# Conservation of Mass and Energy

## Symmetry and Conservation Laws

Even before history began, people must already have noticed certain facts about the sky. The sun and moon both rise in the east and set in the west. Another fact that can be settled to a fair degree of accuracy using the naked eye is that the apparent sizes of the sun and moon don't change noticeably. (There is an optical illusion that makes the moon appear bigger when it's near the horizon, but you can easily verify that it's nothing more than an illusion by checking its angular size against some standard, such as your pinkie held at arm's length.) If the sun and moon were varying their distances from us, they would appear to get bigger and smaller, and since they don't appear to change in size, it appears, at least approximately, that they always stay at the same distance from us.

From observations like these, the ancients constructed a scientific *model*, in which the sun and moon traveled around the earth in perfect circles. Of course, we now know that the earth isn't the center of the universe, but that doesn't mean the model wasn't useful. That's the way science always works. Science never aims to reveal the ultimate reality. Science only tries to make models of reality that have predictive power.

Our modern approach to understanding physics revolves around the concepts of *symmetry* and *conservation laws*, both of which are demonstrated by this example.

The sun and moon were believed to move in circles, and a circle is a very symmetric shape. If you rotate a circle about its center, like a spinning wheel, it doesn't change. Therefore, we say that the circle is *symmetric* with respect to rotation about its center. The ancients thought it was beautiful that the universe seemed to have this type of symmetry built in, and they became very attached to the idea.

A *conservation law* is a statement that some number stays the same with the passage of time. In our example, the distance between the sun and the earth is conserved, and so is the distance between the moon and the earth. (The ancient Greeks were even able to determine that earth-moon distance.)

In our example, the symmetry and the conservation law both give the same information. Either statement can be satisfied only by a circular orbit. That isn't a coincidence. Physicist Emmy Noether showed on very general mathematical grounds that for physical theories of a certain type, every symmetry leads to a corresponding conservation law. Although the precise formulation of Noether's theorem, and its proof, are too mathematical for this book, we'll see many examples like this one, in which the physical content of the theorem is fairly straightforward.

The idea of perfect circular orbits seems very beautiful and intuitively appealing. It came as a great disappointment, therefore, when the astronomer Johannes Kepler discovered, by the painstaking analysis of precise observations, that orbits such as the moon's were actually ellipses, not circles. This is the sort of thing that led the biologist Huxley to say, “The great tragedy of science is the slaying of a beautiful theory by an ugly fact.” The lesson of the story, then, is that symmetries are important and beautiful, but we can't decide which symmetries are right based only on common sense or aesthetics; their validity can only be determined based on observations and experiments.

As a more modern example, consider the symmetry between right and left. For example, we observe that a top spinning clockwise has exactly the same behavior as a top spinning counterclockwise. This kind of observation led physicists to believe, for hundreds of years, that the laws of physics were perfectly symmetric with respect to right and left. This mirror symmetry appealed to physicists' common sense. However, experiments by Chien-Shiung Wu et al. in 1957 showed that right-left symmetry was violated in certain types of nuclear reactions. Physicists were thus forced to change their opinions about what constituted common sense.

## Review of the Metric System and Conversions

### The metric system

Every country in the world besides the U.S. has adopted a system of units known colloquially as the “metric system.” Even in the U.S., the system is used universally by scientists, and also by many engineers. This system is entirely decimal, thanks to the same eminently logical people who brought about the French Revolution. In deference to France, the system's official name is the Système International, or SI, meaning International System. (The phrase “SI system” is therefore redundant.)

The metric system works with a single, consistent set of prefixes (derived from Greek) that modify the basic units. Each prefix stands for a power of ten, and has an abbreviation that can be combined with the symbol for the unit. For instance, the meter is a unit of distance. The prefix kilo- stands for 1000, so a kilometer, 1 km, is a thousand meters.

In this book, we'll be using a flavor of the metric system, the SI, in which there are three basic units, measuring distance, time, and mass. The basic unit of distance is the meter (m), the one for time is the second (s), and for mass the kilogram (kg). Based on these units, we can define others, e.g., m/s (meters per second) for the speed of a car, or kg/s for the rate at which water flows through a pipe. It might seem odd that we consider the basic unit of mass to be the kilogram, rather than the gram. The reason for doing this is that when we start defining other units starting from the basic three, some of them come out to be a more convenient size for use in everyday life. For example, there is a metric unit of force, the newton (N), which is defined as the push or pull that would be able to change a 1-kg object's velocity by 1 m/s, if it acted on it for 1 s. A newton turns out to be about the amount of force you'd use to pick up your keys. If the system had been based on the gram instead of the kilogram, then the newton would have been a thousand times smaller, something like the amount of force required in order to pick up a breadcrumb.

The following are the most common metric prefixes. You should memorize them.

The prefix centi-, meaning 1/100, is only used in the centimeter; a hundredth of a gram would not be written as 1 cg but as 10 mg. The centi- prefix can be easily remembered because a cent is 1/100 of a dollar. The official SI abbreviation for seconds is “s” (not “sec”) and grams are “g” (not “gm”).

You may also encounter the prefixes mega- (a million) and micro- (one millionth).

### Scientific notation

Most of the interesting phenomena in our universe are not on the human scale. It would take about 1,000,000,000,000,000,000,000 bacteria to equal the mass of a human body. When the physicist Thomas Young discovered that light was a wave, scientific notation hadn't been invented, and he was obliged to write that the time required for one vibration of the wave was 1/500 of a millionth of a millionth of a second. Scientific notation is a less awkward way to write very large and very small numbers such as these. Here's a quick review.

Scientific notation means writing a number in terms of a product of something from 1 to 10 and something else that is a power of ten. For instance,

Each number is ten times bigger than the last.

Since 10^{1} is ten times smaller than 10^{2} , it makes sense to use the notation 10^{0} to stand for one, the number that is in turn ten times smaller than 10^{1} . Continuing on, we can write 10^{-1} to stand for 0.1, the number ten times smaller than 10^{0} . Negative exponents are used for small numbers:

A common source of confusion is the notation used on the displays of many calculators. Examples:

The last example is particularly unfortunate, because 3.2^{6} really stands for the number 3.2 × 3.2 × 3.2 × 3.2 × 3.2 × 3.2= 1074, a totally different number from 3.2 × 10^{6}=3200000. The calculator notation should never be used in writing. It's just a way for the manufacturer to save money by making a simpler display.

A student learns that 10^{4} bacteria, standing in line to register for classes at Paramecium Community College, would form a queue of this size:

The student concludes that 10^{2} bacteria would form a line of this length:

Why is the student incorrect?

### Conversions

I suggest you avoid memorizing lots of conversion factors between SI units and U.S. units. Suppose the United Nations sends its black helicopters to invade California (after all who wouldn't rather live here than in New York City?), and institutes water fluoridation and the SI, making the use of inches and pounds into a crime punishable by death. I think you could get by with only two mental conversion factors:

1 inch = 2.54 cm

An object with a weight on Earth of 2.2 pounds-force has a mass of 1 kg.

The first one is the present definition of the inch, so it's exact. The second one is not exact, but is good enough for most purposes. (U.S. units of force and mass are confusing, so it's a good thing they're not used in science. In U.S. units, the unit of force is the pound-force, and the best unit to use for mass is the slug, which is about 14.6 kg.)

More important than memorizing conversion factors is understanding the right method for doing conversions. Even within the SI, you may need to convert, say, from grams to kilograms. Different people have different ways of thinking about conversions, but the method I'll describe here is systematic and easy to understand. The idea is that if 1 kg and 1000 g represent the same mass, then we can consider a fraction like

to be a way of expressing the number one. This may bother you. For instance, if you type 1000/1 into your calculator, you will get 1000, not one. Again, different people have different ways of thinking about it, but the justification is that it helps us to do conversions, and it works! Now if we want to convert 0.7 kg to units of grams, we can multiply kg by the number one:

If you're willing to treat symbols such as “kg” as if they were variables as used in algebra (which they're really not), you can then cancel the kg on top with the kg on the bottom, resulting in

To convert grams to kilograms, you would simply flip the fraction upside down.

One advantage of this method is that it can easily be applied to a series of conversions. For instance, to convert one year to units of seconds,

#### Should that exponent be positive or negative?

A common mistake is to write the conversion fraction incorrectly. For instance the fraction

**equation**

does not equal one, because 10^{3} kg is the mass of a car, and 1 g is the mass of a raisin. One correct way of setting up the conversion factor would be

**equation**

You can usually detect such a mistake if you take the time to check your answer and see if it is reasonable.

If common sense doesn't rule out either a positive or a negative exponent, here's another way to make sure you get it right. There are big prefixes, like kilo-, and small ones, like milli-. In the example above, we want the top of the fraction to be the same as the bottom. Since *k* is a big prefix, we need to *compensate* by putting a small number like 10^{-3} in front of it, not a big number like 10^{3}.

##### Discussion Question

Each of the following conversions contains an error. In each case, explain what the error is:

## Homework Problems

**1**. (solution in the pdf version of the book) Convert 134 mg to units of kg, writing your answer in scientific notation.

**2**. Compute the following things. If they don't make sense because of units, say so.

(a) 3 cm + 5 cm

(b) 1.11 m + 22 cm

(c) 120 miles + 2.0 hours

(d) 120 miles / 2.0 hours

**3**. Your backyard has brick walls on both ends. You measure a distance of 23.4 m from the inside of one wall to the inside of the other. Each wall is 29.4 cm thick. How far is it from the outside of one wall to the outside of the other? Pay attention to significant figures.

**4**. The speed of light is 3.0×10^{8} m/s. Convert this to furlongs per fortnight. A furlong is 220 yards, and a fortnight is 14 days. An inch is 2.54 cm.(answer check available at lightandmatter.com)

**5**. Express each of the following quantities in micrograms:

(a) 10 mg, (b) 10^{4} g, (c) 10 kg, (d) 100×10^{3} g, (e) 1000 ng. (answer check available at lightandmatter.com)

**6**. In the last century, the average age of the onset of puberty for girls has decreased by several years. Urban folklore has it that this is because of hormones fed to beef cattle, but it is more likely to be because modern girls have more body fat on the average and possibly because of estrogen-mimicking chemicals in the environment from the breakdown of pesticides. A hamburger from a hormone-implanted steer has about 0.2 ng of estrogen (about double the amount of natural beef). A serving of peas contains about 300 ng of estrogen. An adult woman produces about 0.5 mg of estrogen per day (note the different unit!). (a) How many hamburgers would a girl have to eat in one day to consume as much estrogen as an adult woman's daily production? (b) How many servings of peas? (answer check available at lightandmatter.com)

**7**. You jump up straight up in the air. When do you have the greatest gravitational energy? The greatest kinetic energy? (Based on a problem by Serway and Faughn.)

**8**. Anya and Ivan lean over a balcony side by side. Anya throws a penny downward with an initial speed of 5 m/s. Ivan throws a penny upward with the same speed. Both pennies end up on the ground below. Compare their kinetic energies and velocities on impact.

**9**. (a) If weight B moves down by a certain amount, how much does weight A move up or down?

(b) What should the ratio of the two weights be if they are to balance? Explain in terms of conservation of energy.

**10**. (a) You release a magnet on a tabletop near a big piece of iron, and the magnet leaps across the table to the iron. Does the magnetic energy increase, or decrease? Explain.

(b) Suppose instead that you have two repelling magnets. You give them an initial push towards each other, so they decelerate while approaching each other. Does the magnetic energy increase, or decrease? Explain.

**11**. For an astronaut sealed inside a space suit, getting rid of body heat can be difficult. Suppose an astronaut is performing vigorous physical activity, expending 200 watts of power. An energy of 200 kJ is enough to raise her body temperature by 1°C. If none of the heat can escape from her space suit, how long will it take before her body temperature rises by 6°C (), an amount sufficient to kill her? Express your answer in units of minutes.(answer check available at lightandmatter.com)

**12**. The multiflash photograph below shows a collision between two pool balls. The ball that was initially at rest shows up as a dark image in its initial position, because its image was exposed several times before it was struck and began moving. By making measurements on the figure, determine whether or not energy appears to have been conserved in the collision. What systematic effects would limit the accuracy of your test? (From an example in PSSC Physics.)

**13**. How high above the surface of the earth should a rocket be in order to have 1/100 of its normal weight? Express your answer in units of earth radii.(answer check available at lightandmatter.com)

**14**. As suggested on page 30, imagine that the mass of the electron rises and falls over time. (Since all electrons are identical, physicists generally talk about “the electron” collectively, as in “the modern man wants more than just beer and sports.”) The idea is that all electrons are increasing and decreasing their masses in unison, and at any given time, they're all identical. They're like a litter of puppies whose weights are all identical on any given day, but who all change their weights in unison from one month to the next. Suppose you were the only person who knew about these small day-to-day changes in the mass of the electron. Find a plan for violating conservation of energy and getting rich.

**15**. A typical balance like the ones used in school classes can be read to an accuracy of about plus or minus 0.1 grams, or 10^{-4} kg. What if the laws of physics had been designed around a different value of the speed of light? To make mass-energy equivalence detectable in example 15 on page 32 using an ordinary balance, would *c* have to be smaller than it is in our universe, or bigger? Find the value of *c* for which the effect would be just barely detectable.(answer check available at lightandmatter.com)

**16**. (a) A free neutron (as opposed to a neutron bound into an atomic nucleus) is unstable, and decays radioactively into a proton, an electron, and a particle called an antineutrino, which fly off in three different directions. The masses are as follows:

Find the energy released in the decay of a free neutron.(answer check available at lightandmatter.com)

(b) Neutrons and protons make up essentially all of the mass of the ordinary matter around us. We observe that the universe around us has no free neutrons, but lots of free protons (the nuclei of hydrogen, which is the element that 90% of the universe is made of). We find neutrons only inside nuclei along with other neutrons and protons, not on their own.

If there are processes that can convert neutrons into protons, we might imagine that there could also be proton-to-neutron conversions, and indeed such a process does occur sometimes in nuclei that contain both neutrons and protons: a proton can decay into a neutron, a positron, and a neutrino. A positron is a particle with the same properties as an electron, except that its electrical charge is positive (see chapter 5). A neutrino, like an antineutrino, has negligible mass.

Although such a process can occur within a nucleus, explain why it cannot happen to a free proton. (If it could, hydrogen would be radioactive, and you wouldn't exist!)

**17**. (a) A 1.0 kg rock is released from rest, and drops 1.0 m. Find the amount of gravitational energy released.(answer check available at lightandmatter.com)

(b) Find the rock's kinetic energy at the end of its fall.(answer check available at lightandmatter.com)

(c) Find the rock's velocity at the end of its fall.(answer check available at lightandmatter.com)

### References

- [1] You may also see this referred to in some books as gravitational potential energy.
- [2] See example 12 on page 52.
- [3] Example 12 on page 52 shows the type of reasoning that Newton had to go through.
- [4] This is not the form in which Newton originally wrote the equation.
- [5] Our present technology isn't good enough to let us pick the planets of other solar systems out from the glare of their suns, except in a few exceptional cases.
- [6] In 2002, there have been some reports that the properties of atoms as observed in distant galaxies are slightly different than those of atoms here and now. If so, then time symmetry is weakly violated, and so is conservation of energy. However, this is a revolutionary claim, and it needs to be examined carefully. The change being claimed is large enough that, if it's real, it should be detectable from one year to the next in ultra-high-precision laboratory experiments here on earth.
- [7] The paper appeared in
*Social Text*#46/47 (1996) pp. 217-252. The full text is available on professor Sokal's web page at www.physics.nyu.edu/faculty/sokal/.

### Contributors

- Benjamin Crowell,
**Conceptual Physics**