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(15) Energy


12b. Orbital Motion

12c. Venus transit (1)

Newtonian Mechanics

13. Free Fall

14. Vectors

15. Energy

16. Newton's Laws

17. Mass

17a. Measuring Mass
        in Orbit
17b. Inertial balance

18. Newton's 2nd Law

18a. The Third Law

18b. Momentum

18c. Work

Potential and Kinetic

downhill skier

   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.

Devices or processes that convert energy
from one form (column) to another (row)
- Kinetic Potential Heat Light Chemical Electric
Kinetic ***** Pendulum Rocket Nozzle Solar sail Muscles Electric motor
Potential Pendulum ***** Steam boiler x x Elevator winch
Heat Friction x ***** Solar heater Fire Electric stove
Light x x Lightbulb, Sun ***** Firefly light Light emitting diode
Chemical x x Quicklime kiln Green plants ***** Car battery
Electric Windmill power Hydroelectric power Thermocouple Solar Cell Flashlight battery *****

    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.


    Like currencies, different kinds of energy have a certain exchange rate: in the exchange between kinetic or potential energy and heat, for instance, one calorie equals about 4.18 joules. The chemical energy of food is also measured in calories, though one should note these are "big" calories or kilocalories, each equal to 1000 of the small kind.

    The rate at which energy is supplied or used is called power and is measured in watts, after the inventor of the modern steam engine, the Scotsman James Watt (1736-1819): an energy supply providing one joule per second gives one watt of power. Thus a 60-watt bulb provides 60 joules each second--about as much as a biker pedaling up a hill. Bills for electric energy, sent out by power companies, are usually calculated at so many dollars, pesos, yen etc. per kilowatt-hour (kwh)--the energy of one kilowatt or 1000 watt, supplied for one hour. Since the hour has 3600 seconds, it follows that one kwh equals 3 600 000 joules.

Chemical Energy--especially, that of Foods

    The energy of foods, as often printed on food packages, is measured in calories--"big" calories or kilo-calories, each equal to 4180 joule. It helps to get some feeling about what those numbers mean.

    Assuming g=10 meter/sec2, the weight mg of a 70 kg person--the pull of gravity on that person--is about 700 newtons. The energy of one calorie is therefore enough to raise him or her by about 6 meters (6 x 700 = 4200 joule). Of course, if the efficiency of converting the chemical energy of food to muscle energy is only 10%, the person would only rise 0.6 meter or about 2 feet. Still, a typical American may consume 3000 calories per day, enough fuel for quite a bit of climbing.

    Most food belongs to one of three classes. Carbohydrates like sugar and starch consist of relatively simple compounds of oxygen, hydrogen and carbon (wood and paper are closely related, but not digestible). They give about 4 calories per gram. Fats and oils are a different family formed by those atoms, and contain about 9 calories per gram. And proteins, which also contain nitrogen (and sometimes sulfur) give a net gain of about 4 calories per gram, after one subtracts the energy needed for their break-up from the amount they can supply. Proteins are less important as fuel than as raw material for the complex molecules that maintain life.

    Compare that to the energy obtained by burning gasoline which gives about 11.5 calories per gram (but is poisonous). Oil and fat come pretty close! Ethyl alcohol, the stuff in beer and whiskey, has about 7 cal/gr. Some people might not classify it as a poison, but its chemical relatives in the alcohol family (e.g. methyl alcohol) are indeed dangerous poisons.

    Interestingly, the explosive TNT--trinitrotoluene--only releases 3.8 calories per gram. Its energy release may be extremely sudden, but per unit weight, TNT contains less energy than sugar.

    How come? Well, sugar contains carbon and hydrogen (also some oxygen), and as these combine with the oxygen breathed in by our lungs, energy is released. The chemical process is complex, not a simple combination with oxygen, but the end produces are the same--hydrogen combines to H2O or water, carbon to CO2 or carbon dioxide.

    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.


    When a bank changes money from one currency to another, it usually charges some percentage as the cost of the transaction. The same happens in the exchange of energy from one form to another: you always get less than what you put in. The marble sliding down the side of the bowl, for instance, always rises on the other side to a lower elevation than the one it started from.

    The missing energy is not lost, however, but ends up as heat. Heat is the "soft currency" of the energy universe: it is possible to convert heat to other forms of energy (in a steam engine, for instance), but one can never get the full value. That, essentially, is the "second law of thermodynamics," a fundamental law related to the nature of heat. The missing amount not only remains as heat, but becomes heat at a lower temperature, from which only an even smaller fraction can be converted into other forms. It would solve all of mankind's energy problems if, for instance, one could extract the heat energy of the oceans, leaving them slightly cooler and converting the extracted heat to electricity; but the second law tells that it cannot be done.


    Packaged foods in the US carry labels listing their nutrients and the amount of calories (actually, kilocalories) per portion and per given weight. In other countries the information may differ, and the British "Cadbury's Wispa" milk chocolate bar offers a choice of kilocalories and kilojoules:

. Bar100 gr.
Energy Kj 885 2300
..........Kcal 210  550
Protein  2.7 gr  7.1
Carbohyd.20.8 53.9
Fat 13.2 34.2

Questions from Users:  
***       Can kinetic energy be converted to work?
      ***       Energy from the Earth's Rotation?
           ***       Teaching about energy in 8th grade

Next Stop: #16 Newton and his Laws

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Author and Curator:   Dr. David P. Stern
     Mail to Dr.Stern:   stargaze("at" symbol) .

Last updated: 11 May 2005
Reformatted 24 March 2006