.Newton's theory of motion
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In the "Principia" ("Mathematical Principles of Natural Philosophy"), Newton's monumental work published in 1687, he presented the motions of planets and things terrestrial in a form both compact and comprehensive. He generalized the work of Galileo into two laws, added a third law, and explained the law by which the gravitational force between massive objects work. Because of this the workings of the world became thought of as a giant clockwork which, once started, could be predicted from that point forward by a simple application of his laws of motion. Few others in the history of mankind contributed more to its direction. In formulating his laws he first defined some basic quantities:
Mass: "The quantity of matter is the measure of the same, arising from its density and bulk conjointly."
In modern terms, the mass of an object is its density times its volume (m = D V).Momentum: "The quantity of motion is the measure of the same, arising from the velocity and quantity of matter conjointly."
In modern terms, the quantity of motion is called momentum, and we write momentum as the product of mass and velocity (p = m v).Force: "An impressed force is an action exerted upon a body, in order to change its state, either of rest, or of uniform motion in a right (we would say straight) line."
Thus the tendency of objects to maintain their state of motion (inertia) is overcome by the application of a force.
He then postulates his three laws of motion:
The law of inertia (Newton's first law):
"Every body continues in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed upon it."The means by which an object's motion is changed (Newton's second law):
"The change of motion is proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed."The law of action and reaction (Newton's third law):
"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."
In terms of momentum (the quantity of motion) we can say that a force F applied to an object (of some mass m) over some time interval )t, will change the object's momentum. As an equation, F )t = )(m v). If this change in momentum involves a change in the velocity of an object of fixed mass (the usual case of, for example, someone throwing a baseball), we can divide our equation by )t and have F = m )v/)t. Since )v/)t, the change of an object's velocity over a time interval, is just that object's acceleration, we have the common expression of Newton's second law, F = m a..
These three laws form the basis for the entire subject we call mechanics today. Except for corrections pointed out by Einstein, they are all we need.
To see how his law of gravitation is formulated, we need to look at what is called "uniform circular motion": Think of a ball going around on the end of a string or, as Newton did, the Moon going around the Earth. If the motion is uniform (i.e. at constant speed as it goes around) it nonetheless must be accelerating because its direction of motion is changing, even if its magnitude is not. It is accelerating without changing speed - how can that be? It must be accelerating at right angles to its direction of motion (think about it)! Conclusion: If an object moves in a circle with uniform motion it must accelerate towards the center of its circular orbit. What is the magnitude of this central (centripetal) acceleration? Well, that takes a bit of mathematics (click here if you want to see how it is done), resulting in the expression ac = v2/r. In order for this acceleration to occur (according to Newton's 2nd law), there must be a force. thus, if an object of some mass (m) is to move in a circular path, a center-seeking force of magnitude Fc = m v2/r must be acting on it. This is the famous centripetal force you may have heard of. In the case of the ball and string it is the tension in the string that provides the force; for the Earth and Moon, gravity does the job. Just how gravity provides the necessary force Newton deduced from Kepler's laws combined with his second law, along with the assumption that Nature didn't care if the forces were between Sun and planet (Kepler), ball and string (on Earth), or Earth and Moon (Newton's quest). The result is Newton's universal law of gravitation, written as
.
Here the left side represents the force
due to gravity - the mutual attraction of any two objects of masses m and
M. The right side states that this force is proportional to the product of
the two masses (m and M) and inversely proportional to the square
of the distances between them. With this single equation the motions of
the heavenly bodies become known - Plato's problem is solved (without the
circles). How accurate is this universal law? Well, we know that on
24 July in the year 3991 there will be a total eclipse of the sun beginning at
22:20 O’clock (Universal Time), and it will last 7 minutes, 18 seconds (we
also know just where on Earth one should be to see it). The power of Newton’s
laws should be obvious.