notwithstanding the mutual actions of the bodies among themselves, always persevere in its state, either of rest, or of proceeding uniformly in a right line, unless it is forc'd out of this state by the action of some power impress'd from without upon the whole system. And therefore the same law takes place in a system, consisting of many bodies, as in one single body, with regard to their persevering in their state of motion or of rest. For the progressive motion whether of one single body or of a whole system of bodies, is always to be estimated, from the motion of the centre of gravity.
COROLLARY V.
The motions of bodies included in a given space are the same among themselves, whether that space is at rest, or moves uniformly forwards in a right line without any circular motion.
For the differences of the motions tending towards the same parts, and the sums of those that tend towards contrary parts, are at first (by supposition) in both cases the same; and it is from those sums and differences that the collisions and impulses do arise with which the bodies mutually impinge one upon another. Wherefore (by Law 2.) the effects of those collisions will be equal in both cases; and therefore the mutual motions of the bodies among themselves in the one case will remain equal to the mutual motions of the bodies among themselves in the other. A clear proof of which we have from the experiment of a ship: where all motions happen after the same manner, whether the ship is at rest, or is carried uniformly forwards in a right line.
COROLLARY VI.
If bodies, any how moved among themselves are urged in the direction of parallel lines by equal accelerative forces; they will all continue to move among themselves, after the same manner as if they had been urged by no such forces.
For these forces acting equally (with respect to the quantities of the bodies to be moved) and in the direction of parallel lines, will (by Law 2.) move all the bodies equally (as to velocity) and therefore will never produce any change in the positions or motions of the bodies among themselves.
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