Mechanics and thermodynamics are two closely related areas of physics. Mechanics, the science of motion, is the oldest area in physics, dating back to the ancient Greeks. Thermodynamics, the science of heat, dates back to the seventeenth century and early work on establishing a temperature scale.
The ancient Greeks gave to the world much of the physics that was known up to AD 1400. In their written work, one finds the germ of fundamental concepts such as the conservation of matter, inertia, atomic theory, the finite speed of light, and so on. Thales of Miletus (625-547 BC) knew about the attractive power of magnets and of rubbed amber. Pythagoras (560-480 BC) held that both Earth and the entire universe were spherical. This may have been based on the idea that the sphere is the most perfect of three-dimensional shapes. Democritus (fl. late fifth century BC) postulated that the universe consists of empty space and an (almost) infinite number of indivisible and invisible particles. He argued that the creation of matter was impossible, since nothing can come from nothing and, moreover, nothing that is can cease to exist. Aristarchus (320-230 BC) had a heliocentric model of the solar system nearly 2000 years before Copernicus.
ARISTOTLE AND THE MOTION OF BODIES
According to Aristotle (384-322 BC), all substances were composed of four elements: earth, fire, water, and air. The Aristotelian view of matter is opposed to the view that all matter is composed of atoms. Aristotelian physics was based on the axiom that every motion presupposes a mover that is either present in the moving body or in direct contact with it. Action at a distance was taken to be impossible.
Aristotle divided the motion of inanimate bodies into natural motion and violent motion. Examples of natural motion were the falling of a stone or the rising of smoke. Examples of violent motion were the throwing of a stone or the shooting of an arrow. The fundamental law of Aristotelian dynamics is that a constant force imparts uniform motion to a body. Aristotle stated that the speed of a body is directly proportional to the force exerted on it and inversely proportional to the resistance offered to the motion. The fundamental dynamical law of Aristotelian physics may be written as v = f · F/R, where f is a constant of proportionality.
Aristotle also understood the law of the lever. For a seesaw (a form of lever) to be balanced, the weights at each end must be inversely proportional to their respective distances from the balance point, or fulcrum, of the seesaw.
Archimedes (287-212 BC) was probably the most famous Greek scientist of antiquity. Born in Syracuse, Sicily, he invented the endless screw, the water screw, a pulley block, and the burning mirror. He studied both mathematics and mechanics, and demonstrated many theorems in geometry. For example, he showed that "the surface of any sphere is equal to four times the greatest circle in it," which in modern formula mathematical form is given by S = 4πr2. He also proved the formula for the volume of a sphere, V = 4/3πr3. He showed that π, the ratio of circumference to diameter of a circle, is less than 3 1/7 and greater than 3 10/71.
Archimedes was a founder of statics, and in his book On Floating Bodies, he laid the foundations of hydrostatics, including his famous Archimedes' principle. A well-known but probably apocryphal story recounts how the Greek king Hieron asked Archimedes to determine whether a certain crown, or wreath, was made of pure gold. The supposition was that the metal was perhaps not pure gold but an alloy of gold and some less precious metal. According to the story, Archimedes found the solution while he was sitting in a bathtub in ancient Syracuse (a Greek city-state in Sicily). Thereupon he ran naked through the streets, shouting "Eureka, eureka," Greek for "I have found it, I have found it."
If a body is immersed in a fluid, it takes the place of the fluid.
Take the case of Archimedes, where the body immersed in the fluid is more dense than the fluid. Since it is heavier, it has to be supported by a rope, or cord. The object is in force equilibrium since it is not accelerated; it is simply hanging in the fluid. The forces on the body sum to zero. What are these forces? There is, first, the weight of the body, that is, the gravitational attraction of Earth on the body (w). Second, there is the upward pull of the rope (T). Third, there is the buoyant force of the water. This buoyant force is the result of all the upward forces exerted by the water all around the body. The force equilibrium on the body can be expressed in symbols by w = T + FB, where FB is the buoyant force. What this symbolic equation says is that the sum of the downward forces is equal to the sum of the upward forces.
Archimedes focused on the buoyant force. This is probably the chain of reasoning that led to his (supposedly) jumping out of the tub and running through the streets naked. If the body were totally removed from the water, the water that now occupied the place of the body would certainly be in force equilibrium. The weight of that water would equal the buoyant force FB. Hence the buoyant force simply equals the weight of the displaced fluid. Any attempt to make a direct calculation of the buoyant force by adding up all the separate upward forces exerted by the water would be horribly complicated. Archimedes solved a difficult mathematical problem in one simple step.
Archimedes' principle holds for both submerged bodies and floating bodies. This principle was Proposition 7 of his work On Floating Bodies. The difference, of course, is that the fluid displaced by the floating body is less than the volume of the floating body. If one keeps this in mind, the floating-body situation is easy to understand. There are only two forces on the floating body, and they are in balance. The upward buoyant force of the fluid equals the downward force exerted by the weight of the body. The buoyant force is still equal to the weight of the displaced fluid, but now the volume of the displaced fluid is only a fraction of the volume of the body. The easiest way to see this is to inspect a diagram of the situation.
Pressure in a Fluid
Consider a column of fluid (water) of area A and height y at rest in a fish tank.
This column of fluid is at rest, hence it is in force equilibrium. Now consider the vertical forces on this column. They are in force equilibrium, that is, they sum to zero. The force on the upper surface is due to the atmospheric air pushing down on the surface. The pressure is defined as the force per unit area, or
p = F/AThe force on the upper surface, for example, is the atmospheric pressure pa multiplied by the area A of the upper surface of our column of fluid. The downward arrow on the upper surface of the block is paA. Now, why is this shown as a downward arrow? The pressure exerted by a fluid is, in a sense, in all directions, and only assumes a direction when a surface exists. Since a fluid flows and cannot exert any force in a direction along any surface that is placed in the fluid, the force on any surface in a fluid (and the atmospheric air is also considered a fluid) is perpendicular to that surface.
The upward force on the lower surface of the column is pA, where p is the pressure in the fluid a distance y below the surface. The weight w of the block of fluid is the weight density ωw of the fluid multiplied by the volume Ay of the block. Since the forces on the block are in equilibrium,
pA = paA + w = paA + ωwAyor
p = pa + ωwYWhat this equation says is that the pressure a distance y down in a fluid equals the atmospheric pressure plus the pressure due to a fluid column of height y. The symbol ωw was used in this formula to indicate explicitly that we were dealing with the weight density rather than the mass density, which is the mass per unit volume. Atmospheric pressure, pa, is 14.7 pounds per square inch (1.03 kilograms per square centimeter).
THEORY OF IMPETUS
The impetus theory was an early attempt to explain motion and to try to connect it with the cause of motion. From direct observation, motions tend to sustain themselves after a mover has started them. A rock that is flung horizontally tends to maintain the horizontal motion after it is flung. Thus the moving object retains the "impetus" with which it was originally thrown. The impetus theory is important as the forerunner to the modern idea of inertia.
The Italian Giovanni Battista Benedetti (1530-90) was the first to state explicitly that the impetus that has been impressed on a body always makes it perform its motion in a straight line. Benedetti found that a stone swung around on the end of a rope in a circle and then released moves along a tangent line to the circle in which it was originally traveling.
In 1586, the Dutch mathematician and physicist Simon Stevin (1548-1620) published a book on statics titled Elements of the Art of Weighing. Stevin is credited with being the originator of the principle that two blocks of different weights connected by a string will be in equilibrium on two inclines when the ratio of the weights equals the ratio of the lengths of the two inclines. The figure illustrates Stevin's law of the inclined plane. The chain STV will be in equilibrium when the pull on section ST equals the pull on section TV. It is best to visualize two weights w1 and w2 on two inclines.
The angles made by the two inclines with the horizontal are q1 and q2. The length of the incline AB is l1, and that of the incline BC is l2. The two weights are connected by a light string or cord. Since the two weights are each in equilibrium, the sum of the forces on each weight is zero. Each weight has three forces on it, the weight force, the normal upward push of the incline, and the pull of the cord. Call the two pulls F1 and F2. Then F1 = F2, and since F1 = w1 sin q1 and F2 = w2 cos q2, then w1 sin q1 = w2 cos q2. This leads to
Since cot q2 is constant, it may be seen that weights w1 and w2 are proportional to lengths l1 and l2 of the inclines.
PASCAL AND HYDROSTATICS
Blaise Pascal (1623-62), a French mathematician, physicist, and philosopher, is an interesting example of that rare person who made contributions to both science and the humanities. The Scientific Revolution was only just beginning, so it is not surprising that his contributions to science were undervalued by his contemporaries in comparison with his work in the humanities. The historian of science E. J. Dijksterhuis (1892‐1965) quoted an apologetic comment made by one of the coeditors of Pascal's scientific work as follows: "It isn't that these Treatises aren't complete in their genre, nor that it is hardly possible to do a better job; but rather that this particular genre is so much beneath him, that those that would judge him only by these writings, would only be able to form a very weak and imperfect idea of the greatness of his genius and of the quality of his spirit." Dijksterhuis commented that "this is very characteristic of the complete lack of understanding of the significance of scientific work prevalent in the seventeenth century (and long after) among representatives of the humanities."
Pascal contributed in important ways to hydrostatics, formulating an important law of physics known as Pascal's principle. The significant factor in determining the pressure in a fluid at rest is the height of fluid above the point at which the pressure is to be determined. Consider fluid in an L-shaped container.
The three points a, b, and c are all at the same level. The pressure at these three points is the same because they are all at the same depth below the surface of the fluid. The pressure at point a must be the same as the pressure at point b because the fluid is in static equilibrium. If the pressure at point b were different from that at point a, fluid would move from the higher pressure to the lower pressure.
Now the pressure at point b is the sum of atmospheric pressure and the pressure due to a liquid column whose height is the vertical distance from point b to the liquid surface. Suppose further that the atmospheric pressure changes--say that it increases by 5 pounds per square inch (0.4 kilograms per square centimeter). The pressure at all points in the liquid would correspondingly increase by 5 pounds per square inch (0.4 kilograms per square centimeter). This statement about liquids is Pascal's principle: A change in pressure applied to an enclosed liquid is transmitted undiminished to every point of the liquid and to the walls of the containing vessel.
GALILEO AND THE COPERNICAN SYSTEM
The first giant of the Scientific Revolution was Galileo Galilei (1564-1642). His Two New Sciences (1638) was a major contribution to physics. The part of this book on the motion of bodies can be considered to represent the beginning of the study of mechanics as we know it today. The two most important reference frames at the time of Galileo were the Sun and Earth. The Copernican viewpoint adopted the Sun as the reference frame, whereas the Aristotelian-Ptolemaic viewpoint adopted Earth as the reference frame. Since one habitually thinks of a reference frame as fixed, it is natural to think of the Copernican system as a fixed Sun and moving planets. In the Ptolemaic viewpoint, Earth is fixed and the Sun, Moon, and planets move around Earth. The Catholic Church backed the stationary-Earth viewpoint, in particular because it believed in a sharp division between Earth and heaven.
Galileo, who was convinced that the Copernican system was correct, wrote a book entitled Dialogue Concerning the Two Chief World Systems. It was published in Florence in March 1632, when Galileo was 68 years old. In the Dialogue, Galileo covertly defended the Copernican system by propounding his views through Salviati, one of the three participants in the dialogue. Despite this attempted concealment, the Church stopped further sales and ordered Galileo to come to Rome in October 1632 to present himself to the Inquisition. Galileo, who was in poor health, ultimately appeared in Rome in March 1633. The outcome of the trial was that Galileo was sentenced to life imprisonment for defending the Copernican "heresy." The Dialogue was placed on the Index of Forbidden Books. In an extrajudicial procedure, Galileo abjured the Copernican heresy. (The Church removed the book from the Index in 1822 and officially apologized for its actions against Galileo in 1992.)
Was this famous trial of Galileo only about differing choices of reference frame? Not quite. Granting that all motion is relative, there is still a desire to associate motion with movers. Although the nature of the gravitational force was unknown in its details at the time of Galileo, there was an implicit understanding that in the Copernican system the Sun was the mover of the planets. Thus the Copernicans, including Galileo, believed that the planets had an absolute motion around a stationary Sun. Earth does move. The unbelievable complexities of describing the solar system from the standpoint of a fixed Earth necessitate the complete rejection of the Ptolemaic system. The choice of a fixed-Sun reference frame was a choice of simplicity, which is a powerful guiding force in physics. The orbits of the planets as viewed from the Sun were simple circles (more accurately, ellipses that were almost circular), whereas these same orbits as viewed from Earth were significantly more complex. They can still be described, but nobody is willing to put up with the complexity.
It must also be remembered that the issue of the authority of the Catholic Church was an important one in the "crime" of Galileo. The Church believed that there was a separation between Earth and heaven. Earth was the domain of man and the animals; heaven was the domain of God and the angels. This separation, people believed, should also extend to the laws of motion. Heavenly motions had to be built around the circle, which had already been recognized in antiquity as the perfect form. In contrast, earthly motions could take any form, as imperfect as humans were imperfect.
But the laws of physics do not distinguish between Earth and heaven. They are universal and extend to the farthest reaches of the universe. These laws will condition the very language we use to describe phenomena. These laws tell us that there is a certain class of reference frames, called the inertial reference frames, which, for reasons of simplicity, are used to describe motion. An inertial frame is one in which Newton's first law of inertia holds. A frame fixed to the Sun is very close to being a perfect inertial reference frame. Thus such a frame is the ideal frame to describe planetary motion. A frame fixed to Earth is an extremely poor choice for describing planetary motions because the major force on the planets is the gravitational pull of the Sun. For movements that are terrestrial, such as the flight of a golf ball or the movement of an automobile, the gravitational pull of the Sun on the objects is not significant for the motion of the objects. In the latter instances, Earth is close enough to being an inertial reference frame and so can safely be defined as such.
At the time of Nicholas Copernicus (1473-1543), it would have seemed unreasonable to want to describe motion in terms of a reference frame fixed to the Sun, which was millions of miles away. Earth was near at hand. It seemed to provide a natural reference frame. The motions of the distant stars were circular as viewed from Earth, as was the motion of the nearest star, the Sun. The planets moved in a region called the zodiac. The "wanderer" planets posed problems. Mars proceeded through the zodiac mostly in one direction. However, sometimes it exhibited retrograde motion in the opposite direction [see Astronomy and Cosmology: Pre-Eighteenth Century]. As seen from a sun-centered (heliocentric) reference frame, however, there was no retrograde motion. As seen from a geocentric reference frame, there was retrograde motion. Mars simply swung smoothly around the Sun in an ellipse. This should have been a telltale sign that the major force on Mars's motion was due to the Sun. And that, in turn, should have given the Copernican system immediate preference over an Earth-centered system for describing the motion of Mars. But people were not yet ready to associate motion with movers, or a force that occasioned the motion.
TWO NEW SCIENCES
Galileo was very interested in compound motion, motion that can be thought of as compounded of two motions. Galileo's study of motion is contained in his masterpiece, the Two New Sciences. The Two New Sciences covers four "days" of dialogue--the same format as the Dialogue itself. The first two days are devoted to the first new science, that of the properties of solid materials. The third and fourth days are devoted to the new science of motion. It is these last two days that stand out as truly remarkable in the history of physics. In them, Galileo analyzes uniform motion, motion with constant acceleration, and projectile motion as a composition of uniform motion and motion with constant acceleration. The treatment of motion is mathematical, which differentiates the discussion from previous treatments. Galileo also deals here with many important issues in physics, such as the approximations needed to describe reality in terms of a usable mathematical model. In many ways, the third and fourth days of the Two New Sciences can be considered to be the very first physics textbook.
In the third day of his Two New Sciences, Galileo discusses uniform motion and "motion as we find it accelerated in nature." This is the beginning of physics, and Galileo senses his role as innovator. He points out that he is the first to set down the odd number rule (see below) for the distances traversed by bodies falling freely from rest and the parabolic paths of projectiles. He proudly proclaims his position as founder of the science of motion: "This and other facts, not few in number or less worth knowing, I have succeeded in proving; and what I consider more important, there have been opened up to this vast and most excellent science, of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore its remote corners."
Galileo opens with a definition of uniform motion: "By steady or uniform motion, I mean one in which the distances traversed by the moving particle during any equal intervals of time, are themselves equal." He cautions the reader as to the importance of the adjective "any" before the "equal intervals of time." Clearly, a particle may travel, say, 1 foot every second, and yet not be traveling with uniform motion. It can speed up or slow down during individual seconds, and yet wind up traveling exactly 1 foot every second. Such motion is not uniform motion.
Since he wants to develop his discussion in the Euclidean style of axioms and propositions, he next sets down his four axioms of uniform motion and proceeds to derive his propositions. His first proposition is a good example of his method of proof. It will seem very cumbersome to modern eyes accustomed to the power of algebraic proofs, which were unknown at the time of Galileo. The statement of Proposition I
is as follows: "If a moving particle, carried uniformly at a constant speed, traverses two distances, the time intervals required are to each other in the ratio of these distances. Let a particle move uniformly with constant speed through two distances AB, BC, and let the time required to traverse AB be represented by DE; the time required to traverse BC, by EF; then I say that the distance AB is to the distance BC as the time DE is to the time EF."
As can be seen, Galileo takes the base distance AB and lays it off n times to generate BG, and similarly takes the base time DE (time to cover AB) and lays it off n times to generate time EI. He does likewise for the other base distance BC and corresponding time EF, generating BH and EK. He then argues that since the motion is assumed uniform if BG equals BH, then EI equals EK, whereas if BG is greater than BH, then EI will be greater than EK, and if less, then EI will be less than EK. Galileo then proceeds to prove five more propositions on uniform motion. They are all derivable from x = vt, where x is a distance traveled, v is a uniform speed, and t is an elapsed time.
In Proposition I, Galileo discusses uniform motion from a purely theoretical viewpoint, unrelated to experience. Now he turns to nature and says: "First of all it seems desirable to find and explain a definition best fitting natural phenomena. For anyone may invent an arbitrary type of motion and discuss its properties." In choosing a definition of uniformly accelerated motion, he is guided by the simplicity of nature. He therefore chooses a definition in which equal increments of velocity are added to the motion of the particle in equal increments of time. His model is that of a freely falling stone.
GALILEO'S THEOREMS ON UNIFORMLY ACCELERATED MOTION
Galileo's first theorem on uniformly accelerated motion is as follows: "The time in which a certain space is traversed by a moveable [object] in uniformly accelerated movement from rest is equal to the time in which the same moveable [object] carried in uniform motion whose degree of speed is one-half the maximum and final degree of speed of the previous, uniformly accelerated motion." Galileo's proof of this first theorem is a geometric proof. The diagram accompanying his proof is shown here.
The time AB shown is that in which the body covers the space CD with a uniformly accelerated motion. The velocity, which increases uniformly with the time, is shown by drawing horizontal lines from AB that increase uniformly with time. Thus the triangle AEB is formed. Let F be the midpoint of EB. Draw FG parallel to AB and GA parallel to FB. The intersection of FG with AE is at the point I, and the horizontal line at I represents the average speed. The distance covered by the uniformly accelerated motion in the time AB is the triangular area AEB, and an equal distance would be covered in that time by a uniform motion whose speed is the mean speed (the rectangular area AGFB). Thus Galileo has proven his first theorem.
In his Two New Sciences, Galileo develops the equations for a uniformly accelerated motion using his geometric proofs. The results, of course, are the same as our modern ones. For example, his Corollary I to Proposition II is that the spaces traversed from rest in equal time intervals go as the odd integers, 1, 3, 5, 7, ... This can readily be deduced from the modern equation s = 1/2at2 (where s is the distance traversed and t is the elapsed time) by noting that the successive coordinates are
The successive spaces traversed therefore go as s1, s2 - s1, s3 - s2, and so on, that is, as the numbers 1, 3, 5, ... This is Galileo's law of odd numbers.
HUYGENS AND COLLISIONS
Christiaan Huygens (1629-95) was born into a well-to-do Dutch family. His father was a diplomat, secretary to the prince of Orange. Huygens quickly developed a formidable reputation as a scientist and mathematician. He was invited to Paris by Louis XIV to organize the Académie des Sciences, which was founded in 1666. Huygens remained in France during a war that broke out between France and Holland. He finally returned to Holland for reasons of health. He became ill again in 1694 and died on June 8, 1695.
The central problem of mechanics in the seventeenth century was the formulation of a general law connecting forces and motion. Two of the main physical phenomena that were studied in an effort to arrive at such a general formulation between forces and motion were collisions and circular motion. Huygens began work on the problem of collisions in the early 1650s, when he was in his 20s. In January 1652, he wrote that he doubted whether the French scientist René Descartes's (1596-1650) rules on collisions were correct, except for Descartes's first rule, which concerned a collision between two equal bodies that move toward one another with equal speeds. It stated that after the collision each body rebounded with the same speed as it had before the collision (assuming that the collision was a "hard" one). The collision was seen by Huygens as a perfect example of symmetry. He consistently sought a way to view collisions from the standpoint of symmetry.
Huygens's manuscript on hard or elastic collisions was titled On the Motion of Colliding Bodies. This treatise dates from 1656, but was not published until 1703, eight years after Huygens's death. The editors of the complete works of Huygens stated that he did not publish his results on the mechanics of collisions, which he obtained in the early 1650s, because "aside from the laws of collisions, there remained other things concerning the nature of motion which he had not yet sufficiently studied and which required prolonged study." One can only guess that "the other things" were the connection between forces and motion. Huygens knew that he had correctly solved the problem of hard collisions through a mathematical approach, that of symmetry. But he also knew that he had not provided a general approach to dynamics. During the years in which he was striving to find a general approach, Isaac Newton (1642-1727), 13 years his junior, found such an approach and published it in 1687 in the first edition of his Philosophiae naturalis principia mathematica (usually referred to simply as the Principia).
In 1656, Huygens organized his work on collisions into a treatise in the classical mode, with hypotheses, propositions, and lemmas. This treatise, or rather something extremely close to it, became what was eventually published posthumously in 1703. In the meantime, however, the essential Huygensian solution of the elastic collision of two bodies was published by him as a letter in the Journal des sçavans of March 18, 1669. It should be noted, by way of historical comparison, that Newton's early unpublished work in mechanics was done in the 1660s and that his first publication of his solution for an elastic collision was in the 1687 first edition of his Principia.
The first proposition in Huygens's On the Motion of Colliding Bodies provides an illustration of his method of symmetry.
A boat is moving to the left, and an experimenter whose hands are labeled A and B is standing on the boat. He is conducting a collision experiment with two equal identical bodies E and F, suspended by strings from his hands. He moves the two bodies toward each other with equal speeds, and they collide and rebound from each other. From the symmetry of the collision, they rebound with their original speeds reversed. There is another observer on the shore who assists in the experiment, and whose hands C and D also hold the strings that are attached to the bodies. It is assumed in the proposition that this observer sees ball E as stationary. Thus he sees the boat moving to the left with exactly the same speed as the observer on the boat is moving ball E to the right. The observer on the shore thus sees ball E as originally stationary and ball F moving to the left with speed 2v, where v is the speed of the boat to the left. The statement of Proposition I is as follows: "When a body at rest is struck by an equal body, after the collision the latter will be at rest, but the one which was at rest will acquire the same speed as that which the striking body originally had."
Huygens's proof of his proposition consists of appealing to his relativity hypothesis, which says that "the motion of bodies, and their equal or unequal speeds, should be understood as being with respect to other bodies which are assumed to be at rest, although, perhaps, the latter as the former, are subject to some other common motion. In consequence, when two bodies collide, even though the two experience some other common movement, they will not act otherwise with respect to an observer who also has this common movement, than if this common movement were absent from bodies and observer."
The application of this relativity hypothesis to the collision on the boat is that the balls will have the same equal rebounds as they would have had if the boat were stationary. This being the case, ball E, which rebounds with speed v as seen by the boat observer, is seen by the shore observer to move with speed 2v to the left, and ball F after the collision is seen to be at rest with respect to the shore observer. Thus Huygens has proved his first proposition. He continues to explore symmetry in the propositions that follow. To handle the more complicated case of a hard collision between unequal bodies, he defines the product of mass and velocity as a new quantity whose symmetry will permit the determination of the outcome of a hard (elastic) collision between two bodies. Thus his symmetry becomes a symmetry in what is called the center-of-mass frame.
The publication in 1687 of the first edition of Newton's Principia was a major event in the history of physics. In the part of this work titled "Axioms or Laws of Motion," we find Newton's three laws:
- Law I: 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.
- Law II: 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.
- 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.
The first law is the law of inertia. The second law is the heart of Newtonian mechanics. In modern form, the change of motion is replaced by the time rate of change of motion, ma, and the motive force is replaced by the instantaneous force F. The modern form of the second law of motion is F = ma. The force and the acceleration are treated in the modern manner as vectors. The third law is Newton's action-reaction law. It was essential to Newton's analysis of collisions.
The unit of force equals a mass unit times an acceleration unit. This unit is called a newton, in honor of Isaac Newton. In terms of the more familiar pound unit of force, 1 N = 2.2 pounds.
Early on, before the Principia, Newton had proved Kepler's second law for any centripetal force between a planet and the Sun. This law states that a line drawn from the Sun to a planet sweeps out equal areas in equal times. This proof was given pride of place as Proposition I of Book I of Principia. It read as follows: "The areas which revolving bodies describe by radii drawn to an immovable centre of force do lie in the same immovable planes, and are proportional to the times in which they are described."
The figure shown is Newton's diagram for this proposition. The point S represents the Sun, the source of the gravitational force that keeps the planets in their orbits. The polygonal path ABCDEF represents an instantaneous impulse approximation to the actual elliptical curved path of a planet around the Sun, with each leg traversed in time Δt. Newton shows that the triangles SAB, SBC, SCD, and so on, are equal in area. So for the polygonal path Newton has shown that equal areas are swept out in equal times. The time interval is then progressively shrunk to zero, the polygonal path becomes the continuous curved path of the body, and Newton's proposition is proven.
The "Kepler problem" was the determination of the force needed to move the planets in their elliptical orbits around the Sun. In Proposition VI of Book I of the Principia, Newton showed how to obtain the instantaneous measure of a centripetal force. In Proposition XI he considered the problem of determining the centripetal force when the orbit is an ellipse and the force center is at one focus of the ellipse. The actual problem to which Newton planned to apply his Proposition XI was the problem of planetary motion in an ellipse. The point S is the Sun at one focus of the ellipse and P is the planet; r is the distance from S to P. Newton proceeded to prove that the force goes inversely as the square of the distance, that is, the centripetal force is F ∝ 1/r2.
THE FAMILY BERNOULLI
The Bernoullis were a very well known Swiss family who made major contributions to mathematics and physics. The best-known members of the family were Jakob (1654-1705), Johann (1667-1748), and Daniel (1700-82). Jakob and Johann were brothers who were brilliant mathematicians. Daniel is remembered for Bernoulli's equation of laminar fluid flow in mechanics.
Jakob became professor of mathematics at the University of Basel. He studied and mastered Leibniz's infinitesimal calculus. Johann studied medicine and mathematics with his older brother. He obtained the chair of mathematics at the University of Groningen in the Netherlands, but later returned to Basel. The two brothers disliked each other and worked in a spirit of intense competition on scientific problems of common interest to them both. Following Jakob's death in 1705, Johann became the preeminent active mathematical scientist in Europe, a position he maintained for more than four decades.
Jakob and Johann Bernoulli made very important contributions to the "brachistochrone problem," first posed by Johann in 1696. In this problem, one seeks the curve that a body would have to follow to descend in minimum time starting from rest in a constant gravitational field. Both brothers solved the problem using different methods. The solution is a cycloid, the curve traced out by a point on the perimeter of a circle as it rolls without slipping along a straight line. The methods used by the Bernoullis formed the basis for a new field of mathematics called the calculus of variations.
D'ALEMBERT'S PRINCIPLE, LAGRANGE, AND VARIATIONAL MECHANICS
D'Alembert's principle, developed by Jean le Rond d'Alembert (1717-83) in his Treatise on Dynamics published in 1743, is a way of rewriting Newton's second law by bringing the ma term to the other side as a fictitious force. Hence the sum of all the real forces on a system minus the fictitious ma force is equal to zero. This is a way of reducing a problem in dynamics to a problem in statics.
The general analytic form of d'Alembert's principle was made the basis for classical mechanics by Joseph Louis Lagrange (1736-1813). Lagrange used d'Alembert's principle rather than Newton's laws in his Analytical Mechanics, which was published in 1788. Classical mechanics depends on using a system of coordinates that are usually, but not always, the Cartesian coordinates x, y, and z. Using the calculus of variations, Lagrange developed a sophisticated alternative formulation of mechanics, called Lagrangian mechanics, involving generalized coordinates q and p. The p coordinates are generalized momentum coordinates, and the q coordinates are generalized position coordinates.
The equations of motion in Lagrange's formulation of mechanics are called Lagrange's equations. These equations do not represent a new physical theory; they are merely a different and equivalent way of writing Newton's laws of motion.
In 1740, Pierre Louis Maupertuis (1698-1759) reasoned that a certain mathematical integral, which he called the action, is always a minimum. The action is the integral ∫ mv ds. Here m is the mass of the particle, v is its speed, and ds is an element of the trajectory. In 1744, Maupertuis published two memoirs based on this minimum principle, which he called the principle of least action. The variation of the action is the change in the action from one path to an immediately adjacent path. In the calculus of variations, a curve that maximizes or minimizes a definite integral is called an extremal. The variation of an extremal is zero. In 1744, Leonhard Euler (1707-83) published a proof that the action integral for central force orbits was either a maximum or a minimum. In 1788, Lagrange showed in his Analytical Mechanics that the action integral was either a maximum or a minimum for conservative forces (these are forces for which the work or potential-energy function of the system depends only on the positions of the bodies).
CALORIC THEORY OF HEAT
It is well known that heat flows from a hot body to a cold body. What is the nature of this heat? In the theory popular in the late eighteenth century, heat was thought of as a weightless fluid called caloric. This theory has long since been rejected in favor of heat as microscopic energy transfers, but the old picture of the flow of caloric has been retained in the term heat flow.
The American-born English scientist Count Rumford (Benjamin Thompson, 1753-1814) performed experiments in the 1790s that showed that a hotter body did not weigh more than the same body at a lower temperature, thus confirming that caloric was weightless. He also produced experimental evidence that raised fundamental questions about the caloric theory. Rumford showed that heat can be produced without limit at the expense of mechanical energy. Heat could not be a conserved weightless fluid if it could be created. He observed, while engaged in boring cannon, that the cannon continued to get hot while the boring continued, even after the cutting tool was no longer sharp. It had been thought that the fine pieces of iron transmitted caloric, which was an invisible, microscopic fluid. The caloric was the heat. This heat made the cannon warm. In this theory, the fine particles were needed to surround, penetrate, and warm the iron. A fluid medium was needed to accomplish the temperature rise in the iron of the cannon. But if the fine particles of iron were no longer present because the cutting tool had become dull, there had to be a problem with the caloric theory of heat. Moreover, the supply of heat was inexhaustible as long as the boring tool continued to exert a frictional force on the iron. Rumford said: "anything which any insulated body or system of bodies, can continue to furnish without limitation, cannot possibly be a material substance, and it appears to me to be extremely difficult, if not quite impossible, to form any distinct idea of anything capable of being excited and communicated in the manner the Heat was excited and communicated in these experiments, except it be MOTION." Rumford and others gradually became convinced that heat was not a fluid, but something mechanical. Thus there arose in the early nineteenth century what was called the mechanical theory of heat.
Previously, it had seemed as if the subject of heat was independent of mechanics. A separate unit, the calorie, was therefore devised for the measurement of heat. A calorie was defined as the heat required to raise the temperature of 1 gram of water by 1 degree centigrade (now Celsius). The corresponding unit in the English system was a British thermal unit, defined as the heat required to raise 1 pound of water by 1 degree Fahrenheit. If heat was really a form of energy, as nineteenth-century physicists eventually came to believe, it should be possible to carry out an experiment whose result would determine how many mechanical units of energy equaled one heat unit. As a result of such experiments and various difficulties with the caloric theory, acceptance of the mechanical theory of heat became universal. Heat became "energy in transit."
SADI CARNOT AND THE MOTIVE POWER OF FIRE
Sadi Carnot (1796-1832) may rightfully be considered to be the founder of the science of thermodynamics, despite his having used the later discredited caloric theory of heat in the only work he published in his lifetime, his Reflections on the Motive Power of Fire (1824). He continued his research after publication of the Reflections, and his notes indicate that he abandoned the caloric theory for the mechanical theory of heat. These notes were published only posthumously. Carnot's theoretical ideas on heat engines in his slim (118 pages), partly incorrect book secured for him a major position in the history of physics.
Carnot had the shortest lifetime, 36 years, of any of the giants in the history of physics. His period was that of the Industrial Revolution and the steam engine. He was the son of Lazare Carnot (1753-1823), a prominent figure in the era of the French Revolution and Napoleon, and a prominent scientist in his own right. The Carnot scholar Eric Mendoza (1919-2007) attributed Sadi's ability to generalize to his father and said of Carnot's Reflections memoir that "the Memoir transcended technical details because Sadi had inherited from his father the capacity to generalize, to see the fundamental processes animating a complicated mechanism. Thus he saw that in an engine--any engine--an amount of caloric fell from a high to a low temperature; he extended some of his father's ideas on mechanics to apply to thermal processes--the impossibility of perpetual motion, the need to avoid irreversible changes."
Carnot was trained at the prestigious École Polytechnique from 1812 to 1814. Following this, he studied military engineering at the École du Génie at Metz for two years. In 1819 he was granted a permanent leave of absence from the army and embarked on a highly productive period of study and research in Paris. On June 12, 1824, his Reflections was published by Bachelier. After that, except for a short period when he returned to military service, Carnot continued his research in thermodynamics. He died on August 24, 1832, a victim of a cholera epidemic.
Carnot's Reflections was generally ignored by his contemporaries in science when it was published. One might hazard the guess that this was because Carnot was not a member of the scientific establishment. He had been trained as a military engineer and served in the army. How could a book written by such a person command the interest of the trained physicists of the period? Quite possibly it was this connection of the name of Sadi Carnot with engineering, as distinct from science, that delayed the recognition of Carnot's fertile ideas on thermodynamics.
Once Carnot's work had belatedly been recognized, people had to deal with his use of the caloric theory in the Reflections, and more important, with the absence of the conservation of energy principle in that work. The caloric theory is wrong. However, at the time Carnot wrote his Reflections, he believed that this theory was correct. It was recognized that after the publication of this book, he came to accept both the mechanical theory of heat and the first law of thermodynamics. Carnot's book was republished in 1878 in a new edition, which also contained a letter from Hippolyte Carnot (1801-88) to the French Academy of Sciences, and some of the unpublished notes. Hippolyte saw his brother as the founder of the science of thermodynamics. He said: "We are therefore justified in saying that if, in his first work, published in 1824, he formulated the principle to which his name is preserved, by his later work he also discovered the principle of equivalence, which makes up, with the first, the fundamental basis of thermodynamics."
Carnot opens his Reflections by stressing the practical importance of heat engines: "The study of these engines is of the greatest interest, their importance is enormous, their use is continually increasing, and they seem destined to produce a great revolution in the civilized world." This kind of opening is what one would expect from a practical engineer enthusiastic about practical progress. But, shortly thereafter, he writes: "The phenomenon of the production of motion by heat has not been considered from a sufficiently general point of view. ... In order to consider in the most general way the principle of the production of motion by heat, it must be considered independently of any mechanism or any particular agent. It is necessary to establish principles applicable not only to steam-engines, but to all imaginable heat-engines, whatever the working substance and whatever the method by which it is operated." This is the voice of the scientist, not that of the engineer; and in the very next paragraph, Carnot calls for a physical theory that can be used to achieve for heat engines what has been achieved for mechanical devices such as waterfalls or windmills: "We shall have it only when the laws of physics shall be extended enough, to make known beforehand all the effects of heat acting in a determined manner on any body." Sadi Carnot is calling for the establishment of the science of thermodynamics. The remainder of his book is his contribution to this new science.
Carnot fixed his attention on the "reestablishment of equilibrium in the caloric." By this he meant that what was characteristic of a heat engine was the passage of the caloric from a higher temperature to a lower temperature. This passage of the caloric from a warm body to a cold body was what he saw as a general guiding principle that was independent of the particular heat engine employed and of the working substance used in the heat engine cycle. Carnot saw the motive power of heat in its ability to produce volume changes in substances. These volume changes produced the motive power; they "drove" the piston of the heat engine. He said: "To heat any substance whatever requires a body warmer than the one to be heated; to cool it requires a cooler body. We supply caloric to the first of these bodies that we may transmit it to the second by means of the intermediary substance."
The most important theoretical development in the Reflections is Carnot's proof that there is a limit to the efficiency of a heat engine that is independent of the working substance, depending only on the temperatures of the reservoirs between which the engine operates. In the Reflections, Carnot gives two proofs of his theorem. The first proof is for a steam engine cycle, and the second, more rigorous proof uses a gas as the working substance.
In the proofs presented by Carnot, he imagined that a more efficient engine existed and coupled this more efficient engine with the less efficient cycle run backward (a refrigerator). The net effect would be to produce excess work by the more efficient cycle. This continuous production of work, with no other effect, would be a perpetual motion machine. Carnot rejected this as impossible and said (in his first proof utilizing the steam cycle): "Such a creation is entirely contrary to ideas now accepted, to the laws of mechanics and of sound physics. It is inadmissible. We should then conclude that the maximum of motive power resulting from the employment of steam is also the maximum of motive power realizable from any means whatever."
Carnot stressed that the use of reversible processes was the criterion for knowing that a heat engine was giving the maximum motive power obtainable from a set of given hot and cold reservoirs. A reversible process is not possible in the real world. It is a theoretical process that can never be fully realized (e.g., a process in which there is no friction). Carnot stated that the condition for a maximum is "that in the bodies employed to realize the motive power of heat there should not occur any change in temperature which may not be due to a change of volume." This requirement by Carnot is equivalent to requiring reversible processes only. Carnot declares: "Every change of temperature which is not due to a change of volume or to chemical action (an action that we provisionally suppose not to occur here) is necessarily due to the direct passage of caloric from a more or less heated body to a colder body," that is, due to an irreversible flow of heat.
Carnot is honored for being the first to call attention to a general consideration of engine cycles, for proving that there was a theoretical limit to their efficiency, and for realizing the significance of reversible heat processes.
CONSERVATION OF ENERGY: JOULE AND THE MECHANICAL EQUIVALENT OF HEAT
James P. Joule (1818-89) was the son of a prosperous brewery owner. At the age of 16, he was sent to study with the famous chemist John Dalton (1766-1844). Although this arrangement lasted only for a short time because of the illness of Dalton, it probably contributed to Joule's interest in science. Since he was financially independent, Joule felt free to pursue experimental research. He converted one of the rooms in his father's house into a laboratory for experimental investigations. In 1840, he presented a paper to the Royal Society of London in which he showed that the heat generated by an electric current was proportional to the square of the current, with the constant of proportionality being the resistance of the conductor. In 1850, he published a memoir in the Philosophical Transactions in which he described his famous paddlewheel experiment (see below) and the precise value of the mechanical equivalent of heat that he had obtained with that apparatus. Following the publication of that classic paper, he was elected to fellowship in the Royal Society. Many honors were bestowed on him, including the naming of the joule as a unit of energy by the Second International Congress on Weights and Measures. Economic misfortune overtook him later in life, and in 1878, he was granted a pension of £200 a year, which permitted him to continue his scientific work.
The idea that heat was a form of energy achieved general acceptance around the middle of the nineteenth century. An important ingredient in this acceptance was the measurement of the mechanical equivalent of heat in a series of experiments conducted by Joule during the 1840s. The mechanical equivalent of heat is the number of mechanical units (joules) equal to one heat unit (1 calorie).
Joule measured the mechanical work needed to raise a measured quantity of water through a measured temperature rise. The figure shows the essential elements of his experiment to measure the mechanical equivalent of heat. The falling weights shown on the right and left do a measured amount of work as they fall through a measured distance. The work of the weights is used to rotate a brass paddlewheel in the copper calorimeter vessel shown in the center. The copper vessel contains a measured weight of water. This water is heated by the friction between the water and the rotating paddlewheel. A thermometer inserted through the copper lid of the vessel is used to measure the temperature rise of the water in the vessel. Joule took elaborate precautions to avoid heat losses to and from the copper vessel. He equated the mechanical work done by the falling weights to the heat that raised the temperature of the water, the copper vessel, and the brass paddlewheel. He listed his final results as follows:
- The quantity of heat produced by the friction of bodies, whether solid or liquid, is always proportional to the quantity of force expended.
- The quantity of heat capable of increasing the temperature of a pound of water [weighed in vacuo (in a vacuum) and taken at between 55 and 60°] by 1° Fahrenheit, requires for its evolution the expenditure of a mechanical force represented by the fall of 772 pounds through the space of 1 foot.
1 British thermal unit = 778 foot-pounds
1 kilocalorie = 4187 joulesIt is indeed remarkable that Joule's final result of 772 foot-pounds is only about 1 percent different from the modern accepted value.
Heat and work are similar in that they both change the energy of a system. The distinction between heat and work is that in the case of heat, the energy is transferred microscopically, whereas in the case of work, the energy transferred can be measured macroscopically in terms of a force moving through a distance. The generalization of the conservation of energy to include heat is called the first law of thermodynamics.
CLAUSIUS AND ENTROPY
Rudolf Clausius (1822-88) was first to introduce the concept of entropy (see below) into thermodynamics. He stated the second law of thermodynamics in terms of his entropy concept as: "The entropy of the universe tends to a maximum." This was set forth in a brilliant early paper on the fundamental nature of heat. It is an excellent illustration of the general method of science. In this particular case, Clausius wished to retain the proven ideas of Carnot about heat engines but also wanted to deal with unanswered questions about the fundamental nature of heat. On the basis of Joule's experiments, he rejected the idea that heat is an indestructible substance that can never be created or destroyed.
Clausius examined the principle of the equivalence of heat and work. He said that it was a common assumption that the total heat of a body depends on the physical properties of the body, such as its temperature, density, and so on. If a body is brought back to its original state, the work done in the closed cycle shall be zero. But he found that Carnot's assumption that heat is conserved in a closed cycle must be rejected.
BOLTZMANN AND STATISTICAL MECHANICS
The entropy concept is connected with the idea of system disorder. The greater the disorder of a system, the greater its entropy. Using a pack of playing cards as an example, if the pack is arranged by suit, the order is greater than in a jumbled pack. If, in addition, the cards are arranged in serial order within a suit, the order is greater than if they are simply arranged by suit.
Now consider a gas confined to a container. There are billions of gas molecules, and nobody would ever attempt to describe the gas by means of the positions and velocities of individual molecules in the gas container. One would think of describing the gas in terms of its pressure, volume, and temperature. These overall properties of the gas are called its macroscopic properties. But somehow the macroscopic properties must be related to the billions of microscopic coordinates of position and velocity of individual molecules of the gas. The study of connections between the two levels is called the kinetic theory of gases.
Kinetic theory was built up from gas laws such as Boyle's law and Charles's law. Boyle's law states that if a gas is trapped in a vessel of variable volume, and if the temperature of the gas is kept constant, then the product of pressure and volume equals a constant. Charles's law states that for a constant mass of a gas held at constant pressure, the volume of the gas is proportional to the temperature. The ideal gas model of a gas was described by the familiar expression pV = nRT, where p is the pressure, V is the volume, T is the temperature, n is the number of moles, and R is a constant of proportionality.
In the kinetic theory of gases, one describes the statistical distribution of the speeds of the individual gas molecules. A distribution function of molecular velocities was developed in the nineteenth century by James Clerk Maxwell (1831-79) and Ludwig Boltzmann (1844-1906). It is called the Maxwell-Boltzmann distribution of molecular speeds. One can develop expressions for macroscopic properties, such as pressure, based on assuming a very large collection of gas molecules banging around inside a container of volume V. For example, one can derive that the average kinetic energy of the gas molecules is 3/2 kT, where T is the absolute temperature and k is called Boltzmann's constant.
PLANCK AND BLACKBODY RADIATION
It is well known that heat conduction occurs by one of three thermal processes: conduction, convection, and radiation. Every system in thermal equilibrium emits electromagnetic radiation that is characteristic of both its temperature and other properties of the system (such as the nature of the system surface). One type of system, called a blackbody, emits radiation that is characterized only by the temperature of the system. A blackbody is an ideal absorber of radiation. Since it absorbs all the radiation incident on it, and reflects none of this incident radiation, it appears black. The characteristic radiation emitted by a blackbody at a given temperature can serve as a measure of its temperature. Blackbody radiation is also known as cavity radiation. If one has a closed cavity at a certain absolute temperature and one makes a small hole in the cavity, the radiation that escapes through the small hole is blackbody radiation and is characterized only by the absolute temperature of the cavity. To model the radiation inside the cavity, one can think of possible standing-wave patterns inside the cavity. Clearly, there are many such standing-wave patterns that can fit inside such a cavity. For example, think of a one-dimensional box. The wavelength (λ) of the wave has to fit inside the box. So n wavelengths must fit inside a box of length L = n λ, where n is any integer. The quantization condition for a one-dimensional box is that n is any positive integer. The historical significance of Max Planck's (1858-1947) model for blackbody radiation is that it quantized the standing-wave modes in a cavity.
At the end of the nineteenth century, one of the unsolved problems of physics was to offer an explanation and model for blackbody radiation. Formulas that were correct at the short wavelength end (Wien's law) and at the long wavelength end (the Rayleigh-Jeans law) were known. But what law would work for the entire range of wavelengths in the blackbody spectrum? This figure shows the radiation spectrum for a blackbody at several different temperatures.
The x-axis shows the wavelengths of the various components of the radiation, and the y-axis shows the amount of radiation emitted by the blackbody for each wavelength interval. Note that the general pattern is that the radiation emitted by the blackbody shifts to lower wavelengths as the temperature increases. At room temperature, the great bulk of blackbody radiation is at very long wavelengths, far beyond the visible region of 4000 to 7000 angstroms (1 angstrom = 10-10 meter). At room temperature, the blackbody indeed appears black. As the temperature is raised from room temperature, the blackbody begins to give off appreciable light. First it appears red hot as it emits appreciable radiation in the red or long-wavelength end of the visible spectrum. Then, as it begins to emit appreciable radiation throughout the visible range, it appears white hot. Our Sun, whose radiation is approximately that of a blackbody, appears white hot and has a surface temperature of about 6000 kelvin.
In the year 1900, Planck offered a model for blackbody radiation that indeed led to the empirically observed curve. In analyzing this problem, one counts standing-wave modes inside a cavity. These standing waves are the model for the blackbody radiation. The energy of a cavity oscillator is proportional to its frequency, with the constant of proportionality designated by h, which is Planck's constant. When a jump occurs between oscillator energy levels, a quantum of radiation is emitted or absorbed. Planck's blackbody radiation law of 1900 marks the starting point for the quantum theory that developed over the next 30 years.
The peak wavelength λmax of the blackbody radiation can be used as an indication of its temperature. It can be shown that the relationship between this wavelength and the temperature T in kelvin is given by λmax T = 0.2898 cm · K. This relationship is called the Wien displacement law, where displacement refers to the shifting of λmax to lower wavelengths as the temperature T rises.
In 1965, radiation peaking at λmax = 0.107 centimeter (in the microwave region of the spectrum) was observed to be coming to Earth in all directions from outer space. This background radiation of the universe was interpreted as residual radiation from a big bang that occurred some 15 billion years ago [see Astronomy and Cosmology: Twentieth Century]. The kelvin temperature corresponding to this background radiation was T = 2.7 kelvin.
Twentieth Century: Einstein and Relativity
Albert Einstein (1879-1955) was born in Ulm, Germany. In 1900, he obtained a diploma from the Federal Institute of Technology in Zurich, Switzerland. In 1905, he published three important scientific papers: on the quantum of light and the photoelectric effect, on Brownian motion of particles and atomic theory, and on the special theory of relativity. In the same year, he also published an important paper about the relation between mass and energy, E = mc2 [see Overview: Physics].
In 1916, Einstein published his general theory of relativity [see Overview: Physics]. He was awarded the Nobel Prize in Physics in 1922 "for his services to theoretical physics and in particular for his discovery of the law of the photo-electric effect." In 1930, he made an extended visit to the United States, and in 1932, he was appointed professor at the Institute for Advanced Study in Princeton. When the Nazis came to power in 1933, Einstein settled in the United States. He died at Princeton at age 76 on April 18, 1955.
In his 1905 paper on special relativity, Einstein extended classical mechanics into the domain of very high speeds. He was motivated to write his paper because he thought the principles of physics should be the same in all inertial reference frames. The principles of mechanics did not depend on which inertial frame the observer was in, but the equations of electricity and magnetism did depend on the reference frame. A simple example of this is the electromagnetic field due to a point charge. If the point charge is stationary, only an electrostatic field exists, whereas if the point charge is moving, both an electric field and a magnetic field exist. For the principles of physics to be the same in all inertial frames, it was necessary to postulate that the speed of light was the same in all inertial frames. With that change, the equations of electricity and magnetism would be of the same form in all inertial reference frames.
This was the dramatic revolution of special relativity. The speed of a light beam would always be measured at 186,000 miles per second (3 x 108 meters per second), irrespective of the motion of the observer. Making c, the speed of light in free space, a constant in this way changed the transformation equations connecting coordinate and time transformations between inertial flames. They were now mixed together so that there was no longer an absolute time, or an absolute simultaneity. Distant simultaneity of events required light signals. Scuttling the concept of absolute time produced all sorts of startling predictions. The new transformation equations that emerged between space and time coordinates were called the Lorentz transformation equations.
The Lorentz transformation equations connect space and time in a way that mixes them together:
Consider a light flash emitted at the common origin when t = t′ = 0. Where will that flash be at a later time? Since Einstein defined time in terms of a distant simultaneity convention, one should no longer think in terms of a common absolute time. One has to think of events. The arrival of the light pulse wavefront on the x-axis at time t is such an event and is defined by x′ = ct′ (where the constant c is equal to the speed of light in a vacuum). That event also has space-time coordinates in the primed reference frame. That event is defined by x = ct. Substituting the Lorenz transformation above in x′ = ct′, we do indeed come up with x = ct.
The special theory of relativity makes predictions about the length of moving objects and the time intervals recorded on clocks. One of the surprising results of the postulate about the constancy of c is that the measured length of moving objects is contracted in the direction of motion. It should be clearly understood that length contraction is the direct result of the convention about assigning coordinates and times in the manner directed by the Lorentz transformation. A length Lo, for example, can be assigned to a stationary length by the usual method of comparing it to a stationary standard length, such as a meter stick. But if you want to measure a moving length, burn marks made at both ends of the meter stick would have to be made at precisely the same instant in that reference frame in order to measure the distance between them.
Consider a length Lo in the primed reference frame, where the left end is x′1 and the right end is x′2. The length Lo is fixed in the primed frame. In the unprimed frame, that length is seen as a moving length. According to the Lorentz transformation,
where b is defined by . It can be seen that Lo = L/b or L = Lob. Moving lengths are therefore seen as length-contracted.
The length contraction of a moving object is a direct result of paying closer attention to the measurement process plus the postulate that the round-trip speed of light is c in all inertial frames. The idea of a one-way speed of light (or one-way speed of anything) is undefined in the special theory. If you tried to measure a one-way speed, you would need synchronized clocks at two different locations. To synchronize these clocks, you would have to send a light signal between the two locations and you would have to assume that the one-way speed of light is C1, the same as the round-trip speed of light. Suppose, on the other hand, that we want to compare the time intervals on a moving clock with those on a stationary clock. Let us fix the moving clock in the primed frame. This clock is stationary as seen in the primed frame. Let a time interval ΔTo be read on it. In the unprimed frame, it is a moving clock moving with speed v. We have to use the Lorentz time transformation to read time intervals ΔT on such a clock.
ΔT = ΔTo/bThe interval ΔT is longer than ΔTo by a factor of . An observer in S would say "moving clocks run slowly." This is referred to as time dilation: ΔT = ΔTo/b.
Another idea that emerged from special relativity is that the mass of a particle is not constant, as was supposed in classical mechanics. Instead, it increases with increasing speed according to the relation
where m is known as the relativistic mass. The total energy of a particle is a function of its mass, according to Einstein's famous relation E = mc2.
GENERAL THEORY OF RELATIVITY
The special theory of relativity is a theory that involves reference frames that are unaccelerated with respect to each other. Einstein later tackled the issue of accelerated reference frames. The theory that he constructed for accelerated reference frames is called the general theory of relativity.
When a reference frame is accelerated, such as an elevator is accelerated, there is an apparent effect on the weight of objects. When an elevator is accelerated upward, a rider in the elevator feels heavier; when it is accelerated downward, the rider feels lighter. So in accelerated reference frames the apparent force of gravity changes. Einstein reasoned that there should be no difference between gravity and the effect of an accelerated frame. He adopted the equivalence of these two effects as a postulate. This led to the prediction of some physical effects that had not been observed before. For example, the trajectory of a light beam should be bent by a gravitational field.
The theory of space, time, and gravitation is called the general theory of relativity. Attention has been focused on three effects predicted by Einstein's general theory of relativity. There is a very small effect on the orbits of the planets, light rays passing close to the Sun should be bent very slightly toward it, and physical processes should take place more slowly in regions of low gravitational potential. This last effect is called the gravitational red shift.
BOHR MODEL OF THE HYDROGEN ATOM
Niels Bohr (1885-1962) was one of the giants of modern physics. Bohr produced a model of the atom that successfully predicted the lines of the hydrogen spectrum. The Bohr model of the hydrogen atom was a great triumph.
Bohr's father was professor of physiology at the University of Copenhagen, where Bohr obtained his Ph.D. in 1911. His mother came from a wealthy Jewish family. The Jews were a principal target for Adolf Hitler and the Nazis. When the Nazis came to power in 1933, their rise ultimately forced Bohr and his family to escape from Denmark in 1943. Einstein, who was fully of Jewish descent, left Germany for the United States around 1933. Bohr and Einstein were the protagonists in the famous Bohr-Einstein debates about the interpretation of quantum mechanics.
Bohr is best known for his model of the hydrogen atom. A hydrogen atom, or any atom for that matter, is far removed from our immediate sensory experiences. That individual atoms do indeed exist was widely accepted at the time that Bohr was growing up. Bohr's atomic model arose when he worked in 1911 with Ernest Rutherford (1871-1937) at the University of Manchester [see Atomic and Nuclear Physics].
Quantum mechanics is the name given to the mechanics of the very small. Atoms are very small particles of matter. The simplest of atoms is the hydrogen atom. The earliest successful model of the hydrogen atom was the Bohr model. It was known that the hydrogen atom was electrically neutral, so that whatever the atomic model was, it had to consist of an electron and a proton. Bohr proposed a model for the hydrogen atom that was like a miniature solar system. The proton at the center was like the sun. The much less massive electron revolved about the central proton in one of a set of permitted Bohr orbits.
At the time, the only data that hinted at the internal workings of hydrogen atoms were the data from the hydrogen sharp-line spectrum. When hydrogen atoms are excited from their ground state, or the state of lowest energy, they emit various spectral lines. These lines had been studied in the nineteenth century and divided into spectral series. The part of the hydrogen spectrum that lay in the visible region (4000 to 7000 angstroms) was called the Balmer series.
The central idea of the Bohr theory was that the orbits were to be discrete, or quantized. The radiation emitted when the hydrogen atom changed from one discrete state to another was to be quantized so as to obtain a match with the known sharp-line spectrum.
The angular momentum of the orbiting electron was quantized as an integral multiple of Planck's constant h divided by 2π. The permitted nonradiating orbits were characterized by a quantum number n. When the electron in a hydrogen atom made a transition from one nonradiating orbit to another, the difference in energy of the atom showed up as a photon whose energy was equal to the energy difference. The predicted energies of the photons compared quite closely with the measured line spectra of hydrogen. This was a spectacular triumph for the Bohr theory.
The total energy of the Bohr hydrogen atom is negative, as it should be for a bound system (the electron is bound to the proton nucleus by the electrostatic force of attraction). It takes positive work to separate the electron from the proton. The energy of the lowest level is - 13.6 electron-volts.
During the nineteenth century, the wave model became the dominant model for light. Experiments on the interference of light from two slits, and all the evidence from diffraction, supported the wave model for light.
In the wave model, light has a frequency and a wavelength. The electromagnetic spectrum extends from very short wavelengths in the x-ray portion of the spectrum to very long wavelengths in the radio portion of the spectrum. Visible light, the very small portion of the vast electromagnetic spectrum to which the human eye is sensitive, occupies the range from 4000 to 7000 angstroms.
Experimental evidence from the photoelectric effect could not be accommodated successfully by the wave model for light. In 1905, Einstein provided a successful particle explanation for light in the photoelectric effect. Einstein's explanation of the release of photoelectrons from a metal surface viewed this phenomenon as a simple collision between light "particles" and the photoelectric surface.
The light particles were called photons. The energy of a photon was directly proportional to its frequency, and the proportionality constant was Planck's constant h. To release an electron from that surface requires a certain minimum amount of energy. This minimum energy depends upon the particular photoelectric surface and is called the work function of the photoelectric surface. The Greek lowercase letter phi, φ, is usually assigned to represent the work function.
Consider the release of a photoelectron from a surface on the basis of this Einstein model. The photoelectron receives energy of hf, uses up energy f in escaping from the surface, and is left with the balance as a kinetic energy of 1/2mv2. So the maximum kinetic energy of the photoelectrons is hf - f. Einstein's famous photoelectric equation is 1/2mv2max = hf - f.
If the light shining on the photoelectric surface is of a frequency such that the photons have energy equal only to the work function energy, the electrons come off the photoelectric surface with zero kinetic energy. This is the threshold for photoelectron emission by that particular surface. This threshold can be expressed as a threshold frequency or a threshold wavelength: f = hfthreshold = hc/λthreshold.
DE BROGLIE'S HYPOTHESIS AND THE WAVE BEHAVIOR OF MATTER
Light is dualistic: It possesses both wave and particle properties. Its wave properties are in evidence when one constructs models for interference and diffraction. Its particle properties are in evidence when one constructs a model to explain such things as the photoelectric effect.
In his doctoral dissertation in 1924, Louis de Broglie (1892-1987) hypothesized that matter, such as electrons, which had been classically considered only as particles, would also possess wave properties. The wave properties were associated with a wavelength of h/p, where p is the momentum of the matter and h is Planck's constant. This wavelength has come to be known as the de Broglie wavelength: λ = h/p. It is of some interest that the Bohr quantum condition for the permitted angular momenta in a stationary orbit of the hydrogen atom can be interpreted in terms of the de Broglie wavelength. One finds that an integral number of de Broglie wavelengths fit into the permitted Bohr orbits.
WAVE MECHANICS AND QUANTUM MECHANICS
The period of the 1920s is called the "heroic period" in physics. Bohr had tried unsuccessfully to extend his model of the hydrogen atom to the next element in the periodic table, helium. Younger physicists such as Louis de Broglie, Erwin Schrödinger (1887-1961), and Werner Heisenberg (1901-76) dominated the heroic period. Schrödinger developed a wave equation for de Broglie's matter waves. This is the famous Schrödinger equation. The wave amplitude in this equation is called a probability amplitude. A variant form of quantum mechanics was developed by Werner Heisenberg, who is also well known for the Heisenberg uncertainty principle.
To be sure, there are those who still struggle with trying to understand this abstraction in terms of physical reality. Einstein, much older now, proposed a famous paradox, called the Einstein-Podolsky-Rosen paradox, which sharpened the philosophic debate about the new quantum mechanics. Bohr disputed Einstein in a famous series of debates. But nobody can dispute the successes of the new theory of quantum mechanics or the agreement of its predictions with experimental reality.
Galileo's Pendulums and Planes
Galileo believed that the period of a pendulum was independent of angular amplitude. In the Two New Sciences, he claimed that the periods of a 60° pendulum and a 6° pendulum of the same length were exactly the same. If Galileo had conducted a simple experiment to test this, he would surely have noticed that the period of the larger-amplitude pendulum is longer than the period of the smaller-amplitude pendulum. In a vacuum, the period of a 60° pendulum is about 7 percent longer than that of a 6° pendulum.
One powerful reason for Galileo to believe that a pendulum had exactly the same period for all arcs from 90° down to the smallest arc was an intellectual leap from isochronous plane motion along the planes corresponding to circular chords to isochronous circular motion along the arcs corresponding to these chords. He believed that in a constant gravitational field, the time for a frictionless particle to descend to the bottom (point A) would be the same for all planes, such as BA and CA. Galileo thought that the descent along circular arc BA would take the same as along circular arc CA.
The demonstration referred to is found in Theorem VI of the third day: The descent times along all chords that terminate at the lowest point A of the circle are equal. Thus a body descends along all planes BA, CA, DA, EA, and so on, in the same time. After referring to the demonstration of equal times for chords, Galileo makes the leap to experience, showing equal times for all arcs of 90° or less.
Why did Galileo refer to experience here, whereas previously he spoke about demonstration? Almost certainly because he had tried to do a demonstration and failed. Indeed, as we now know, the demonstration of such a theorem is impossible because it is not true that equal times are taken in all descents along arcs of 90° or less. A pendulum has a period that is a function of amplitude, and this period increases with amplitude up to a maximum period at 90° amplitude. But Galileo had seen from experience that relatively small-angle pendulums are isochronous (very closely as we now know), so he made the leap to a general statement about all pendulums with amplitudes of 90° or less.
Words to Know
- A statement that is assumed to be true, without being proved.
- A statement that is to be proved.
- A physical quantity that has both magnitude and direction.
- Centripetal force
- A force that is directed toward the center of a system.
- The gram-molecular weight of a substance. One mole of any substance has M grams of the substance, where M is numerically equal to the molecular weight.
- A discrete packet or particle of radiation.
- The energy change that occurs when an electron is carried through a difference of potential of 1 volt.
- The bending of light around obstacles.