Notes from Under Sky

A Modicum of Energy

How did we come up with the notion of energy?

Someone asked about Einstein's equation E = mc2:

But I'm thrown by the SPEED of light thing. If something has a mass of 10 grams and I multiply it by 300,000 km/s, squared, it doesn't make sense. Is there a scientific conversion from speed to some other unit? How do you multiply mass times speed squared? Or is it just representational? Can the explanation be simplified?

The problem is that energy is a pretty abstract concept. You can't point to some place on a brick and say, "There's its energy," the way you can spread your hand against it and say, "That's its width," or pick it up and say, "That's how much it weighs."

Nevertheless, if you drop a brick from shoulder height onto your foot, it'll hurt a lot and may break some bones if it's heavy enough, whereas the same brick if dropped from a height of an inch will merely make you look foolish. The only real difference between the two situations is the speed with which the brick hits your foot. In one case, the brick strikes your foot with a gentle thud, and in the other, it has had some time to build up speed and land on your foot with considerable impact.

Some enterprising soul may have decided to try to quantify this effect, and rather than using his foot as the impact meter, decided to use some other absorbing substance. We might use styrofoam, for instance. If you drop a brick onto a block of styrofoam, it sinks partway in before coming to a stop. That's because the styrofoam exerts an upward force on the brick, and it take some time before enough force has been applied to bring the brick to a complete stop.

If you stack two bricks, one on top of the other, before dropping them onto the styrofoam, the pile will sink further in (assuming you drop it from the same height as before), because the styrofoam exerts just as much force as before, and that same force has to be applied for more time to stop the greater brickage. For "ideal" styrofoam, we might determine that the indentation is just twice as deep as before; for a pile of three bricks (with therefore three times the brickage), it would be three times as deep; and so on.

Alternatively, we could drop one brick as before, but from a greater height, and see how much higher we have to go before we also get twice as deep an indentation, or three times, or whatever. Again assuming that we have ideal styrofoam, we find that we need only drop it from twice as great a height, or three times, or whatever.

As a result of these observations, we might define a new quantity, called energy, which is a measure of the impact with which the brick hits the styrofoam. Since it's proportional to brickage for a given height, and also proportional to height for a given brickage, we might assume that it would be proportional to brickage times height, and we would be right—except that we probably want a more dignified and general term for "amount of stuff" in the brick than "brickage." Following Newton, we use the term "mass." In that case, we can write

E = kmh

where E, m, and h stand for energy, mass, and height, respectively, and k is the constant of proportionality. What would that be? With hindsight, it is not so hard to see. If we were on the Moon, for instance, dropping a brick is not as painful as it is on the Earth, since the Moon's gravity is weaker. We might guess, then, that it is the Earth's gravity, denoted g, that is the constant of proportionality. The value of g is about 9.8 meters per second per second, meaning that an object dropped near the Earth's surface accelerates downward, increasing its velocity by 9.8 meters per second every second until it hits the ground. With that in mind, we can now write

E = gmh

I emphasize that there is no obvious physical interpretation of this product of three quantities. It's not as though you're measuring the volume of the brick and therefore multiply the three dimensions. The energy is just a quantity we defined to measure a property of interest.

Then again, we need not drop the brick. We can throw it from the side, and as long as we attain whatever speed was derived from gravity when we dropped it, we should get the same effect. To find out what that effect is, we need to find out what speed a brick achieves when dropped from a height of h. It turns out to be

v = sqrt(2gh)

where "sqrt" is the square root function. In that case, we also have

v2 = 2gh
v2/2 = gh

and then

E = mv2/2

which you might notice has the same units as mc2 in Einstein's equation. In the so-called mks metric system (meters/kilograms/seconds), mass is measured in kilograms, and velocity in meters per second. Thus, energy is measured in units of kilogram-meters-squared-per-second-squared, an ungainly mouthful that is given the special unit name of "joule." For instance, according to this formula, a brick with a mass of 1 kilogram and a velocity of 6 meters per second (about 13 mph) is 18 joules, since half of 6 squared is 18. (There is also a cgs system, for centimeters/grams/seconds, in which the unit of energy is the "erg." There are 10 million ergs in each joule, as you can verify if you do the unit conversion.)

You may also have noticed that these formulas describe the energy of the brick at two different moments in time. The first gives the energy of the brick before it's been dropped, whereas the second gives the energy just before it hits the styrofoam (or your foot). We can put it another way: the first gives the energy due to the brick's position, which we call its potential energy; and the second gives the energy due to the brick's velocity, which we call its kinetic energy. It's better therefore to write them as we often do, as

PE = mgh
KE = (1/2) mv2

where I've slightly reformatted the equations according to tradition.

Incidentally, those equations can characterize the brick at different points throughout its fall, not just at the start and at the end. At the start, a brick's PE might be 18 joules, but since it's not moving, its KE must be zero. Conversely, as it strikes the styrofoam, its KE is 18 joules, but since its height is zero, its PE has to be zero, too. It's tempting to think that throughout the fall, the KE plus the PE must be 18 joules, and that thought turns out to be true: Although both KE and PE are constantly changing, with KE increasing and PE decreasing, they change in such a way that their sum is constant.

Another way of saying this is that the brick's total energy is conserved. There's no reason why this has to be true—it's just our common experience that it is true. You can raise the brick back to its original height, thus restoring its PE—but then in order to do that, you always use up at least 18 joules worth of energy. (In fact, you invariably exert more, the rest of it being wasted as heat.)

There are lots of conservation laws, by the way: conservation of energy, of linear momentum, of angular momentum, and so forth. Before Einstein formulated his theory of relativity, there was also a law of the conservation of mass. This seems pretty straightforward, since in everyday experience, you can't just destroy mass; we merely shuffle it from one place to another. We can burn wood, it is true, and in so doing, it seems to vanish, but careful chemical experiments demonstrated that what really happened was that the wood was oxidized and the loss of mass in the wood we burned was really transformation into carbon dioxide and other molecules that dispersed into the air. So it seemed that mass really was conserved.

But then Einstein came along and demonstrated convincingly that mass was just another form of energy—albeit extremely concentrated. This was his famous formula E = mc2. Incidentally, that mass is relativistic mass. Physicists generally prefer to deal with proper mass, also known as rest mass, in which case the formula is the somewhat less pithy E2 = p2c2 + m2c4, where p is the object's momentum. Note, though, that this formula still gives energy in good old units of kilogram-meters-squared-per-second-squared, or joules. It is therefore entirely consistent with the definitions of PE and KE.

By Einstein's law, we can make matter vanish, but in doing so, we produce an enormous amount of energy. In order to build up enough velocity to create 18 joules upon impact, we must drop a 1-kilogram brick from a height of about 1.8 meters (about 6 feet), assuming g = 10 meters per second per second. That same brick, if converted entirely into energy according to Einstein's formula, would yield an amazing 90 million billion joules. This conversion is what allows the Sun, for instance, to get such wholesale returns of energy from such a trifling investment of hydrogen (to paraphrase Twain for a moment).

It works the other way around, too: We can create matter, provided we throw in enough energy. There's a catch, though. Whenever we create a particle, it seems we have to create its corresponding antiparticle. We can create an electron if we put in about a million electron volts of energy (electron volts are also units of energy, which are much smaller than joules and therefore convenient when dealing with subatomic particles), but we have to create the anti-electron, too, also known as the positron. There doesn't seem to be any way around this, and—you guessed it—this too has to do with conservation laws.

Finally, an electron volt is called that because it's the amount of energy required to push an electron "the wrong way"—say, from a battery's positive terminal to its negative terminal—across a potential of one volt. In other words, to push an electron through a wire from the nubby end of a AA battery to its flat end requires an investment of 1.5 eV. You can see why electron volts are so tiny.

Nevertheless, it's the same kind of energy, which is why you can create energy by using gravity (usually falling water instead of falling bricks), and using that energy to push electrons the wrong way—uphill, so to speak. Then you let them go the natural way, and they can power your appliances.

Copyright (c) 2005 Brian Tung