Newtonian Mechanics

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We are now ready to discuss the contributions of Isaac Newton (1642-1727) to the science of mechanics. His work presents fairly complete answers to the questions first raised by Aristotle, which have been the subject of this chapter. In fact, Newton's work represents one of the greatest contributions made by an individual to human understanding of the physical universe. It is difficult to overestimate the impact of his work on Western thought, as we will discuss further below. He developed a picture of the universe as a subtle, elaborate clockwork, working according to well-defined rules.

1. Historical Background

Isaac Newton was born in England in 1642, the same year that Galileo died. Newton continued the work of Galileo in several important ways, especially with regard to mechanics. Like Galileo, Newton was the dominant scientist of his generation. Newton made important contributions in mathematics, optics, wave phenomena, mechanics, and astronomy. Any of his several most important contributions would have established him as one of the most important scientists in history.

Newton became interested in experimental devices as a child and demonstrated an ability to make original designs of windmills, water clocks, and sundials. Because of his great potential, he was able to attend Cambridge University, where he showed exceptional mathematical ability. Following his graduation in 1665, Newton returned to his boyhood home at Woolsthorpe in Lincolnshire, where he lived with his widowed mother. This was the time of the Great Plague throughout Europe, including England. Some 31,000 people in London alone perished from the Black Death over a two-year period. (This was also the time of a great fire that devastated much of London.) Newton spent the two years from 1665 to 1667 essentially in seclusion at Woolsthorpe. It is clear now that during those two years he formulated the well-known binomial theorem of mathematics, developed calculus, studied the decomposition of white light into its spectrum (using a prism), and began his study of mechanics, including the universal law of gravitation. None of this work was published immediately, and some of it not for 30 years. But these were surely two of the most productive years ever spent by a single scientist.

In 1667 Newton returned to Cambridge as a lecturer. Newton's master at Cambridge, a noted mathematician named Isaac Barrow, became so impressed with Newton's work and abilities that he resigned his own chair of mathematics (in 1669) so that it might go to Newton. Newton was thus put into a position to freely follow his various studies. Unfortunately, Newton's first publications in the area of optics were not well-received by other English scientists. Newton quickly became disillusioned and was reluctant to publish at all. He retreated into studies of alchemy, theology, and Biblical prophecy (Newton was an ardent Christian).

Later, following some suggestions from Robert Hooke and at the urging of an astronomer friend named Edmund Halley (the discoverer of the famous comet), Newton finished his work on mechanics begun nearly 20 years earlier. In 1687 he published his most important work, Principia Mathematica Philosophia Naturalis or in English, The Mathematical Principles of Natural Philosophy, or simply the Principia. (It was written in Latin, as were most scientific works at that time.) The Principia represents one of the greatest accomplishments of the human mind, and perhaps the single most important work in the history of physics.

Certainly Newton's work was the continuation of the earlier work of Aristotle, Kepler, and especially Galileo. Newton himself said, ' 'If I have been able to see so far, it is because I have stood on the shoulders of giants." Nevertheless, Newton's work was more than the great synthesis of all the thinking that had been accomplished earlier, and as a result of the publication of the Principia, he was established as the outstanding natural philosopher (i.e., physicist) of his time. He later served in Parliament (as the representative of Cambridge University) and became Warden of the Mint in London. Eventually Newton completely overshadowed all his early denigrators and even became president of the prestigious Royal Society, with unparalleled power in the realm of science.

As a person, Isaac Newton was jealous, egotistical, complex, and troubled. An absent-minded, confirmed bachelor, he was cared for by his niece. He was recognized as the leading scientist of his time and was the first scientist to be knighted. He had heated rivalries with other scientists, especially John Flamsteed, an English astronomer, and Gottfried Leibnitz, the German philosopher who co-invented calculus. Newton was admired by Voltaire and Alexander Pope, despised by Jonathan Swift and William Blake. He made some of the most important contributions to science ever and his work ushered in a new wave of optimism regarding the ability of man to understand his universe.

2. The Principia Mathematica

Newton's Principia was written in a rigorously logical, axiomatic style, following the example set by the ancient Greek mathematician Euclid in his book on geometry. Newton presented a number of basic definitions and assumptions, defined what he called "fundamental" and "derived" quantities, and then presented his three laws of motion, and a few other laws, including his universal law of gravitation, all based on induction from experiment. Then by logical deduction he showed that all the observed motions of objects in the physical universe, including celestial motion, natural local motion, and projectile motion, were simply consequences of his definitions and laws of motion. The Principia stands as a simple but truly monumental exposition of the nature of motion. Only in modern times, with the development of relativity and quantum mechanics, have any limitations on Newton's system been substantiated. These limitations are important only for systems involving extreme conditions: high speeds, small size, or very low temperatures.

Newton's definitions begin with the quantity of matter, which he described as the product of the density and volume of an object. This quantity describes ' 'how much stuff'' there is in an object, and is now called mass. The quantity of motion is defined as the product of the quantity of matter of an object and its velocity—that is, its mass times its velocity (mv)—and is a vector quantity called momentum. Note that both mass and velocity contribute to momentum. A 10-ton truck traveling at 20 mph has a greater magnitude of momentum than a subcompact automobile traveling at 50 mph, because the truck is about 10 times more massive than the automobile, whereas the automobile has a speed only 2.5 times greater than the truck.

Inertia is introduced as an inherent property of mass that describes its resistance to a change of its state of uniform motion. This will be discussed more fully below. Finally, Newton defined impressed force as being an action that may change the state of motion (i.e., the momentum) of a body. Normally, one thinks of an impressed force as likely to speed up or slow down an object. Newton recognized that it is also possible for a force to change the direction of motion of an object, without changing its speed. Such a force is called a centripetal force. An example is an object spun in a circle fastened to a string. The object may swing around with a constant speed, but its direction is continually changing. This change of direction is a form of acceleration and is caused by the string.

Newton next introduced his "fundamental quantities," which were required to be measurable and objective, that is, independent of the state of mind of the individual observer. Moreover, he desired that there be only a small number of these quantities, so that the complete description of science would be based on as few basic ideas as possible. He needed only three such fundamental quantities: time, length, and mass. These were to be measured in terms of fundamental units. The modern scientific units are the second, the meter (39.37 in.), and the kilogram (equivalent to 2.2 lb), respectively. The studies of electricity and heat have necessitated the introduction of two additional fundamental quantities. Other so-called derived quantities are formed as combinations of the fundamental quantities, for example, kinetic energy is half the product of momentum and velocity (hence its units are mass times the square of the ratio of length to time).

Having defined various concepts and quantities carefully, Newton was able to introduce his three laws of motion very simply. These laws were intended to specify the relationship between impressed forces and the changes in the motions of an object.

FIRST LAW OF MOTION—LAW OF INERTIA

In the absence of a net external force, an object will continue in a state of uniform motion (including rest) in a straight line.

As already discussed, this law had been recognized by Galileo before Newton was born, and represented a recasting of Aristotle's question, "Why do objects keep moving?" into the form, "Why do objects stop moving?" This law simply states that all objects with mass have a comfnon property, called inertia, which "keeps them doing what they have been doing." (In fact, mass is a measure of that inertia.) Newton, however, stated the law correctly in recognizing that inertial motion is straight-line motion, not circular motion.

SECOND LAW OF MOTION—LAW OF ACCELERATION

The time rate of change of motion (momentum) of an object is directly proportional to the magnitude of the impressed force and is in the direction of the impressed force.

This law relates acceleration of an object to the impressed force. Note that momentum is mass times velocity, so that if the mass of an object is not changing, then change of momentum implies change in the velocity, which is called acceleration. Hence, this law says that the acceleration of an object (with fixed mass) is proportional to the impressed force. Doubling the force doubles the acceleration, and tripling the force triples the acceleration. Mathematically, the second law becomes mass X acceleration = impressed force or symbolically, m • ~a = F

where the arrows over the a and F are just to remind us that both acceleration and force are vector quantities, with specific directions, which, according to the law, must be in the same direction. If we divide both sides of this equation by the mass m, we obtain the expression for the acceleration,

In this form, Newton's second law serves to tell us how the acceleration of an object depends on both the impressed force and the mass of the object. In words, it says that the acceleration of an object is directly proportional to the net impressed force (and in the same direction) and inversely proportional to the mass of the object.

It is important to understand what "inversely proportional to" means. Symbolically, if A is inversely proportional to B we write

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