(15) Energy

Potential and Kinetic

   An interesting thing about the final speed of an object descending (with no friction) from some given height h, along a sloping surface: one can change the slope, one can even change the shape of the surface--yet the final speed v with which it reaches the bottom will always be the same. If it were not for friction, any skier gliding down a snowy hill from the summit to the base should arrive with the same speed (though not necessarily in the same time!), whether the path taken is the easy beginner's slope or the steep experts-only route.

   Reducing the slope of the surface reduces the acceleration a, but it also lengthens the time of descent, and these two variations cancel, leaving the final speed unchanged. The same speed is also obtained if the object falls vertically from that height h, and in that case it is easily derived, as follows. The duration t of the fall is given by

h = g t²/2

Multiply both sides by g:

gh = g²t²/2

Then since the final velocity is

v = gt

one gets

gh = v²/2

    By the last equation, as the object loses elevation--assuming nothing interferes with its motion--v² grows, and as noted, this growth does not depend on the path taken.

    This exchange between h and v² also works in the opposite direction: an object rolling up an incline loses v² in direct proportion to the elevation h it gains. A marble rolling down the inside of a smooth bowl gathers speed as it approaches the bottom, then as it shoots up the other side it loses all of it again. If no friction existed, it would rise again to the same height as the one from which it had started.

   A simple pendulum, or a child on a swing, also trades height for v² and back again, in the same manner. And bicycle riders are well aware that the speed gained rolling down a hillside can be traded for height when climbing the next slope. It is as if height gave us something with which we could purchase speed, and which later, if the occasion demanded, could be converted back to height.

That "something" is called energy. It was already briefly discussed in an earlier section.

This back and forth trading suggests that perhaps the sum

gh + v²/2

has a constant value: if one part decreases, the other part must get bigger. Is that sum the energy? Not quite. The effort of getting heavy load up a height h is greater than that of lifting a light one. Let us now call the amount of matter in an object its "mass. " It is obviously proportional to the object's weight, but as will later be seen, the concept of mass is more complicated than that.

   If energy is to measure the effort in lifting a load, it should also be proportional to its mass m. We thus multiply everything by m and write

Energy = E = mgh + mv²/2

    A well-known fact--already hinted at--is that in a system which does not interact with its surroundings, the total energy (denoted here by the letter E) stays the same ("is conserved"). In a pendulum at the extreme point of its swing, v = 0 and therefore the second term above vanishes, while the first term is at its biggest. Then as the mass descends descends, mv²/ 2 increases and mgh drops, until at the bottom of the swing the first term is at its smallest and the second reaches maximum. On the upswing the process reverses, and the sequence is repeated for every swing.

    Both terms in the equation above have names: mgh is the potential energy, the energy of position, and mv²/2 is the kinetic energy, the energy of motion.

    The exact number representing E will obviously depend on where h is measured from (the floor? sea level? the center of the Earth? ). Different choices are possible, and each leads to a different value of E: the formula is thus meaningful only if a certain reference height is chosen where h=0.

Other Kinds of Energy

    Textbooks define energy as "the ability to do work" and they define work as "overcoming resistance over a distance". For instance, if m is the mass of a brick, the force on it is mg and lifting it against gravity to a height h, against the pull of gravity, requires the performance of work W, with

W = mgh

Dragging that brick a distance x along level ground against the force of friction F similarly requires the performance of work

W = Fx

    The above two types of work are discussed again, in more detail, in section 18c: Work. A third type is treated in the following section #18d: Work Against an Electric Force: The Van De Graaff generator. That section also covers the generation of lightning and the clinging of projector transparencies after they emerge from a copying machine.

    For the record, work is measured in joules, after James Prescott Joule (1818-89), a brewer in Manchester, England, whose experiments helped establish the fact that heat was a form of energy (see further below) and not some mysterious fluid permeating matter. Since any such work can be performed by a machine, one can also loosely define energy as anything that can cause a machine to turn.

    Energy is also measured in joules. In many ways it resembles money: it is a currency in which all processes in nature must be paid for. Just as money can come in dollars, pesos, yen, rubles or liras, so energy can come in many forms--electricity, heat, light, sound, chemical, nuclear. The expression for the total energy of a system of objects can be written

E = (potential) + (kinetic) + (electric) + (heat) + . . .

    where "kinetic" for instance stands for the sum of mv²/2 for all the component parts. And it is still true that if the system does not interact with the outside, the total value of E is conserved.

    It is also true that in most cases, with proper tools, one form of energy can be converted into another: light shining on solar cells generates electricity, which can turn the motor of a fan, providing the kinetic energy of rotating blades, or run a radio, producing sound.

    The approximate relative weights of atoms are H=1, C=12, O=16, so 2 grams of H combine with 16 grams of O, and 12 grams of C with 32 grams of O. Each gram of hydrogen combines with 8 times its weight in oxygen, each gram of carbon with nearly 3 times its weight in oxygen. Molecules of TNT also contain carbon and hydrogen, which is where their energy comes from (and nitrogen, whose role is different). But in order for these atoms to combine rapidly with oxygen, the required oxygen must be already present inside the TNT molecule, kept apart from the "fuel" by the nitrogen atoms. That oxygen takes up about 45% of the weight of the material, and nitrogen atoms take an additional 20%. Thus only about 1/3 of the weight of TNT consists of energy-releasing atoms.

    In fact, exploding pure TNT creates only about a quarter of the energy listed above, because the 6 atoms of oxygen in TNT are too few to combine with 9 atoms of carbon (each of which can hold 2 atoms of oxygen), not even counting the hydrogen (4.5 atoms of oxygen more). The oxygen in TNT can link to only about a quarter of the atoms that can combine with it. It can get more powerful by mixing it with oxygen-rich compounds (see here).

Heat