“To develop a complete mind: Study the art of science; study the science of art. Learn how to see. Realize that everything connects to everything else”
SECTION 1: Elements of the Physics of Motion
This section is for those students who don’t remember (or perhaps never were taught) elementary physics. I hope to give some qualitative notions of some basic concepts in physics and to do so with a minimum of mathematics, using pictures, animations and links to available explanations on the web. So, dear reader, imagine you’re living in pre-Renaissance Europe, and are listening to those Medieval monks explain what they think about motion, and how it differs from what Aristotle had to say.
1.2 DISTANCE, VELOCITY, ACCELERATION
First, let’s consider distance. I believe all you readers have an intuitive notion of what distance is: you draw a straight line between point A and point B and the length of that line is the distance between points A and B.¹
What is velocity, then? Velocity is a rate, distance per time. (And, to be fussy, velocity has direction; “speed” is the magnitude of velocity; you don’t care what the direction is; velocity is “speed” plus direction.)
Now I ask your pardon, dear reader to bear with me while I inject just a little math to make the concept clear. Suppose it’s four miles to the nearest rest stop on the thruway and you must get there in five minutes (or less–I won’t ask why. How fast do you have to travel or what should your car’s velocity be? Your rate of travel, speed, must be four miles in five minutes, or 4miles/ 5 minutes, or as it would be written conventionally, 4/5 miles/minute; in other words, distance divided by time. Since there are 60 minutes in an hour, a little arithmetic shows you would have to travel 60x (4/5) miles/hour or 48 mph². And here’s an equation (again, pardon)
v= d/t where v is velocity, d is distance and t is time to travel that distance
What is acceleration? It’s also a rate, the change in velocity divided by the corresponding change in time. Let’s turn again to an example with some numbers. Fresh out of grad school I bought a MG TD (red, no less!). The MG was not, to use my grandson’s lingo, “zippy.” From a standing start, it could get to a speed of 42 mph in about 20 seconds (real sports cars take only about 5 seconds to get to 60 mph). This acceleration rate corresponds happily (for nice numbers) to about 1 (m/s)/s or 1 m/s². So we have acceleration, a, given by the gain in velocity over the time, t, it takes to achieve that change:
a = (change in v) / t
Here’s an illustration to give you some notion of what acceleration and velocity look like. It’s the MG TD performing as above, going from 0 to 42 mph in 20 s and thereafter at the constant speed of 42 mph. The shots correspond to 4 s intervals from 8s to 28 s.
Velocities at 4 second intervals from 8s to 28 s. Acceleration is 1 m/s^2, to get to 42 mph in 20 s. Acceleration ceases at 20 seconds, so velocity is constant from 20 seconds to 28 seconds; the speed is listed above each car image; the arrow length corresponds (roughly) to the velocity:
An easy way to think about constant acceleration is that the distance covered in a given time is average velocity multiplied by the time. The average velocity is just (1/2) (v_beginning + v_end).³
As pointed out in ESSAY 1, SECTION 3.2, Nicolas Oresme had derived these relations between velocity, distance and acceleration by a graphical analysis, 100 years before Galileo. However, it was Galileo who did the science: confirmed the theory by experiment.
How did Galileo set up an experiment where the motion would be slow enough for him to measure time, distances and speed? Acceleration of falling bodies would be too fast. Here’s the experiment, done in elementary physics lab classes. An inclined plane, as in the illustration below, length L, is set up so that the top end of the plane is a height h above the ground. A ball or cylinder rolls down the plane and you measure distance traveled in given times. Now if the plane were to be vertical (h=L), the ball would fall with an acceleration that of gravity (9.8 m/s²) and that would be too fast. If the plane is flat (h=0), the ball would not roll at all (hey! that’s poetry?). Clearly the acceleration is going to vary as the height h changes. It turns out that the acceleration is proportional to h/L. It will be the same–independent of size or material–for a given shape sliding or rolling down the plane.
How do objects acquire velocity, that is accelerate? Buridan in the 14th Century had ideas about velocity that anticipated Galileo and Newton centuries later. He said that a moving body had “impetus,” the heavier the body moving at a given velocity, the more impetus it had. If you threw a ball, the motion of your arm gave the ball its impetus. “Impetus” is what we now call “momentum” and define as
momentum = mass x velocity
Mass is what we ordinarily think of as weight, but to be fussy, weight is really mass times the force of gravity. You can think of mass as resistance to change in motion, what would technically be termed “inertia.”
Here’s an example to give you some intuitive notion about momentum: the MG TD referred to above is a very light car, weighing only about 1/2 ton (1000 pounds); a late model Cadillac is much heavier, weighing about two tons. Accordingly, the mass of the Caddy is about four times greater than that of the MG. So, if the MG were traveling at 40 mph and the Caddy at (1/4)x 40 mph = 10 mph, they would have equal momentum (if they were traveling in the same direction–remember, velocity has direction, speed does not). This is illustrated below.
What causes a body to accelerate, acquire velocity? Again, Buridan had the right qualitative notion: the body acquired impetus because of an action by an agent, you, throwing the ball with your arm. In this notion there is an implied notion of force, which Newton (17th century) made explicit by his Second Law of Motion:
Force = mass x acceleration
more generally if mass doesn’t stay constant (think of an example involving liquids!)
Force = change of momentum/change of time
For the first definition, go back to the example of the accelerating MG: the force is provided by friction between the tires and the road, the tires—wheels—are made to go round by the engine turning a drive-shaft.
For the second definition, think of a pitcher winding up and releasing a baseball moving at 90 mph as depicted in this video . The baseball has a mass of about 0.15 kg (or about 0.3 lbs) If you go frame by frame in the video, you’ll see that it takes less than 10 ms (0.01 s) for the pitcher to start his windup and release the ball; that’s the change in time for the baseball to acquire its velocity of 90 mph (we’ll neglect air friction slowing the ball down). So, fussing with units—I don’t need for you all to mess with the arithmetic—you get a force of about 650 Newtons required.
For comparison, the force of gravity on the baseball is about 1.5 Newtons. If air friction is neglected, from what height would the ball have to fall to get this 90 mph velocity? About 100 yards. Why the greater force to throw the ball this fast? Because the force of the throw is acting for only a short period of time, during the pitcher’s windup, whereas gravity will be acting all during the fall.
1.5 KINDS OF ENERGY; CONSERVATION OF ENERGY
There are two other physics concepts, as important as velocity, acceleration and force, that bear on motion, and those are energy and work. I’ll talk about “Work” in Section 2, below, but here are some ideas about the different kinds of energy. To get an intuitive idea of this, let’s go into more detail about how the MG acquires velocity.
First, fuel is burnt in the cylinders to move the pistons up and down and thereby rotate the shaft that turns the rear wheels around, moving them against the friction of the road. We have then chemical energy from the gasoline combining with oxygen (burning) converted to mechanical energy. The energy of motion is called “kinetic energy” and is given by the formula
Kinetic Energy = (1/2) mass x velocity^2 (the “^2” means “squared”)
Another important form of energy is “potential energy,” energy a body has by virtue of its position. Let’s think about what this means. When you let a ball roll down an inclined plane it has zero kinetic energy at the top and kinetic energy at the bottom after it’s accelerated due to gravity and acquired velocity. So where does that kinetic energy come from? To balance the energy books we say the ball at the top of the plane has potential energy that can be converted to kinetic energy. This potential energy is given (for gravity at the surface of the earth) by
Potential Energy = mass x g x h= mgh
where g is the acceleration due to gravity (9.8 m/s^2), h is the height above the bottom
This is illustrated below:
An important principle of physics is that energy is conserved. What does that mean? It means that energy doesn’t disappear into nowhere, for example:
- kinetic energy, energy of motion is lost due to friction, but is converted to the same amount of heat energy;
- kinetic energy, energy of motion is lost due to work done, moving the MG up a hill–the work done is equal to the amount of kinetic energy lost; the work done is equal to the gain in potential energy at the end;
- chemical energy of the gasoline is converted to kinetic energy less friction losses in the engine and drive shaft.
Accordingly, the energy bank account balances: input (at the beginning) of chemical energy, gasoline in the fuel tank = kinetic energy at the end of the drive, when the fuel tank is empty + energy lost due to friction of the tires with the road, engine and drive shaft friction + work done due to a net change in height level at the end or gain in potential energy. One important concept that deals with how energy is lost or gained is “Work,” discussed next.
What do we mean in physics by the term “work”? It means applied force times distance moved. If you apply a force—push against a stone wall—but don’t move the wall, you may work up a sweat, but you haven’t done any work. These ideas are illustrated below. In the two diagrams below, a basket is moved up a distance d. The force applied is the weight, mg, due to gravity: F=mg; the distance moved is “d.” So the work W is given by
Work = applied force times distance moved or W = mg X d
basket after being lifted a distance d. the basket is now a height h+d above the ground and the potential energy is mg(h+d)
In the next two diagrams the basket is moved across a table against a resisting frictional force, Fr. Again the basket moves a distance d, so the Work done on the basket is W=FrXd.
I should emphasize that the examples given are for “mechanical work. I also want to emphasize again that doing work is more than exerting a force. Work is force times distance force moved.
1.7 WORK, HEAT AND ENERGY
To repeat: there are many kinds of energy: for example, mechanical; electrical; magnetic, chemical, heat. All these forms of energy can be converted to work and work can be changed into these several forms of energy. (See this interesting video about conversion of different forms of energy.)
In the first example above, a basket is pulled up a distance d against the force of gravity, mg.
- before the lift the potential energy was mgh;
- after the lift the potential energy was mg (h+d) (the height above the ground of the basket has increased to h+d)
- so the difference (after – before) is just mgd, is the increase in potential energy;
- but this is just the work done, mgd, force x distance, done in lifting the basket.
In the second example the work done does not increase the potential energy of the basket—it’s still at the same height. Where has the energy which should have been produced by the work gone? Recall that the basket moved against a frictional force. What form of energy is produced by friction? Heat! An account of Joule’s experiment on the conversion of work to heat:is given in Section 2.2
In SECTION 2, I’ll have more to say about the science of energy, “Thermodynamics,” particularly these two important laws: The First and Second Laws of Thermodynamics.
¹Let me add a cautionary note physicswise: if you are traveling between A and B (home and the local fastfood place, let’s say) and you wander around, make side-trips, the distance is still the length of the line between beginning and ending points. If you want to get total mileage traveled, then you have to draw straight lines between each of the intermediate starting and stopping points and add the lengths up.
² Since each hour contains 60 minutes, you would have to go 60 (
minutes/hour) x (4/5) (miles/ minute) or 60 x (4/5) (miles/hour)= 48 (miles/hour).
³For our example, the distance covered by the accelerating MG between 12 seconds (v_beginning = 25 mph) and 16 seconds (v_end=33mph) is just
(1/2) (25+33) (miles/
hour) x(1 hour/ (3600 seconds) x (16-12) seconds or about 56 yards
SECTION 2: Thermodynamics, the Science of Energy
“It looks full of hard words and signs and numbers, not very entertaining or understandable looking, and I wonder whether it will make people wiser or better.’ So wrote a cousin of Josiah Willard Gibbs when she happened onto a copy of his most famous paper on thermodynamics lying on his desk.”
—As quoted from Order and Chaos, by Stanley Angrist and Loren Hepler.
From the uncoiling energetics of DNA to the information lost into black holes, thermodynamics enters into every field of science. The Second Law of Thermodynamics, all about order and disorder—you can’t (realistically) unscramble eggs—is perhaps the most fundamental of those principles at the inner core of the Lakatos sphere. Einstein’s comment about thermodynamics says it all:
“A theory is the more impressive the greater the simplicity of its premises, the more different kinds of things it relates, and the more extended its area of applicability. Therefore the deep impression that classical thermodynamics made upon me. It is the only physical theory of universal content which I am convinced will never be overthrown, within the framework of applicability of its basic concepts.”
—Albert Einstein (author), Paul Arthur, Schilpp (editor). Autobiographical Notes. A Centennial Edition. Open Court Publishing Company.
In this section I’ll try to explain some fundamental concepts in thermodynamics and to explore what the First and Second Laws of thermodynamics tell us about the world. Before doing that a brief account of how thermodynamics developed is in order.
2.2 A BRIEF HISTORY OF THERMODYNAMICS
Benjamin Thompson, Count Rumford (1753-1814), showed that heat was not a substance (the "caloric") by his cannon-boring experiments
Sadi Carnot, "The Father of Thermodynamics" (1796-1832)
Carnot was an engineer who wanted to understand how steam engines worked. He gave us the "Carnot Cycle" which explained how heat was converted to useful work.
Rudolf Clausius (1822-1888), German Physicist/
Clausius proposed the First and Second Laws of Thermodynamics.
Sir James Joule, (1822-1889) , British physicist, established the convertibility of work to heat, which lead to the First Law of Thermodynamics, the conservation of energy.
Hermann von Helmholtz (1821-1894), German physician and physicist, who developed applications of thermodynamics to chemistry and electrodynamics. The thermodynamic "Work Function, A" .is also called the Helmholtz Free Energy.
Ludwig Boltzmann (1844-1906), Austrian Physicist, who tried to explain thermodynamics from kinetic theory, the theory of how large numbers of molecules interact. He developed "statistical mechanics," using probability and statistical arguments, to explain and justify the Second Law of Thermodynamics.
James Clerk Maxwell (1831-1879), English theoretical physicist, is more renowned for his theory of electromagnetism ("Maxwell's Equations"), but he, as did Boltzmann, developed fundamental equation in statistical mechanics to show that that heat was a form of random molecular motion.
Josiah Willard Gibbs (1839-1903), American physicist and mathematician, developed a rigorous theory to explain thermodynamics on a molecular level, statistical mechanics. He showed how thermodynamics could be applied to physical chemistry to make it a mathematically rigorous discipline. The thermodynamic function, G, the Gibbs Free Energy, shows how energy and entropy combine to determine chemical stability.
The pictures above are of scientists who developed thermodynamics in the 19th century: beginning with the American (but a Loyalist) Benjamin Thompson, Count Rumford, who showed by his cannon-boring experiments that heat was not a substance (the “caloric”) but something else, not conserved; and ending with the American, Josiah Willard Gibbs, who developed a theory, statistical mechanics, that explained thermodynamics in terms of molecular motions and probability (capping theories by Maxwell and Boltzmann). Gibbs also developed an elegant mathematical form for the laws of thermodynamics.
I’ll discuss briefly how each of these scientists contributed to the development of thermodynamics.
History of Thermodynamics: Count Rumford: Cannon Boring —> Heat Not Conserved.
In 1798 Benjamin Thompson, Count Rumford, submitted a paper to the Royal Society about his experiments in which boring a cannon could make water boil, and boring with a blunt instrument produced more heat than with a sharp one (more friction with the blunt). The experiments showed that repeated boring on the same cannon continued to produce heat, so clearly heat was not conserved and therefore could not be a material substance.
This experiment disproved the then prevalent theory of heat, that it was a fluid transmitted from one thing to another, “the caloric.” The results validated another theory of heat, the kinetic theory, in which heat was due to the motion of atoms and molecules. However the kinetic theory, despite Rumford’s groundbreaking experiment, still did not hold sway until years later, after James Joule showed in 1845 that work could be quantitatively converted into heat.
History of Thermodynamics: James Joule: Work—>Heat
As the weight falls, the potential energy of the weight is converted into work done (a paddle stirs the water in the container against a frictional force due to water viscosity). The temperature rise corresponding to a given fall of weights (work done) yields the amount of heat rise (in calories) of the known mass of water.¹ Since the temperature rise is very small, the measurements have to be very accurate.
It took 30 to 50 years after Joule’s definitive experiment (and subsequent refinements and repetitions) for the kinetic theory of heat—heat caused by random, irregular motion of atoms and molecules–to be fully accepted by the scientific community. James Clerk Maxwell published in 1871 a paper, “Theory of Heat”. This comprehensive treatise and advances in thermodynamics convinced scientists finally to accept that heat was a form of energy related to the kinetic energy (the energy of motion) of the atoms and molecules in a substance.
2.3 CONSERVATION OF ENERGY—THE FIRST LAW OF THERMODYNAMICS
The conservation of mechanical energy was discussed in Section 1: the potential energy of a body a height h above the ground is equal to its kinetic energy just before it hits the ground, where the potential energy is zero. The First Law of Thermodynamics states the conservation of energy in a more general way:
ΔE = Q + W
We focus here on a “system.” The system might be a container of water, it might be the earth, or anything of interest with some boundaries that are closed (by “closed” we mean that no matter crosses the boundaries of the system). “Q” is the heat absorbed by the system; W is the work done on the system; ΔE is the change in energy of the system.² (The “Δ” is a symbol for “change of.”)
An early statement (1850) of the First Law was given by the German physicist Rudolf Clausius:
“In all cases in which work is produced by the agency of heat, a quantity of heat is consumed which is proportional to the work done; and conversely, by the expenditure of an equal quantity of work an equal quantity of heat is produced.
Clausius also gave a definitive statement for the Second Law, but before discussing that I’d like to talk about how the Second Law developed and the concept of entropy came to be.
2.4 THE SECOND LAW: HEAT ENGINES.
The diagram below illustrates how steam engines work. Water is heated in the boiler to make steam, gaseous water. The steam passes through a pipe into a cylinder and expanding, moves a piston up, doing mechanical work. The steam then passes through a pipe and is cooled by condensing water to form liquid water. The water is pumped back into the boiler by a pump. Less work is used to pump the water from the condenser into the boiler than is done by the expanding steam in the engine cylinder.
History of Thermodynamics: Carnot’s Cycle
Carnot devised an abstract scheme for the working of a heat engine (for example, the steam engine) that laid the foundation for the later development of thermodynamics in his book “Reflections on the Motive Power of Heat.” This scheme was the “Carnot Cycle,” which set discrete stages for what happened to the water as it went from the boiler to the piston to the condenser and then back to the boiler.
Here’s how it works:
Heat (QH) is transferred from the hight temperature source, the boiler, at temperature TH, to the water. (T is the symbol used for absolute temperature.) The water vaporizes (steam) and goes into the piston expanding and doing work (W); the steam is condensed to liquid water in the condenser at temperature TL, and gives off heat (QL) to the condenser and is then pumped back into the boiler. That’s the cycle.
History of Thermodynamics: State Functions (Gibbs)
So, in this cycle the water goes back into the boiler at the same temperature, pressure, etc. as when it started to heat up and boil off. It’s like someone making a trip around the city from his home and coming back to the home. In 1876 Willard Gibbs set forth the concepts of states and state functions (see Note 3, below) to yield the following important and useful relation:
- if a system starts off in some initial state and ends up in some final state, then the value for a change in state function depends ONLY on what the initial and final states were, not on the path taken to get from initial to final state.
- Thus, if the initial and final states are the same—if you”re dealing with a “cycle”—then the change in the state function is zero, since the state function has the same value for the initial and final states—they are the same state.
Then we can say that since the initial and final states of the water (the system) in this heat engine cycle are the same, and since the Energy E of the system is a state function, the change in energy for this cycle is zero: ΔECYCLE = 0. (Recall, the “Δ” is a symbol for “change of.”) What is this change in E for the cycle in terms of the heat transferred and net work done? It’s net heat input minus work done by the system:
ΔE = QH – QL – W = 0 (1)
Notice that there’s a minus sign in front of QL because the system (the water) is transferring energy in the form of heat to its environment, the condenser. Notice also the minus sign in front of W; W is work done BY the system against the environment (pushing the piston against a resisting pressure) so the system has to lose energy if W is positive. So we get a relation between work done in the cycle, W, and the net heat transferred to the system, QH – QL :
W = QH – QL (2)
Is there any more information we can get about this? Yes, but we have to learn about entropy and the 2nd Law of thermodynamics in order to do so.
History of Thermodynamics: Clausius’s Definition of Entropy
Rudolf Clausius noticed something very important about heat: it flows spontaneously from a high temperature to a lower temperature, as, for example, if you drop an ice cube into a cup of hot coffee, heat will flow from the hot liquid to the cold ice cube and melt it. The greater the difference in temperature, the faster the heat flows from hot to cold.
So, here’s how Clausius might have thought about this: “Ach, so! (heavy German accent here, please!). Vat can I write that vill have heat and temperature in it? Let’s call this new function “entropy” from the Greek ‘εν τροπε, ‘in trope’ or ‘in change’ or ‘transformation.’ And l’ll denote it by the letter S.” (Why S? I don’t know.) “So, if we have a little bit of heat, and a high temperature, the transformation would be small, so let’s say that a little bit of S equals a little bit of heat transferred divided by temperature.” Actually, Clausius used arguments from calculus to arrive at his definition. See the 1867 English translation of the work in which he defined entropy.
Then what we get for a change in entropy, ΔS, for some change of states is
Δ S= adding up (little bits of heat/T) (3)
If the temperature stays constant (what’s called an “isothermal process”) you can add up the little bits of heat separately to have Q, total heat transferred to the system (Q>0) or from the system (Q<0) to get
¹Here’s how the amount of heat transferred to the water,Q, is determined. Q is related to the temperature rise, ΔT, as follows: Q = C ΔT. C is the “heat capacity,” which is proportional to the amount of water in the apparatus and a constant, specific heat capacity, c, that depends on the substance. For liquid water at ordinary temperatures, c= 1 calorie/ (gram x degree Centigrade).or 4181 Joules/(kilogram x degree Centigrade).
²If we were to conform strictly to current usage we would use ΔU rather than ΔE, where U is the “internal energy” of the system (as distinct from kinetic energy of the system, for example). This “Internal Energy” is defined by the First Law:.the Change in U is given by Q+W, but the “zero” of U is arbitrary. For example, if you’re concerned with chemical reactions you can define a zero of U for elements in their most stable state under standard conditions (e.g. oxygen as O2, diatomic molecules, at 25 degrees Centigrade and 1 atm pressure—if oxygen were behaving as an ideal gas).But why make things more complicated than necessary? The goal of this discussion is to achieve an intuitive understanding of what thermodynamics is about, not to pass a final exam.