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Table of Contents

Section 1.1 - Mass
Section 1.2 - Equivalence of Gravitational and Inertial Mass
Section 1.3 - Galilean Relativity
Section 1.4 - A Preview of Some Modern Physics


The universe has been recycling its contents ever since the Big Bang, 13.7 billion years ago.

Chapter 1. Conservation of Mass

\renewcommand{\gamma}{\gamma} It took just a moment for that head to fall, but a hundred years might not produce another like it. -- Joseph-Louis Lagrange, referring to the execution of Lavoisier on May 8, 1794

1.1 Mass


a / Portrait of Monsieur Lavoisier and His Wife, by Jacques-Louis David, 1788. Lavoisier invented the concept of conservation of mass. The husband is depicted with his scientific apparatus, while in the background on the left is the portfolio belonging to Madame Lavoisier, who is thought to have been a student of David's.


b / A measurement of gravitational mass: the sphere has a gravitational mass of five kilograms.


c / A measurement of inertial mass: the wagon recoils with the same velocity in experiments 1 and 2, establishing that the inertial mass of the cement block is four kilograms.


d / The time for one cycle of vibration is related to the object's inertial mass.


e / Astronaut Tamara Jernigan measures her inertial mass aboard the Space Shuttle.

Change is impossible, claimed the ancient Greek philosopher Parmenides. His work was nonscientific, since he didn't state his ideas in a form that would allow them to be tested experimentally, but modern science nevertheless has a strong Parmenidean flavor. His main argument that change is an illusion was that something can't be turned into nothing, and likewise if you have nothing, you can't turn it into something. To make this into a scientific theory, we have to decide on a way to measure what “something” is, and we can then check by measurements whether the total amount of “something” in the universe really stays constant. How much “something” is there in a rock? Does a sunbeam count as “something?” Does heat count? Motion? Thoughts and feelings?

If you look at the table of contents of this book, you'll see that the first four chapters have the word “conservation” in them. In physics, a conservation law is a statement that the total amount of a certain physical quantity always stays the same. This chapter is about conservation of mass. The metric system is designed around a unit of distance, the meter, a unit of mass, the kilogram, and a time unit, the second. Numerical measurement of distance and time probably date back almost as far into prehistory as counting money, but mass is a more modern concept. Until scientists figured out that mass was conserved, it wasn't obvious that there could be a single, consistent way of measuring an amount of matter, hence jiggers of whiskey and cords of wood. You may wonder why conservation of mass wasn't discovered until relatively modern times, but it wasn't obvious, for example, that gases had mass, and that the apparent loss of mass when wood was burned was exactly matched by the mass of the escaping gases.

Once scientists were on the track of the conservation of mass concept, they began looking for a way to define mass in terms of a definite measuring procedure. If they tried such a procedure, and the result was that it led to nonconservation of mass, then they would throw it out and try a different procedure. For instance, we might be tempted to define mass using kitchen measuring cups, i.e., as a measure of volume. Mass would then be perfectly conserved for a process like mixing marbles with peanut butter, but there would be processes like freezing water that led to a net increase in mass, and others like soaking up water with a sponge that caused a decrease. If, with the benefit of hindsight, it seems like the measuring cup definition was just plain silly, then here's a more subtle example of a wrong definition of mass. Suppose we define it using a bathroom scale, or a more precise device such as a postal scale that works on the same principle of using gravity to compress or twist a spring. The trouble is that gravity is not equally strong all over the surface of the earth, so for instance there would be nonconservation of mass when you brought an object up to the top of a mountain, where gravity is a little weaker.

Although some of the obvious possibilities have problems, there do turn out to be at least two approaches to defining mass that lead to its being a conserved quantity, so we consider these definitions to be “right” in the pragmatic sense that what's correct is what's useful.

One definition that works is to use balances, but compensate for the local strength of gravity. This is the method that is used by scientists who actually specialize in ultraprecise measurements. A standard kilogram, in the form of a platinum-iridium cylinder, is kept in a special shrine in Paris. Copies are made that balance against the standard kilogram in Parisian gravity, and they are then transported to laboratories in other parts of the world, where they are compared with other masses in the local gravity. The quantity defined in this way is called gravitational mass.

A second and completely different approach is to measure how hard it is to change an object's state of motion. This tells us its inertial mass. For example, I'd be more willing to stand in the way of an oncoming poodle than in the path of a freight train, because my body will have a harder time convincing the freight train to stop. This is a dictionary-style conceptual definition, but in physics we need to back up a conceptual definition with an operational definition, which is one that spells out the operations required in order to measure the quantity being defined. We can operationalize our definition of inertial mass by throwing a standard kilogram at an object at a speed of 1 m/s (one meter per second) and measuring the recoiling object's velocity. Suppose we want to measure the mass of a particular block of cement. We put the block in a toy wagon on the sidewalk, and throw a standard kilogram at it. Suppose the standard kilogram hits the wagon, and then drops straight down to the sidewalk, having lost all its velocity, and the wagon and the block inside recoil at a velocity of 0.23 m/s. We then repeat the experiment with the block replaced by various numbers of standard kilograms, and find that we can reproduce the recoil velocity of 0.23 m/s with four standard kilograms in the wagon. We have determined the mass of the block to be four kilograms.1 Although this definition of inertial mass has an appealing conceptual simplicity, it is obviously not very practical, at least in this crude form. Nevertheless, this method of collision is very much like the methods used for measuring the masses of subatomic particles, which, after all, can't be put on little postal scales!

Astronauts spending long periods of time in space need to monitor their loss of bone and muscle mass, and here as well, it's impossible to measure gravitational mass. Since they don't want to have standard kilograms thrown at them, they use a slightly different technique (figures d and e). They strap themselves to a chair which is attached to a large spring, and measure the time it takes for one cycle of vibration.


f / Example 1.

1.1.1 Problem-solving techniques

How do we use a conservation law, such as conservation of mass, to solve problems? There are two basic techniques.

As an analogy, consider conservation of money, which makes it illegal for you to create dollar bills using your own inkjet printer. (Most people don't intentionally destroy their dollar bills, either!) Suppose the police notice that a particular store doesn't seem to have any customers, but the owner wears lots of gold jewelry and drives a BMW. They suspect that the store is a front for some kind of crime, perhaps counterfeiting. With intensive surveillance, there are two basic approaches they could use in their investigation. One method would be to have undercover agents try to find out how much money goes in the door, and how much money comes back out at the end of the day, perhaps by arranging through some trick to get access to the owner's briefcase in the morning and evening. If the amount of money that comes out every day is greater than the amount that went in, and if they're convinced there is no safe on the premises holding a large reservoir of money, then the owner must be counterfeiting. This inflow-equals-outflow technique is useful if we are sure that there is a region of space within which there is no supply of mass that is being built up or depleted.

Example 1: A stream of water
If you watch water flowing out of the end of a hose, you'll see that the stream of water is fatter near the mouth of the hose, and skinnier lower down. This is because the water speeds up as it falls. If the cross-sectional area of the stream was equal all along its length, then the rate of flow (kilograms per second) through a lower cross-section would be greater than the rate of flow through a cross-section higher up. Since the flow is steady, the amount of water between the two cross-sections stays constant. Conservation of mass therefore requires that the cross-sectional area of the stream shrink in inverse proportion to the increasing speed of the falling water.

Suppose the you point the hose straight up, so that the water is rising rather than falling. What happens as the velocity gets smaller? What happens when the velocity becomes zero?

(answer in the back of the PDF version of the book)

How can we apply a conservation law, such as conservation of mass, in a situation where mass might be stored up somewhere? To use a crime analogy again, a prison could contain a certain number of prisoners, who are not allowed to flow in or out at will. In physics, this is known as a closed system. A guard might notice that a certain prisoner's cell is empty, but that doesn't mean he's escaped. He could be sick in the infirmary, or hard at work in the shop earning cigarette money. What prisons actually do is to count all their prisoners every day, and make sure today's total is the same as yesterday's. One way of stating a conservation law is that for a closed system, the total amount of stuff (mass, in this chapter) stays constant.

Example 2: Lavoisier and chemical reactions in a closed system

The French chemist Antoine-Laurent Lavoisier is considered the inventor of the concept of conservation of mass. Before Lavoisier, chemists had never systematically weighed their chemicals to quantify the amount of each substance that was undergoing reactions. They also didn't completely understand that gases were just another state of matter, and hadn't tried performing reactions in sealed chambers to determine whether gases were being consumed from or released into the air. For this they had at least one practical excuse, which is that if you perform a gas-releasing reaction in a sealed chamber with no room for expansion, you get an explosion! Lavoisier invented a balance that was capable of measuring milligram masses, and figured out how to do reactions in an upside-down bowl in a basin of water, so that the gases could expand by pushing out some of the water. In a crucial experiment, Lavoisier heated a red mercury compound, which we would now describe as mercury oxide (HgO), in such a sealed chamber. A gas was produced (Lavoisier later named it “oxygen”), driving out some of the water, and the red compound was transformed into silvery liquid mercury metal. The crucial point was that the total mass of the entire apparatus was exactly the same before and after the reaction. Based on many observations of this type, Lavoisier proposed a general law of nature, that mass is always conserved. (In earlier experiments, in which closed systems were not used, chemists had become convinced that there was a mysterious substance, phlogiston, involved in combustion and oxidation reactions, and that phlogiston's mass could be positive, negative, or zero depending on the situation!)

1.1.2 Delta notation

A convenient notation used throughout physics is \(\Delta\), the uppercase Greek letter delta, which indicates “change in” or “after minus before.” For example, if \(b\) represents how much money you have in the bank, then a deposit of $100 could be represented as $\Delta{}b=$100$. That is, the change in your balance was $100, or the balance after the transaction minus the balance before the transaction equals $100. A withdrawal would be indicated by \(\Delta{}b\lt0\). We represent “before” and “after” using the subscripts \(i\) (initial) and \(f\) (final), e.g., \(\Delta{}b=b_f-b_i\). Often the delta notation allows more precision than English words. For instance, “time” can be used to mean a point in time (“now's the time”), \(t\), or it could mean a period of time (“the whole time, he had spit on his chin”), \(\Delta{}t\).

This notation is particularly convenient for discussing conserved quantities. The law of conservation of mass can be stated simply as \(\Delta{}m=0\), where \(m\) is the total mass of any closed system.


If \(x\) represents the location of an object moving in one dimension, then how would positive and negative signs of \(\Delta{}x\) be interpreted?

(answer in the back of the PDF version of the book)
Discussion Questions

If an object had a straight-line \(x-t\) graph with \(\Delta x=0\) and \(\Delta t\ne0\), what would be true about its velocity? What would this look like on a graph? What about \(\Delta t=0\) and \(\Delta x\ne0\)?

1.2 Equivalence of Gravitational and Inertial Mass


a / The two pendulum bobs are constructed with equal gravitational masses. If their inertial masses are also equal, then each pendulum should take exactly the same amount of time per swing.


b / If the cylinders have slightly unequal ratios of inertial to gravitational mass, their trajectories will be a little different.


c / A simplified drawing of an Eötvös-style experiment. If the two masses, made out of two different substances, have slightly different ratios of inertial to gravitational mass, then the apparatus will twist slightly as the earth spins.


d / A more realistic drawing of Braginskii and Panov's experiment. The whole thing was encased in a tall vacuum tube, which was placed in a sealed basement whose temperature was controlled to within 0.02°\textup{C}. The total mass of the platinum and aluminum test masses, plus the tungsten wire and the balance arms, was only 4.4 g. To detect tiny motions, a laser beam was bounced off of a mirror attached to the wire. There was so little friction that the balance would have taken on the order of several years to calm down completely after being put in place; to stop these vibrations, static electrical forces were applied through the two circular plates to provide very gentle twists on the ellipsoidal mass between them. After Braginskii and Panov.

We find experimentally that both gravitational and inertial mass are conserved to a high degree of precision for a great number of processes, including chemical reactions, melting, boiling, soaking up water with a sponge, and rotting of meat and vegetables. Now it's logically possible that both gravitational and inertial mass are conserved, but that there is no particular relationship between them, in which case we would say that they are separately conserved. On the other hand, the two conservation laws may be redundant, like having one law against murder and another law against killing people!

Here's an experiment that gets at the issue: stand up now and drop a coin and one of your shoes side by side. I used a 400-gram shoe and a 2-gram penny, and they hit the floor at the same time as far as I could tell by eye. This is an interesting result, but a physicist and an ordinary person will find it interesting for different reasons.

The layperson is surprised, since it would seem logical that heaver objects would always fall faster than light ones. However, it's fairly easy to prove that if air friction is negligible, any two objects made of the same substance must have identical motion when they fall. For instance, a 2-kg copper mass must exhibit the same falling motion as a 1-kg copper mass, because nothing would be changed by physically joining together two 1-kg copper masses to make a single 2-kg copper mass. Suppose, for example, that they are joined with a dab of glue; the glue isn't under any strain, because the two masses are doing the same thing side by side. Since the glue isn't really doing anything, it makes no difference whether the masses fall separately or side by side.2

What a physicist finds remarkable about the shoe-and-penny experiment is that it came out the way it did even though the shoe and the penny are made of different substances. There is absolutely no theoretical reason why this should be true. We could say that it happens because the greater gravitational mass of the shoe is exactly counteracted by its greater inertial mass, which makes it harder for gravity to get it moving, but that just leaves us wondering why inertial mass and gravitational mass are always in proportion to each other. It's possible that they are only approximately equivalent. Most of the mass of ordinary matter comes from neutrons and protons, and we could imagine, for instance, that neutrons and protons do not have exactly the same ratio of gravitational to inertial mass. This would show up as a different ratio of gravitational to inertial mass for substances containing different proportions of neutrons and protons.

Galileo did the first numerical experiments on this issue in the seventeenth century by rolling balls down inclined planes, although he didn't think about his results in these terms. A fairly easy way to improve on Galileo's accuracy is to use pendulums with bobs made of different materials. Suppose, for example, that we construct an aluminum bob and a brass bob, and use a double-pan balance to verify to good precision that their gravitational masses are equal. If we then measure the time required for each pendulum to perform a hundred cycles, we can check whether the results are the same. If their inertial masses are unequal, then the one with a smaller inertial mass will go through each cycle faster, since gravity has an easier time accelerating and decelerating it. With this type of experiment, one can easily verify that gravitational and inertial mass are proportional to each other to an accuracy of \(10^{-3}\) or \(10^{-4}\).

In 1889, the Hungarian physicist Roland Eötvös used a slightly different approach to verify the equivalence of gravitational and inertial mass for various substances to an accuracy of about \(10^{-8}\), and the best such experiment, figure d, improved on even this phenomenal accuracy, bringing it to the \(10^{-12}\) level.3 In all the experiments described so far, the two objects move along similar trajectories: straight lines in the penny-and-shoe and inclined plane experiments, and circular arcs in the pendulum version. The Eötvös-style experiment looks for differences in the objects' trajectories. The concept can be understood by imagining the following simplified version. Suppose, as in figure b, we roll a brass cylinder off of a tabletop and measure where it hits the floor, and then do the same with an aluminum cylinder, making sure that both of them go over the edge with precisely the same velocity. An object with zero gravitational mass would fly off straight and hit the wall, while an object with zero inertial mass would make a sudden 90-degree turn and drop straight to the floor. If the aluminum and brass cylinders have ordinary, but slightly unequal, ratios of gravitational to inertial mass, then they will follow trajectories that are just slightly different. In other words, if inertial and gravitational mass are not exactly proportional to each other for all substances, then objects made of different substances will have different trajectories in the presence of gravity.

A simplified drawing of a practical, high-precision experiment is shown in figure c. Two objects made of different substances are balanced on the ends of a bar, which is suspended at the center from a thin fiber. The whole apparatus moves through space on a complicated, looping trajectory arising from the rotation of the earth superimposed on the earth's orbital motion around the sun. Both the earth's gravity and the sun's gravity act on the two objects. If their inertial masses are not exactly in proportion to their gravitational masses, then they will follow slightly different trajectories through space, which will result in a very slight twisting of the fiber between the daytime, when the sun's gravity is pulling upward, and the night, when the sun's gravity is downward. Figure d shows a more realistic picture of the apparatus.

This type of experiment, in which one expects a null result, is a tough way to make a career as a scientist. If your measurement comes out as expected, but with better accuracy than other people had previously achieved, your result is publishable, but won't be considered earthshattering. On the other hand, if you build the most sensitive experiment ever, and the result comes out contrary to expectations, you're in a scary situation. You could be right, and earn a place in history, but if the result turns out to be due to a defect in your experiment, then you've made a fool of yourself.

1.3 Galilean Relativity


a / Galileo Galilei (1564-1642).


b / The earth spins. People in Shanghai say they're at rest and people in Los Angeles are moving. Angelenos say the same about the Shanghainese.


d / Foucault demonstrates his pendulum to an audience at a lecture in 1851.


e / Galileo's trial.


f / Discussion question B.


g / Discussion question C.


h / Discussion question E.

I defined inertial mass conceptually as a measure of how hard it is to change an object's state of motion, the implication being that if you don't interfere, the object's motion won't change. Most people, however, believe that objects in motion have a natural tendency to slow down. Suppose I push my refrigerator to the west for a while at 0.1 m/s, and then stop pushing. The average person would say fridge just naturally stopped moving, but let's imagine how someone in China would describe the fridge experiment carried out in my house here in California. Due to the rotation of the earth, California is moving to the east at about 400 m/s. A point in China at the same latitude has the same speed, but since China is on the other side of the planet, China's east is my west. (If you're finding the three-dimensional visualization difficult, just think of China and California as two freight trains that go past each other, each traveling at 400 m/s.) If I insist on thinking of my dirt as being stationary, then China and its dirt are moving at 800 m/s to my west. From China's point of view, however, it's California that is moving 800 m/s in the opposite direction (my east). When I'm pushing the fridge to the west at 0.1 m/s, the observer in China describes its speed as 799.9 m/s. Once I stop pushing, the fridge speeds back up to 800 m/s. From my point of view, the fridge “naturally” slowed down when I stopped pushing, but according to the observer in China, it “naturally” sped up!

What's really happening here is that there's a tendency, due to friction, for the fridge to stop moving relative to the floor. In general, only relative motion has physical significance in physics, not absolute motion. It's not even possible to define absolute motion, since there is no special reference point in the universe that everyone can agree is at rest. Of course if we want to measure motion, we do have to pick some arbitrary reference point which we will say is standing still, and we can then define \(x\), \(y\), and \(z\) coordinates extending out from that point, which we can define as having \(x=0\), \(y=0\), \(z=0\). Setting up such a system is known as choosing a frame of reference. The local dirt is a natural frame of reference for describing a game of basketball, but if the game was taking place on the deck of a moving ocean liner, we would probably pick a frame of reference in which the deck was at rest, and the land was moving.

Galileo was the first scientist to reason along these lines, and we now use the term Galilean relativity to refer to a somewhat modernized version of his principle. Roughly speaking, the principle of Galilean relativity states that the same laws of physics apply in any frame of reference that is moving in a straight line at constant speed. We need to refine this statement, however, since it is not necessarily obvious which frames of reference are going in a straight line at constant speed. A person in a pickup truck pulling away from a stoplight could admit that the car's velocity is changing, or she could insist that the truck is at rest, and the meter on the dashboard is going up because the asphalt picked that moment to start moving faster and faster backward! Frames of reference are not all created equal, however, and the accelerating truck's frame of reference is not as good as the asphalt's. We can tell, because a bowling ball in the back of the truck, as in figure c, appears to behave strangely in the driver's frame of reference: in her rear-view mirror, she sees the ball, initially at rest, start moving faster and faster toward the back of the truck. This goofy behavior is evidence that there is something wrong with her frame of reference. A person on the sidewalk, however, sees the ball as standing still. In the sidewalk's frame of reference, the truck pulls away from the ball, and this makes sense, because the truck is burning gas and using up energy to change its state of motion.

We therefore define an inertial frame of reference as one in which we never see objects change their state of motion without any apparent reason. The sidewalk is a pretty good inertial frame, and a car moving relative to the sidewalk at constant speed in a straight line defines a pretty good inertial frame, but a car that is accelerating or turning is not a inertial frame.


c / Left: In a frame of reference that speeds up with the truck, the bowling ball appears to change its state of motion for no reason. Right: In an inertial frame of reference, which the surface of the earth approximately is, the bowling ball stands still, which makes sense because there is nothing that would cause it to change its state of motion.

The principle of Galilean relativity states that inertial frames exist, and that the same laws of physics apply in all inertial frames of reference, regardless of one frame's straight-line, constant-speed motion relative to another.4

Another way of putting it is that all inertial frames are created equal. We can say whether one inertial frame is in motion or at rest relative to another, but there is no privileged “rest frame.” There is no experiment that comes out any different in laboratories in different inertial frames, so there is no experiment that could tell us which inertial frame is really, truly at rest.

Example 3: The speed of sound

\(\triangleright\) The speed of sound in air is only 340 m/s, so unless you live at a near-polar latitude, you're moving at greater than the speed of sound right now due to the Earth's rotation. In that case, why don't we experience exciting phenomena like sonic booms all the time? \(\triangleright\) It might seem as though you're unprepared to deal with this question right now, since the only law of physics you know is conservation of mass, and conservation of mass doesn't tell you anything obviously useful about the speed of sound or sonic booms. Galilean relativity, however, is a blanket statement about all the laws of physics, so in a situation like this, it may let you predict the results of the laws of physics without actually knowing what all the laws are! If the laws of physics predict a certain value for the speed of sound, then they had better predict the speed of the sound relative to the air, not their speed relative to some special “rest frame.” Since the air is moving along with the rotation of the earth, we don't detect any special phenomena. To get a sonic boom, the source of the sound would have to be moving relative to the air.

Example 4: The Foucault pendulum

Note that in the example of the bowling ball in the truck, I didn't claim the sidewalk was exactly a Galilean frame of reference. This is because the sidewalk is moving in a circle due to the rotation of the Earth, and is therefore changing the direction of its motion continuously on a 24-hour cycle. However, the curve of the motion is so gentle that under ordinary conditions we don't notice that the local dirt's frame of reference isn't quite inertial. The first demonstration of the noninertial nature of the earth-fixed frame of reference was by Foucault using a very massive pendulum (figure d) whose oscillations would persist for many hours without becoming imperceptible. Although Foucault did his demonstration in Paris, it's easier to imagine what would happen at the north pole: the pendulum would keep swinging in the same plane, but the earth would spin underneath it once every 24 hours. To someone standing in the snow, it would appear that the pendulum's plane of motion was twisting. The effect at latitudes less than 90 degrees turns out to be slower, but otherwise similar. The Foucault pendulum was the first definitive experimental proof that the earth really did spin on its axis, although scientists had been convinced of its rotation for a century based on more indirect evidence about the structure of the solar system.

Although popular belief has Galileo being prosecuted by the Catholic Church for saying the earth rotated on its axis and also orbited the sun, Foucault's pendulum was still centuries in the future, so Galileo had no hard proof; Galileo's insights into relative versus absolute motion simply made it more plausible that the world could be spinning without producing dramatic effects, but didn't disprove the contrary hypothesis that the sun, moon, and stars went around the earth every 24 hours. Furthermore, the Church was much more liberal and enlightened than most people believe. It didn't (and still doesn't) require a literal interpretation of the Bible, and one of the Church officials involved in the Galileo affair wrote that “the Bible tells us how to go to heaven, not how the heavens go.” In other words, religion and science should be separate. The actual reason Galileo got in trouble is shrouded in mystery, since Italy in the age of the Medicis was a secretive place where unscrupulous people might settle a score with poison or a false accusation of heresy. What is certain is that Galileo's satirical style of scientific writing made many enemies among the powerful Jesuit scholars who were his intellectual opponents --- he compared one to a snake that doesn't know its own back is broken. It's also possible that the Church was far less upset by his astronomical work than by his support for atomism (discussed further in the next section). Some theologians perceived atomism as contradicting transubstantiation, the Church's doctrine that the holy bread and wine were literally transformed into the flesh and blood of Christ by the priest's blessing.


What is incorrect about the following supposed counterexamples to the principle of inertia?

(1) When astronauts blast off in a rocket, their huge velocity does cause a physical effect on their bodies --- they get pressed back into their seats, the flesh on their faces gets distorted, and they have a hard time lifting their arms.

(2) When you're driving in a convertible with the top down, the wind in your face is an observable physical effect of your absolute motion.

(answer in the back of the PDF version of the book)

◊ Solved problem: a bug on a wheel — problem 12

Discussion Questions

A passenger on a cruise ship finds, while the ship is docked, that he can leap off of the upper deck and just barely make it into the pool on the lower deck. If the ship leaves dock and is cruising rapidly, will this adrenaline junkie still be able to make it?

You are a passenger in the open basket hanging under a helium balloon. The balloon is being carried along by the wind at a constant velocity. If you are holding a flag in your hand, will the flag wave? If so, which way? [Based on a question from PSSC Physics.]

Aristotle stated that all objects naturally wanted to come to rest, with the unspoken implication that “rest” would be interpreted relative to the surface of the earth. Suppose we could transport Aristotle to the moon, put him in a space suit, and kick him out the door of the spaceship and into the lunar landscape. What would he expect his fate to be in this situation? If intelligent creatures inhabited the moon, and one of them independently came up with the equivalent of Aristotelian physics, what would they think about objects coming to rest?

Sally is on an amusement park ride which begins with her chair being hoisted straight up a tower at a constant speed of 60 miles/hour. Despite stern warnings from her father that he'll take her home the next time she misbehaves, she decides that as a scientific experiment she really needs to release her corndog over the side as she's on the way up. She does not throw it. She simply sticks it out of the car, lets it go, and watches it against the background of the sky, with no trees or buildings as reference points. What does the corndog's motion look like as observed by Sally? Does its speed ever appear to her to be zero? What acceleration does she observe it to have: is it ever positive? negative? zero? What would her enraged father answer if asked for a similar description of its motion as it appears to him, standing on the ground?


Self-check D.

1.3.1 Applications of calculus

Let's see how this relates to calculus. If an object is moving in one dimension, we can describe its position with a function \(x(t)\). The derivative \(v=d{}x/d{}t\) is called the velocity, and the second derivative \(a=d{}v/d{}t=d{}^2x/d{}t^2\) is the acceleration. Galilean relativity tells us that there is no detectable effect due to an object's absolute velocity, since in some other frame of reference, the object's velocity might be zero. However, an acceleration does have physical consequences.


i / This Air Force doctor volunteered to ride a rocket sled as a medical experiment. The obvious effects on his head and face are not because of the sled's speed but because of its rapid changes in speed: increasing in (ii) and (iii), and decreasing in (v) and (vi). In (iv) his speed is greatest, but because his speed is not increasing or decreasing very much at this moment, there is little effect on him. (U.S. Air Force)

Observers in different inertial frames of reference will disagree on velocities, but agree on accelerations. Let's keep it simple by continuing to work in one dimension. One frame of reference uses a coordinate system \(x_1\), and the other we label \(x_2\). If the positive \(x_1\) and \(x_2\) axes point in the same direction, then in general two inertial frames could be related by an equation of the form \(x_2=x_1+b+ut\), where \(u\) is the constant velocity of one frame relative to the other, and the constant \(b\) tells us how far apart the origins of the two coordinate systems were at \(t=0\). The velocities are different in the two frames of reference:

\[\begin{equation*} \frac{dx_2}{dt} = \frac{dx_1}{dt} + u , \end{equation*}\]

Suppose, for example, frame 1 is defined from the sidewalk, and frame 2 is fixed to a float in a parade that is moving to our left at a velocity \(u=1\ \text{m}/\text{s}\). A dog that is moving to the right with a velocity \(v_1=d{}x_1/d{}t=3\ \text{m}/\text{s}\) in the sidewalk's frame will appear to be moving at a velocity of \(v_2=dx_2/dt=dx_1/dt+u=4\ \text{m}/\text{s}\) in the float's frame.

For acceleration, however, we have

\[\begin{equation*} \frac{d^2 x_2}{dt^2} = \frac{d^2 x_1}{dt^2} , \end{equation*}\]

since the derivative of the constant \(u\) is zero. Thus an acceleration, unlike a velocity, can have a definite physical significance to all observers in all frames of reference. If this wasn't true, then there would be no particular reason to define a quantity called acceleration in the first place.


The figure shows a bottle of beer sitting on a table in the dining car of a train. Does the tilting of the surface tell us about the train's velocity, or its acceleration? What would a person in the train say about the bottle's velocity? What about a person standing in a field outside and looking in through the window? What about the acceleration?

(answer in the back of the PDF version of the book)

1.4 A Preview of Some Modern Physics

“Mommy, why do you and Daddy have to go to work?” “To make money, sweetie-pie.” “Why do we need money?” “To buy food.” “Why does food cost money?” When small children ask a chain of “why” questions like this, it usually isn't too long before their parents end up saying something like, “Because that's just the way it is,” or, more honestly, “I don't know the answer.”

The same happens in physics. We may gradually learn to explain things more and more deeply, but there's always the possibility that a certain observed fact, such as conservation of mass, will never be understood on any deeper level. Science, after all, uses limited methods to achieve limited goals, so the ultimate reason for all existence will always be the province of religion. There is, however, an appealing explanation for conservation of mass, which is atomism, the theory that matter is made of tiny, unchanging particles. The atomic hypothesis dates back to ancient Greece, but the first solid evidence to support it didn't come until around the eighteenth century, and individual atoms were never detected until about 1900. The atomic theory implies not only conservation of mass, but a couple of other things as well.

First, it implies that the total mass of one particular element is conserved. For instance, lead and gold are both elements, and if we assume that lead atoms can't be turned into gold atoms, then the total mass of lead and the total mass of gold are separately conserved. It's as though there was not just a law against pickpocketing, but also a law against surreptitiously moving money from one of the victim's pockets to the other. It turns out, however, that although chemical reactions never change one type of atom into another, transmutation can happen in nuclear reactions, such as the ones that created most of the elements in your body out of the primordial hydrogen and helium that condensed out of the aftermath of the Big Bang.

Second, atomism implies that mass is quantized, meaning that only certain values of mass are possible and the ones in between can't exist. We can have three atoms of gold or four atoms of gold, but not three an a half. Although quantization of mass is a natural consequence of any theory in which matter is made up of tiny particles, it was discovered in the twentieth century that other quantities, such as energy, are quantized as well, which had previously not been suspected.


Is money quantized?

(answer in the back of the PDF version of the book)

If atomism is starting to make conservation of mass seem inevitable to you, then it may disturb you to know that Einstein discovered it isn't really conserved. If you put a 50-gram iron nail in some water, seal the whole thing up, and let it sit on a fantastically precise balance while the nail rusts, you'll find that the system loses about \(6\times10^{-12}\) kg of mass by the time the nail has turned completely to rust. This has to do with Einstein's famous equation \(E=mc^2\). Rusting releases heat energy, which then escapes out into the room. Einstein's equation states that this amount of heat, \(E\), is equivalent to a certain amount of mass, \(m\). The \(c\) in the \(c^2\) is the speed of light, which is a large number, and a large amount of energy is therefore equivalent to a very small amount of mass, so you don't notice nonconservation of mass under ordinary conditions. What is really conserved is not the mass, \(m\), but the mass-plus-energy, \(E+mc^2\). The point of this discussion is not to get you to do numerical exercises with \(E=mc^2\) (at this point you don't even know what units are used to measure energy), but simply to point out to you the empirical nature of the laws of physics. If a previously accepted theory is contradicted by an experiment, then the theory needs to be changed. This is also a good example of something called the correspondence principle, which is a historical observation about how scientific theories change: when a new scientific theory replaces an old one, the old theory is always contained within the new one as an approximation that works within a certain restricted range of situations. Conservation of mass is an extremely good approximation for all chemical reactions, since chemical reactions never release or consume enough energy to change the total mass by a large percentage. Conservation of mass would not have been accepted for 110 years as a fundamental principle of physics if it hadn't been verified over and over again by a huge number of accurate experiments.


Homework Problems


a / Problem 6.


b / Problem 8.


c / Problem 11.


d / Problem 12.

[Problems] \addcontentsline{toc}{section}{\protect{Problems}}

1. Thermometers normally use either mercury or alcohol as their working fluid. If the level of the fluid rises or falls, does this violate conservation of mass?

2. The ratios of the masses of different types of atoms were determined a century before anyone knew any actual atomic masses in units of kg. One finds, for example, that when ordinary table salt, NaCl, is melted, the chlorine atoms bubble off as a gas, leaving liquid sodium metal. Suppose the chlorine escapes, so that its mass cannot be directly determined by weighing. Experiments show that when 1.00000 kg of NaCl is treated in this way, the mass of the remaining sodium metal is 0.39337 kg. Based on this information, determine the ratio of the mass of a chlorine atom to that of a sodium atom.(answer check available at

3. An atom of the most common naturally occurring uranium isotope breaks up spontaneously into a thorium atom plus a helium atom. The masses are as follows:

uranium395292849times 1025 textupkg
thorium 388638748times 1025 textupkg
helium 6646481times 1027 textupkg

{}Each of these experimentally determined masses is uncertain in its last decimal place. Is mass conserved in this process to within the accuracy of the experimental data? How would you interpret this?

4. If two spherical water droplets of radius \(b\) combine to make a single droplet, what is its radius? (Assume that water has constant density.)

5. Make up an experiment that would test whether mass is conserved in an animal's metabolic processes.

6. The figure shows a hydraulic jack. What is the relationship between the distance traveled by the plunger and the distance traveled by the object being lifted, in terms of the cross-sectional areas?

7. In an example in this chapter, I argued that a stream of water must change its cross-sectional area as it rises or falls. Suppose that the stream of water is confined to a constant-diameter pipe. Which assumption breaks down in this situation?

8. A river with a certain width and depth splits into two parts, each of which has the same width and depth as the original river. What can you say about the speed of the current after the split?

9. The diagram shows a cross-section of a wind tunnel of the kind used, for example, to test designs of airplanes. Under normal conditions of use, the density of the air remains nearly constant throughout the whole wind tunnel. How can the speed of the air be controlled and calculated? (Diagram by NASA, Glenn Research Center.)


10. A water wave is in a tank that extends horizontally from \(x=0\) to \(x=a\), and from \(z=0\) to \(z=b\). We assume for simplicity that at a certain moment in time the height \(y\) of the water's surface only depends on \(x\), not \(z\), so that we can effectively ignore the \(z\) coordinate. Under these assumptions, the total volume of the water in the tank is \begin{displaymath}V = b \int_0^a{y(x) \der{}x} .\end{displaymath}
Since the density of the water is essentially constant, conservation of mass requires that \(V\) is always the same. When the water is calm, we have \(y=h\), where \(h=V/ab\). If two different wave patterns move into each other, we might imagine that they would add in the sense that \(y_{total}-h = (y_1-h) + (y_2-h)\). Show that this type of addition is consistent with conservation of mass.

11. The figure shows the position of a falling ball at equal time intervals, depicted in a certain frame of reference. On a similar grid, show how the ball's motion would appear in a frame of reference that was moving horizontally at a speed of one box per unit time relative to the first frame.

12. (solution in the pdf version of the book) The figure shows the motion of a point on the rim of a rolling wheel. (The shape is called a cycloid.) Suppose bug A is riding on the rim of the wheel on a bicycle that is rolling, while bug B is on the spinning wheel of a bike that is sitting upside down on the floor. Bug A is moving along a cycloid, while bug B is moving in a circle. Both wheels are doing the same number of revolutions per minute. Which bug has a harder time holding on, or do they find it equally difficult?

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

[1] You might think intuitively that the recoil velocity should be exactly one fourth of a meter per second, and you'd be right except that the wagon has some mass as well. Our present approach, however, only requires that we give a way to test for equality of masses. To predict the recoil velocity from scratch, we'd need to use conservation of momentum, which is discussed in a later chapter.
[2] The argument only fails for objects light enough to be affected appreciably by air friction: a bunch of feathers falls differently if you wad them up because the pattern of air flow is altered by putting them together.
[3] V.B. Braginskii and V.I. Panov, Soviet Physics JETP 34, 463 (1972).
[4] The principle of Galilean relativity is extended on page 190.