11.3 A numerical scale of energy by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
11.3 A numerical scale of energy
Energy comes in a variety of forms, and physicists didn't discover all of them right away. They had to start somewhere, so they picked one form of energy to use as a standard for creating a numerical energy scale. (In fact the history is complicated, and several different energy units were defined before it was realized that there was a single general energy concept that deserved a single consistent unit of measurement.) One practical approach is to define an energy unit based on heating water. The SI unit of energy is the joule, J, (rhymes with “cool”), named after the British physicist James Joule. One Joule is the amount of energy required in order to heat 0.24 g of water by `1°C`. The number 0.24 is not worth memorizing.
Note that heat, which is a form of energy, is completely different from temperature, which is not. Twice as much heat energy is required to prepare two cups of coffee as to make one, but two cups of coffee mixed together don't have double the temperature. In other words, the temperature of an object tells us how hot it is, but the heat energy contained in an object also takes into account the object's mass and what it is made of.
Later we will encounter other quantities that are conserved in physics, such as momentum and angular momentum, and the method for defining them will be similar to the one we have used for energy: pick some standard form of it, and then measure other forms by comparison with this standard. The flexible and adaptable nature of this procedure is part of what has made conservation laws such a durable basis for the evolution of physics.
Example 4: Heating a swimming pool
`=>` If electricity costs 3.9 cents per MJ (1 MJ = 1 megajoule = `10^6` J), how much does it cost to heat a 26000-gallon swimming pool from `10°C` to `18°C`?
`=>` Converting gallons to `cm^3` gives
`26000 "gallons"×(3780 cm^3)/(1 gallon)=9.8×10^7 cm^3`.
Water has a density of 1 gram per cubic centimeter, so the mass of the water is `9.8×10^7 g`. One joule is sufficient to heat 0.24 g by 1°C, so the energy needed to heat the swimming pool is
`1 J×(9.8×10^7 g)/(0.24 g)×(8°C)/(1°C)=3.3×10^9 J`
`=3.3×10^3 MJ`.= Conservation laws =
The cost of the electricity is `(3.3×10^3 MJ)($0.039"/"MJ)=$130`.
Example 5: Irish coffee
`=>` You make a cup of Irish coffee out of 300 g of coffee at `100°C` and 30 g of pure ethyl alcohol at `20°C`. One Joule is enough energy to produce a change of `1°C` in 0.42 g of ethyl alcohol (i.e., alcohol is easier to heat than water). What temperature is the final mixture?
`=>` Adding up all the energy after mixing has to give the same result as the total before mixing. We let the subscript `i` stand for the initial situation, before mixing, and `f` for the final situation, and use subscripts `c` for the coffee and `a` for the alcohol. In this notation, we have
`"total initial energy"="total final energy"`
`E_(ci)+E_(ai)=E_(cf)+E_(af)`.
We assume coffee has the same heat-carrying properties as water. Our information about the heat-carrying properties of the two substances is stated in terms of the change in energy required for a certain change in temperature, so we rearrange the equation to express everything in terms of energy differences:
`E_(af)-E_(ai)=E_(ci)-E_(cf)`.
Using the given ratios of temperature change to energy change, we have
`E_(ci)-E_(cf)=(T_(ci)-T_(cf))(m_c)"/"(0.24 g)`
`E_(af)-E_(ai)=(T_(af)-T_(ai))(m_a)"/"(0.42 g)`
Setting these two quantities to be equal, we have
`(T_(af)-T_(ai))(m_a)"/"(0.42 g)=(T_(ci)-T_(cf))(m_c)"/"(0.24 g)`.
In the final mixture the two substances must be at the same temperature, so we can use a single symbol `T_f=T_(cf)=T_(af)` for the two quantities previously represented by two different symbols,
`(T_f-T_(ai))(m_a)"/"(0.42 g)=(T_(ci)-T_f)(m_c)"/"(0.24 g)`.
Solving for `T_f` gives
`T_f=(T_(ci)m_c/0.24+T_(ai)m_a/0.42)/(m_c/0.24+m_a/0.42)`
`=96°C`.
Once a numerical scale of energy has been established for some form of energy such as heat, it can easily be extended to other types of energy. For instance, the energy stored in one gallon of gasoline can be determined by putting some gasoline and some water in an insulated chamber, igniting the gas, and measuring the rise in the water's temperature. (The fact that the apparatus is known as a “bomb calorimeter” will give you some idea of how dangerous these experiments are if you don't take the right safety precautions.) Here are some examples of other types of energy that can be measured using the same units of joules:
| Type of energy | Example |
| chemical energy released by burning | About 50 MJ are released by burning a kg of gasoline. |
| energy required to break an object | When a person suffers a spiral fracture of the thighbone (a common type in skiing accidents), about 2 J of energy go into breaking the bone. |
| energy required to melt a solid substance | 7 MJ are required to melt 1 kg of tin. |
| chemical energy released by digesting food | A bowl of Cheeries with milk provides us with about 800 kJ of usable energy. |
| raising a mass against the force of gravity | Lifting 1.0 kg through a height of 1.0 m requires 9.8 J. |
| nuclear energy released in fission | 1 kg of uranium oxide fuel consumed by a reactor releases2×1012 J of stored nuclear energy. |
It is interesting to note the disproportion between the megajoule energies we consume as food and the joule-sized energies we expend in physical activities. If we could perceive the flow of energy around us the way we perceive the flow of water, eating a bowl of cereal would be like swallowing a bathtub's worth of energy, the continual loss of body heat to one's environment would be like an energy-hose left on all day, and lifting a bag of cement would be like flicking it with a few tiny energy-drops. The human body is tremendously inefficient. The calories we “burn” in heavy exercise are almost all dissipated directly as body heat.
Example 6: You take the high road and I'll take the low road.
`=>` Figure f shows two ramps which two balls will roll down. Compare their final speeds, when they reach point B. Assume friction is negligible.
`=>` Each ball loses some energy because of its decreasing height above the earth, and conservation of energy says that it must gain an equal amount of energy of motion (minus a little heat created by friction). The balls lose the same amount of height, so their final speeds must be equal.
It's impressive to note the complete impossibility of solving this problem using only Newton's laws. Even if the shape of the track had been given mathematically, it would have been a formidable task to compute the balls' final speed based on vector addition of the normal force and gravitational force at each point along the way.
How new forms of energy are discovered
Textbooks often give the impression that a sophisticated physics concept was created by one person who had an inspiration one day, but in reality it is more in the nature of science to rough out an idea and then gradually refine it over many years. The idea of energy was tinkered with from the early 1800's on, and new types of energy kept getting added to the list.
To establish the existence of a new form of energy, a physicist has to
(1) show that it could be converted to and from other forms of energy; and
(2) show that it related to some definite measurable property of the object, for example its temperature, motion, position relative to another object, or being in a solid or liquid state.
For example, energy is released when a piece of iron is soaked in water, so apparently there is some form of energy already stored in the iron. The release of this energy can also be related to a definite measurable property of the chunk of metal: it turns reddish-orange. There has been a chemical change in its physical state, which we call rusting.
Although the list of types of energy kept getting longer and longer, it was clear that many of the types were just variations on a theme. There is an obvious similarity between the energy needed to melt ice and to melt butter, or between the rusting of iron and many other chemical reactions. The topic of the next chapter is how this process of simplification reduced all the types of energy to a very small number (four, according to the way I've chosen to count them).
It might seem that if the principle of conservation of energy ever appeared to be violated, we could fix it up simply by inventing some new type of energy to compensate for the discrepancy. This would be like balancing your checkbook by adding in an imaginary deposit or withdrawal to make your figures agree with the bank's statements. Step (2) above guards against this kind of chicanery. In the 1920s there were experiments that suggested energy was not conserved in radioactive processes. Precise measurements of the energy released in the radioactive decay of a given type of atom showed inconsistent results. One atom might decay and release, say, `1.1×10^(-10) J` of energy, which had presumably been stored in some mysterious form in the nucleus. But in a later measurement, an atom of exactly the same type might release `1.2×10^(-10) J`. Atoms of the same type are supposed to be identical, so both atoms were thought to have started out with the same energy. If the amount released was random, then apparently the total amount of energy was not the same after the decay as before, i.e., energy was not conserved.
Only later was it found that a previously unknown particle, which is very hard to detect, was being spewed out in the decay. The particle, now called a neutrino, was carrying off some energy, and if this previously unsuspected form of energy was added in, energy was found to be conserved after all. The discovery of the energy discrepancies is seen with hindsight as being step (1) in the establishment of a new form of energy, and the discovery of the neutrino was step (2). But during the decade or so between step (1) and step (2) (the accumulation of evidence was gradual), physicists had the admirable honesty to admit that the cherished principle of conservation of energy might have to be discarded.
self-check:
How would you carry out the two steps given above in order to establish that some form of energy was stored in a stretched or compressed spring?
(answer in the back of the PDF version of the book)
Mass Into Energy
Einstein showed that mass itself could be converted to and from energy, according to his celebrated equation `E=mc^2`, in which `c` is the speed of light. We thus speak of mass as simply another form of energy, and it is valid to measure it in units of joules. The mass of a 15-gram pencil corresponds to about `1.3×10^15 J`. The issue is largely academic in the case of the pencil, because very violent processes such as nuclear reactions are required in order to convert any significant fraction of an object's mass into energy. Cosmic rays, however, are continually striking you and your surroundings and converting part of their energy of motion into the mass of newly created particles. A single high-energy cosmic ray can create a “shower” of millions of previously nonexistent particles when it strikes the atmosphere. Einstein's theories are discussed later in this book.
Even today, when the energy concept is relatively mature and stable, a new form of energy has been proposed based on observations of distant galaxies whose light began its voyage to us billions of years ago. Astronomers have found that the universe's continuing expansion, resulting from the Big Bang, has not been decelerating as rapidly in the last few billion years as would have been expected from gravitational forces. They suggest that a new form of energy may be at work.
Discussion Question
A I'm not making this up. XS Energy Drink has ads that read like this: All the “Energy” ... Without the Sugar! Only 8 Calories! Comment on this.
11.3 A numerical scale of energy by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.