Lesson Plan #24

(15)  Energy  

  An intuitive introduction to the concept of energy as "anything that can cause a machine to turn" and "the universal currency in which every physical process must be paid for."
  This topic stresses mechanical energy, potential and kinetic, and also describes conversion between types of energy (while conserving the total amount), units, and the special position of heat.

Part of a high school course on astronomy, Newtonian mechanics and spaceflight
by David P. Stern

This lesson plan supplements: "Energy," section #15   http://www.phy6.org/stargaze/Senergy.htm

Related sections:
            #18c: Work
            #18d: Work Against an Electric Force: The Van De Graaff generator

"From Stargazers to Starships" home page: ....stargaze/Sintro.htm
Lesson plan home page and index:             ....stargaze/Lintro.htm

Goals: The student will learn about

  • The concept of energy, using a variety of examples and also the analogy between energy and money.

  • The conservation of energy in its conversion from one form to another.

  • Units of energy--Joule, calorie, also watt and kilowatt-hour.

  • The special nature of heat, as the "soft currency" of the energy world.

Terms: Energy (potential, kinetic, conservation of), pendulum, joule, calorie, second law of thermodynamics. (kilojoule, kilocalories)

Stories and extras: The energy content of a candy bar. And of TNT

Starting the lesson:

Today we will study energy,so we might just as well start by asking--"What is energy?"

    Solicit answers; stop if someone says "what it takes to run a machine."

    If someone gives the formal definition "ability to do work" ask:
    "What then is work?" If the answer is "force times distance" say, that is correct; however, that definition will have to wait until the class has learned what a force is.

    We could redefine it as "overcoming resistance over a distance"--for instance, lifting a brick (against gravity) from the floor to the table, or dragging it along the floor (against friction), and then work equals resistance times distance.

    Any of these could also be done by a machine, so for a start we will simply say "energy is anything that can run a machine."

Next question to the class:: what kinds of energy did you use today?

    --Electricity, if you used electric lights, or a radio, or TV
    --Light--that was what the electricity in the lightbulb was converted to
    --Sound--that was what the electricity in the radio was converted to
    --Chemical energy--when you ate breakfast, it gave you strength.
    --Heat--if you cooked your food, or heated the house.
    --Nuclear energy--if you enjoyed sunlight, because the Sun gets its energy by combining atomic nuclei of hydrogen to helium, deep inside it.

    That energy becomes heat, and heat causes light to come out.
    In an electric lightbulb, electricity makes a wire very hot, and that heat is what produces light.

Does anyone ride a bicycle? (Maybe to school?).
    On a bike, the chemical energy of your food is turned to motion, through your muscles. You can build up speed--that is kinetic energy--or you can climb up a hill--that is potential energy.

    And you know you can trade one kind against the other: rolling down a hill, you lose height as you gain speed, and that speed helps you get up the next hill (the same in a roller coaster).

Then go to the lesson. The questions below may be used in the lesson, or afterwards. You could also distribute to the students paper copies of the table from "Stargazers", showing the conversion of energies.

        Note to the teacher: a somewhat related discussion forms the opening of the lesson plan on sunlight. That discussion opens with the comment that almost all the energy we use comes from the Sun and students are asked to list types of energy together with the ways the Sun has provided them (e.g.: growing the plant food we eat requires sunlight). A later part of the discussion tries to list the relatively few types of energy which do not come from the Sun.

Guiding questions and additional tidbits with suggested answers.

-- When an object falls down from a height h meters, what is the relation between h and its final velocity v, in meters per second?

    gh = v2/2

--What is interesting about this relation?

        In the absence of friction, the final speed v is the same no matter how the object came down--sliding down a smooth inclined ramp, or even a roller-coaster track, still gives the same final speed.

    [The teacher may note that while the final speed is the same, the time taken to reach bottom isn't.
      For example: sliding down an inclined board like the one used by Galileo, the object gains speed much more slowly, yet the distance it must cover is longer, so that the time required is much longer, too.]

-- Is something kept constant in this motion?

    Yes, the sum (gh +v2/2)

-- Is this the energy? (No) Why?

        You expect a bigger moving body to have more energy--a rolling bowling ball more than a rolling marble. For that reason we must multiply by mass m:

    E = mgh + (1/2)mv2

      We have not yet defined mass; for the time it is understood to be "the amount of matter in motion," which we can measure by weighing.
      Later we will find a way of measuring mass that does not involve gravity.

    A note about units. In any calculation in physics, we must always pay close attention to the units we use. If inappropriate units are used, mistakes easily creep in, fulfilling "Murphy's law"--if anything can go wrong, it will.

    In all our calculations involving Newtonian mechanics (unless explicitely stated otherwise) the so-called MKS units are used--distances in Meters, masses in Kilograms, time in Seconds, and all derived units based on these three. In those units g = 9.81 and energy comes out in joules. Whenever other units are given, be sure to convert them to MKS! One spacecraft sent towards Mars was lost, because engineers giving the command for a final crucial rocket burn got their units confused.

--How does a pendulum or a swing demonstrate the conservation of total energy?

--How does a roller-coaster demonstrate it?

--What is work W? How much work is performed in lifting a mass m by a height h?

    Work is overcoming resistance over a distance--multiplying the resistance by distance. A mass m (see next lesson) has weight mg, and lifting it a distance h performs work

    W = mgh

--If m is in kilograms, h in meters, g = 9.81 meter/sec2, in what units is W, as given by the above formula?


--You have climbed to the second floor, raising yourself by 9 feet, (1 ft=30.5 cm = 0.305 meter). You weigh 150 pounds (1 pound = 0.454 kg). How much work did you perform?

    h = 9*0.305 = 2.745 meters, m = 150*0.454 = 68.1 kg. If g = 9.81 m/s2, then

    W = mgh = 1833.8 joule

--Into what form of energy did this work go?

    Into the potential energy of height.

--From what form of energy did it come?

       From the chemical energy of the food you ate. But note (below) that you do not "burn many calories" by climbing one floor, compared to the number you get from food.

       Suppose you have eaten one square of chocolate weighing 4 grams (1/8 of a bar weighing one ounce). The chocolate contains 2 grams cocoa fat, providing 9 calories per gram (typical for fats), and 2 grams sugar, a carbohydrate with 4 calories per gram, for a total of 18 + 8 = 26 calories. These are "kilocalories" of 4180 joule each, so that piece of chocolate has given you the equivalent of 108,680 joules. If your body can convert it to muscle power with an efficiency of 10% = 0.1, you get 10,868 joules of usable work from that piece of candy, enough to climb 10,868/1833.8 or about 6 floors.

--You jump down from the height of one floor. With what speed v do you hit the ground?

    Your potential energy of 1833.8 joule is converted to kinetic energy of (1/2)mv2 = 34.05 v2 . Then v2 = 53.856 , v = 7.3387 m/s.

    In miles-per hour (1 mile = 1609 meters).

    v = 7.3387*3600 = 26,419 meters/hour = 16.4 miles/hour.

--Even a hospital patient lying in bed all day needs to eat. Why?

       Food energy creates body warmth, also powers the many chemical reactions required by life. In addition, the air we breathe out contains moisture (breathe onto a cool mirror to see it!). Energy is needed to convert the water we drink to vapor.

On the table of energy conversions, which form is converted into which:

-- In an electric fan?

    Electric to kinetic

--In an elevator winch?

    Electric to potential.

--Can we convert it back when the elevator descends?

       In principle, yes, the motor can become a generator when turned from the outside. In practice, it is too complicated to return power to the public supply. But we can absorb electric power generated this way, turning it to heat in a resistor, and that way brake the motor.

--In a light emitting diode?

    Electricity to light.

--Why did we say "light emitting diode" and not "electric lightbulb"?

    In a diode, electricity is directly converted to light. In a lightbulb, it creates heat, and the heat creates light.

--In a car battery?

    Chemical to electric.

-- Can it be converted back to chemical energy?

    Yes, when the battery is charged.

-- In a rocket nozzle?

    Heat to kinetic energy. We will later see that the converging-diverging design of the rocket nozzle is very efficient in converting heat to kinetic energy.

       Has anyone here read "October Sky", or seen the film? It is a true story of high school boys building and flying rockets, and after they discovered the proper design of the nozzle, their rockets flew much higher. The conversion is never complete--heat can never be completely converted to mechanical energy--but the nozzle comes fairly close to the theoretical limit.

--In quicklime? What happens there?
      [This question may not be meaningful enough to students living in a big city.]

    Limestone is heated over fire in a kiln. It is a compound of calcium, oxygen and carbon dioxide. The heat drives off the carbon dioxide and crumbles the stone to calcium oxide, quicklime, which holds more chemical energy.

    For making mortar, builders slake the quicklime with water. It heats up, returning its chemical energy to heat.

-- How do spacecraft get their electric energy?

    Spacecraft near Earth use solar cells, that convert light to electricity--sort of the inverse effect of a light-emitting diode.

    Around the outer planets, sunlight is too dim to provide enough energy in this way. Spacecraft that fly there, e.g. Voyagers 1-2 and Pioneers 10-11, use radioactive sources which generate heat, and thermocouples (special junctions of different metals) convert it to electricity.

    The Russians experimented with small nuclear reactors on spacecraft. One crashed into a lake in Canada, contaminating it with radioactivity and creating great resentment. No such reactors are being flown any more.

-- How is mechanical power defined? What are its units?

    Power is the rate at which energy is supplied, measured in watts. One watt is 1 joule/second, 1 kilowatt = 1000 watt.

-- Your electric bill charges you a certain price per kilowatt-hour (kwh). What do kilowatt-hours measure?

    Energy: 1 kwh = 3,600,000 joule.

--Food energy is measured in calories. How many calories does a gram of sugar contain?

    About 4 calories. These are "kilocalories" of "large calories", each containing 1000 "small" calories.

    A "small" calorie contains 4.18 joule, a "kilocalorie" has 4180 joule, and a gram of sugar--as in a piece of candy--has 16720 joule.

(The example with a square of chocolate suggests this is about enough to let you climb one floor)

--How about other foods?

    Proteins contain about 4 calories per gram, too (actually more, but a fraction of their energy is invested in breaking them up).

    Fats have about 9 cal/gr., alcohol about 7

(Only in case the question comes up: sugar belongs to a family of compounds known as carbohydrates, produced by plants which take apart carbon dioxide and water, and re-arrange their atoms in new combinations. Some carbohydrates are digestible, others (like wood) are not but serve plants as building materials. Fats are a different rearrangement of these atoms, richer in energy. Proteins contain nitrogen and are the basic building blocks of living tissue: they can serve as fuel, but their main role is in forming the complex chemicals of our bodies.)

--Any materials contain more energy?

    Yes, gasoline for instance has about 11.5 cal/gr. However, such substances are poisonous.

--How does TNT (tri-nitro-toluene) compare?

    Less than sugar, only about 3.8 calories/gram. The reason is that TNT must already contain the oxygen with which it combines, and that takes up a big part of its weight (explain if needed).

-- Seward, the port at the end of the Alaska Railroad, has steep Mt. Marathon towering just behind it, to a height of about 900 meters. Every 4th of July a footrace is held, from the town to the top of Mt. Marathon and (with a lot of sliding!) back. The current record is 43 minutes and a fraction.

    Q.:       A runner weighing 60 kg reaches the top of Mt. Marathon in one hour. Approximating g = 10 m/s2 and one horsepower=750 watt (accurate value is 736), how many horsepower must that person develop just to overcome gravity, on the average? ?

(This can be given on the blackboard.)
    A.: Overcoming a force mg = 600 Newton over 900 meters requires 540,000 joule. Dividing by 1 hour = 3600 sec gives a power output of 150 watt or 0.2 HP.

-- Why do we often say "energy is lost as heat"?

    Because we often cannot convert any of it back to other forms (or at most can just get back some part).

--What does the second law of thermodynamics say?

    That one can never convert heat completely back into other forms of energy, some of it is always irrecoverable.

    [Optional: The fraction of heat energy which can be converted to other forms depends on the temperature at which the heat is provided.

       The unrecovered heat is changed to a lower temperature, and the fraction we recover depends on these two temperatures--the one at which the heat is received, and the one at which the remainder can be dumped.

       A power station needs not only a supply of hot steam, but also a way of dumping the heat left at the end of the cycle. Steam locomotives dumped their spent steam into the air, and therefore needed a great amount of of water, carried in their tenders. Electric power stations (of any kind) recycle their steam and cool it with air in cooling towers, like those of 3-mile island which (for some reason) became a symbol of nuclear energy. Other power stations use nearby lakes and rivers to cool and condense their steam, and steamships (of course!) do so with seawater.]

(Only in case the question is asked: the first law of thermodynamics says, essentially, that heat is a form of energy.)

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Author and Curator:   Dr. David P. Stern
     Mail to Dr.Stern:   stargaze("at" symbol)phy6.org .

Last updated: 10-8-2004