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Table of Contents

Contents
Section 11.1 - The search for a perpetual motion machine
Section 11.2 - Energy
Section 11.3 - A numerical scale of energy
Section 11.4 - Kinetic energy
Section 11.5 - Power
Section 11.6 - Summary

Conservation laws

../../../share/mechanics/figs/jupiter-comet

a / In July of 1994, Comet Shoemaker-Levy struck the planet Jupiter, depositing \(7\times10^{22}\) joules of energy, and incidentally giving rise to a series of Hollywood movies in which our own planet is threatened by an impact by a comet or asteroid. There is evidence that such an impact caused the extinction of the dinosaurs. Left: Jupiter's gravitational force on the near side of the comet was greater than on the far side, and this difference in force tore up the comet into a string of fragments. Two separate telescope images have been combined to create the illusion of a point of view just behind the comet. (The colored fringes at the edges of Jupiter are artifacts of the imaging system.) Top: A series of images of the plume of superheated gas kicked up by the impact of one of the fragments. The plume is about the size of North America. Bottom: An image after all the impacts were over, showing the damage done.

Chapter 11. Conservation of energy

11.1 The search for a perpetual motion machine

perpetual-motion-magnet

a / The magnet draws the ball to the top of the ramp, where it falls through the hole and rolls back to the bottom.

perpetual-motion-arms

b / As the wheel spins clockwise, the flexible arms sweep around and bend and unbend. By dropping off its ball on the ramp, the arm is supposed to make itself lighter and easier to lift over the top. Picking its own ball back up again on the right, it helps to pull the right side down.

Don't underestimate greed and laziness as forces for progress. Modern chemistry was born from the collision of lust for gold with distaste for the hard work of finding it and digging it up. Failed efforts by generations of alchemists to turn lead into gold led finally to the conclusion that it could not be done: certain substances, the chemical elements, are fundamental, and chemical reactions can neither increase nor decrease the amount of an element such as gold.

Now flash forward to the early industrial age. Greed and laziness have created the factory, the train, and the ocean liner, but in each of these is a boiler room where someone gets sweaty shoveling the coal to fuel the steam engine. Generations of inventors have tried to create a machine, called a perpetual motion machine, that would run forever without fuel. Such a machine is not forbidden by Newton's laws of motion, which are built around the concepts of force and inertia. Force is free, and can be multiplied indefinitely with pulleys, gears, or levers. The principle of inertia seems even to encourage the belief that a cleverly constructed machine might not ever run down.

Figures a and b show two of the innumerable perpetual motion machines that have been proposed. The reason these two examples don't work is not much different from the reason all the others have failed. Consider machine a. Even if we assume that a properly shaped ramp would keep the ball rolling smoothly through each cycle, friction would always be at work. The designer imagined that the machine would repeat the same motion over and over again, so that every time it reached a given point its speed would be exactly the same as the last time. But because of friction, the speed would actually be reduced a little with each cycle, until finally the ball would no longer be able to make it over the top.

Friction has a way of creeping into all moving systems. The rotating earth might seem like a perfect perpetual motion machine, since it is isolated in the vacuum of outer space with nothing to exert frictional forces on it. But in fact our planet's rotation has slowed drastically since it first formed, and the earth continues to slow its rotation, making today just a little longer than yesterday. The very subtle source of friction is the tides. The moon's gravity raises bulges in the earth's oceans, and as the earth rotates the bulges progress around the planet. Where the bulges encounter land, there is friction, which slows the earth's rotation very gradually.

11.2 Energy

moon-rock

c / A mysterious moon rock makes a perpetual motion machine.

faucet

d / Example 1.

eg-stevin

e / Example 3.

hoover-dam

Discussion question A. The water behind the Hoover Dam has energy because of its position relative to the planet earth, which is attracting it with a gravitational force. Letting water down to the bottom of the dam converts that energy into energy of motion. When the water reaches the bottom of the dam, it hits turbine blades that drive generators, and its energy of motion is converted into electrical energy.

The analysis based on friction is somewhat superficial, however. One could understand friction perfectly well and yet imagine the following situation. Astronauts bring back a piece of magnetic ore from the moon which does not behave like ordinary magnets. A normal bar magnet, c/1, attracts a piece of iron essentially directly toward it, and has no left- or right-handedness. The moon rock, however, exerts forces that form a whirlpool pattern around it, 2. NASA goes to a machine shop and has the moon rock put in a lathe and machined down to a smooth cylinder, 3. If we now release a ball bearing on the surface of the cylinder, the magnetic force whips it around and around at ever higher speeds. Of course there is some friction, but there is a net gain in speed with each revolution.

Physicists would lay long odds against the discovery of such a moon rock, not just because it breaks the rules that magnets normally obey but because, like the alchemists, they have discovered a very deep and fundamental principle of nature which forbids certain things from happening. The first alchemist who deserved to be called a chemist was the one who realized one day, “In all these attempts to create gold where there was none before, all I've been doing is shuffling the same atoms back and forth among different test tubes. The only way to increase the amount of gold in my laboratory is to bring some in through the door.” It was like having some of your money in a checking account and some in a savings account. Transferring money from one account into the other doesn't change the total amount.

We say that the number of grams of gold is a conserved quantity. In this context, the word “conserve” does not have its usual meaning of trying not to waste something. In physics, a conserved quantity is something that you wouldn't be able to get rid of even if you wanted to. Conservation laws in physics always refer to a closed system, meaning a region of space with boundaries through which the quantity in question is not passing. In our example, the alchemist's laboratory is a closed system because no gold is coming in or out through the doors.

Example 1: Conservation of mass
In figure d, the stream of water is fatter near the mouth of the faucet, and skinnier lower down. This is because the water speeds up as it falls. If the cross-sectional area of the stream was equal all along its length, then the rate of flow through a lower cross-section would be greater than the rate of flow through a cross-section higher up. Since the flow is steady, the amount of water between the two cross-sections stays constant. The cross-sectional area of the stream must therefore shrink in inverse proportion to the increasing speed of the falling water. This is an example of conservation of mass.

In general, the amount of any particular substance is not conserved. Chemical reactions can change one substance into another, and nuclear reactions can even change one element into another. The total mass of all substances is however conserved:

the law of conservation of mass

The total mass of a closed system always remains constant. Mass cannot be created or destroyed, but only transferred from one system to another.

A similar lightbulb eventually lit up in the heads of the people who had been frustrated trying to build a perpetual motion machine. In perpetual motion machine a, consider the motion of one of the balls. It performs a cycle of rising and falling. On the way down it gains speed, and coming up it slows back down. Having a greater speed is like having more money in your checking account, and being high up is like having more in your savings account. The device is simply shuffling funds back and forth between the two. Having more balls doesn't change anything fundamentally. Not only that, but friction is always draining off money into a third “bank account:” heat. The reason we rub our hands together when we're cold is that kinetic friction heats things up. The continual buildup in the “heat account” leaves less and less for the “motion account” and “height account,” causing the machine eventually to run down.

These insights can be distilled into the following basic principle of physics:

the law of conservation of energy

It is possible to give a numerical rating, called energy, to the state of a physical system. The total energy is found by adding up contributions from characteristics of the system such as motion of objects in it, heating of the objects, and the relative positions of objects that interact via forces. The total energy of a closed system always remains constant. Energy cannot be created or destroyed, but only transferred from one system to another.

The moon rock story violates conservation of energy because the rock-cylinder and the ball together constitute a closed system. Once the ball has made one revolution around the cylinder, its position relative to the cylinder is exactly the same as before, so the numerical energy rating associated with its position is the same as before. Since the total amount of energy must remain constant, it is impossible for the ball to have a greater speed after one revolution. If it had picked up speed, it would have more energy associated with motion, the same amount of energy associated with position, and a little more energy associated with heating through friction. There cannot be a net increase in energy.

Example 2: Converting one form of energy to another

Dropping a rock: The rock loses energy because of its changing position with respect to the earth. Nearly all that energy is transformed into energy of motion, except for a small amount lost to heat created by air friction.

Sliding in to home base: The runner's energy of motion is nearly all converted into heat via friction with the ground.

Accelerating a car: The gasoline has energy stored in it, which is released as heat by burning it inside the engine. Perhaps 10% of this heat energy is converted into the car's energy of motion. The rest remains in the form of heat, which is carried away by the exhaust.

Cruising in a car: As you cruise at constant speed in your car, all the energy of the burning gas is being converted into heat. The tires and engine get hot, and heat is also dissipated into the air through the radiator and the exhaust.

Stepping on the brakes: All the energy of the car's motion is converted into heat in the brake shoes.

Example 3: Stevin's machine
The Dutch mathematician and engineer Simon Stevin proposed the imaginary machine shown in figure e, which he had inscribed on his tombstone. This is an interesting example, because it shows a link between the force concept used earlier in this course, and the energy concept being developed now.

The point of the imaginary machine is to show the mechanical advantage of an inclined plane. In this example, the triangle has the proportions 3-4-5, but the argument works for any right triangle. We imagine that the chain of balls slides without friction, so that no energy is ever converted into heat. If we were to slide the chain clockwise by one step, then each ball would take the place of the one in front of it, and the over all configuration would be exactly the same. Since energy is something that only depends on the state of the system, the energy would have to be the same. Similarly for a counterclockwise rotation, no energy of position would be released by gravity. This means that if we place the chain on the triangle, and release it at rest, it can't start moving, because there would be no way for it to convert energy of position into energy of motion. Thus the chain must be perfectly balanced. Now by symmetry, the arc of the chain hanging underneath the triangle has equal tension at both ends, so removing this arc wouldn't affect the balance of the rest of the chain. This means that a weight of three units hanging vertically balances a weight of five units hanging diagonally along the hypotenuse.

The mechanical advantage of the inclined plane is therefore \(5/3\), which is exactly the same as the result, \(1/\sin\theta\), that we got on p. 217 by analyzing force vectors. What this shows is that Newton's laws and conservation laws are not logically separate, but rather are very closely related descriptions of nature. In the cases where Newton's laws are true, they give the same answers as the conservation laws. This is an example of a more general idea, called the correspondence principle, about how science progresses over time. When a newer, more general theory is proposed to replace an older theory, the new theory must agree with the old one in the realm of applicability of the old theory, since the old theory only became accepted as a valid theory by being verified experimentally in a variety of experiments. In other words, the new theory must be backward-compatible with the old one. Even though conservation laws can prove things that Newton's laws can't (that perpetual motion is impossible, for example), they aren't going to disprove Newton's laws when applied to mechanical systems where we already knew Newton's laws were valid.

Discussion Question

Hydroelectric power (water flowing over a dam to spin turbines) appears to be completely free. Does this violate conservation of energy? If not, then what is the ultimate source of the electrical energy produced by a hydroelectric plant?

How does the proof in example 3 fail if the assumption of a frictionless surface doesn't hold?

11.3 A numerical scale of energy

high-road-low-road

f / Example 6.

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 \(1°\text{C}\). 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.1

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

\(\triangleright\) 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 \(10°\text{C}\ \) to 18°\textup{C}?

\(\triangleright\) Converting gallons to \(\text{cm}^3\) gives

\[\begin{equation*} 26000\ \text{gallons} \times \frac{3780\ \text{cm}^3}{1\ \text{gallon}} = 9.8\times10^7\ \text{cm}^3 . \end{equation*}\]

Water has a density of 1 gram per cubic centimeter, so the mass of the water is \(9.8\times10^7\) g. One joule is sufficient to heat 0.24 g by 1°\textup{C}, so the energy needed to heat the swimming pool is

\[\begin{align*} 1\ \text{J} \times \frac{9.8\times10^7\ \text{g}}{0.24\ \text{g}} \times \frac{8°\text{C}}{1°\text{C}} &= 3.3\times10^9\ \text{J} \\ &= 3.3\times10^3\ \text{MJ} . \end{align*}\]

The cost of the electricity is (\(3.3\times10^3\) MJ)($0.039/MJ)=$130.

Example 5: Irish coffee

\(\triangleright\) You make a cup of Irish coffee out of 300 g of coffee at \(100°\text{C}\ \) and 30 g of pure ethyl alcohol at 20°\textup{C}. One Joule is enough energy to produce a change of 1°\textup{C}\ in 0.42 g of ethyl alcohol (i.e., alcohol is easier to heat than water). What temperature is the final mixture?

\(\triangleright\) 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

\[\begin{align*} \text{total initial energy} &= \text{total final energy} \\ E_{ci}+E_{ai} &= E_{cf}+E_{af} . \end{align*}\]

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:

\[\begin{equation*} E_{af}-E_{ai} = E_{ci}-E_{cf} . \end{equation*}\]

Using the given ratios of temperature change to energy change, we have

\[\begin{align*} E_{ci}-E_{cf} &= (T_{ci}-T_{cf})(m_c)/(0.24\ \text{g}) \\ E_{af}-E_{ai} &= (T_{af}-T_{ai})(m_a)/(0.42\ \text{g}) \end{align*}\]

Setting these two quantities to be equal, we have

\[\begin{equation*} (T_{af}-T_{ai})(m_a)/(0.42\ \text{g}) = (T_{ci}-T_{cf})(m_c)/(0.24\ \text{g}) . \end{equation*}\]

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,

\[\begin{equation*} (T_f-T_{ai})(m_a)/(0.42\ \text{g}) = (T_{ci}-T_f)(m_c)/(0.24\ \text{g}) . \end{equation*}\]

Solving for \(T_f\) gives

\[\begin{align*} T_f &= \frac{T_{ci}\frac{m_c}{0.24}+T_{ai}\frac{m_a}{0.42}}{\frac{m_c}{0.24}+\frac{m_a}{0.42}}\\ &= 96°\text{C} . \end{align*}\]

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.
\(\triangleright\) Figure f shows two ramps which two balls will roll down. Compare their final speeds, when they reach point B. Assume friction is negligible.

\(\triangleright\) 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, \(1.1\times10^{-10}\) 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 \(1.2\times10^{-10}\) 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 \(1.3\times10^{15}\) 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

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.

11.4 Kinetic energy

dq-pendulum-and-peg

Discussion question B

The technical term for the energy associated with motion is kinetic energy, from the Greek word for motion. (The root is the same as the root of the word “cinema” for a motion picture, and in French the term for kinetic energy is “énergie cinétique.”) To find how much kinetic energy is possessed by a given moving object, we must convert all its kinetic energy into heat energy, which we have chosen as the standard reference type of energy. We could do this, for example, by firing projectiles into a tank of water and measuring the increase in temperature of the water as a function of the projectile's mass and velocity. Consider the following data from a series of three such experiments:




m (kg)v (m/s)energy (J)



1.00 1.00 0.50



1.00 2.00 2.00



2.00 1.00 1.00



Comparing the first experiment with the second, we see that doubling the object's velocity doesn't just double its energy, it quadruples it. If we compare the first and third lines, however, we find that doubling the mass only doubles the energy. This suggests that kinetic energy is proportional to mass and to the square of velocity, \(KE\propto mv^2\), and further experiments of this type would indeed establish such a general rule. The proportionality factor equals 0.5 because of the design of the metric system, so the kinetic energy of a moving object is given by

\[\begin{equation*} KE = \frac{1}{2}mv^2 . \end{equation*}\]

The metric system is based on the meter, kilogram, and second, with other units being derived from those. Comparing the units on the left and right sides of the equation shows that the joule can be reexpressed in terms of the basic units as \(\text{kg}\!\cdot\!\text{m}^2/\text{s}^2\).

Students are often mystified by the occurrence of the factor of 1/2, but it is less obscure than it looks. The metric system was designed so that some of the equations relating to energy would come out looking simple, at the expense of some others, which had to have inconvenient conversion factors in front. If we were using the old British Engineering System of units in this course, then we'd have the British Thermal Unit (BTU) as our unit of energy. In that system, the equation you'd learn for kinetic energy would have an inconvenient proportionality constant, \(KE=\left(1.29\times10^{-3}\right)mv^2\), with \(KE\) measured in units of BTUs, \(v\) measured in feet per second, and so on. At the expense of this inconvenient equation for kinetic energy, the designers of the British Engineering System got a simple rule for calculating the energy required to heat water: one BTU per degree Fahrenheit per pound. The inventor of kinetic energy, Thomas Young, actually defined it as \(KE=mv^2\), which meant that all his other equations had to be different from ours by a factor of two. All these systems of units work just fine as long as they are not combined with one another in an inconsistent way.

Example 7: Energy released by a comet impact

\(\triangleright\) Comet Shoemaker-Levy, which struck the planet Jupiter in 1994, had a mass of roughly \(4\times10^{13}\) kg, and was moving at a speed of 60 km/s. Compare the kinetic energy released in the impact to the total energy in the world's nuclear arsenals, which is \(2\times10^{19}\) J. Assume for the sake of simplicity that Jupiter was at rest.

\(\triangleright\) Since we assume Jupiter was at rest, we can imagine that the comet stopped completely on impact, and 100% of its kinetic energy was converted to heat and sound. We first convert the speed to mks units, \(v=6\times10^4\) m/s, and then plug in to the equation to find that the comet's kinetic energy was roughly \(7\times10^{22}\) J, or about 3000 times the energy in the world's nuclear arsenals.

Is there any way to derive the equation \(KE=(1/2)mv^2\) mathematically from first principles? No, it is purely empirical. The factor of 1/2 in front is definitely not derivable, since it is different in different systems of units. The proportionality to \(v^2\) is not even quite correct; experiments have shown deviations from the \(v^2\) rule at high speeds, an effect that is related to Einstein's theory of relativity. Only the proportionality to \(m\) is inevitable. The whole energy concept is based on the idea that we add up energy contributions from all the objects within a system. Based on this philosophy, it is logically necessary that a 2-kg object moving at 1 m/s have the same kinetic energy as two 1-kg objects moving side-by-side at the same speed.

Energy and relative motion

Although I mentioned Einstein's theory of relativity above, it's more relevant right now to consider how conservation of energy relates to the simpler Galilean idea, which we've already studied, that motion is relative. Galileo's Aristotelian enemies (and it is no exaggeration to call them enemies!) would probably have objected to conservation of energy. After all, the Galilean idea that an object in motion will continue in motion indefinitely in the absence of a force is not so different from the idea that an object's kinetic energy stays the same unless there is a mechanism like frictional heating for converting that energy into some other form.

More subtly, however, it's not immediately obvious that what we've learned so far about energy is strictly mathematically consistent with the principle that motion is relative. Suppose we verify that a certain process, say the collision of two pool balls, conserves energy as measured in a certain frame of reference: the sum of the balls' kinetic energies before the collision is equal to their sum after the collision. (In reality we'd need to add in other forms of energy, like heat and sound, that are liberated by the collision, but let's keep it simple.) But what if we were to measure everything in a frame of reference that was in a different state of motion? A particular pool ball might have less kinetic energy in this new frame; for example, if the new frame of reference was moving right along with it, its kinetic energy in that frame would be zero. On the other hand, some other balls might have a greater kinetic energy in the new frame. It's not immediately obvious that the total energy before the collision will still equal the total energy after the collision. After all, the equation for kinetic energy is fairly complicated, since it involves the square of the velocity, so it would be surprising if everything still worked out in the new frame of reference. It does still work out. Homework problem 13 in this chapter gives a simple numerical example, and the general proof is taken up in problem 15 on p. 376 (with the solution given in the back of the book).

Discussion Questions

Suppose that, like Young or Einstein, you were trying out different equations for kinetic energy to see if they agreed with the experimental data. Based on the meaning of positive and negative signs of velocity, why would you suspect that a proportionality to mv would be less likely than \(mv^2?\)

The figure shows a pendulum that is released at A and caught by a peg as it passes through the vertical, B. To what height will the bob rise on the right?

11.5 Power

A car may have plenty of energy in its gas tank, but still may not be able to increase its kinetic energy rapidly. A Porsche doesn't necessarily have more energy in its gas tank than a Hyundai, it is just able to transfer it more quickly. The rate of transferring energy from one form to another is called \(power\). The definition can be written as an equation,

\[\begin{equation*} P = \frac{\Delta E}{\Delta t} , \end{equation*}\]

where the use of the delta notation in the symbol \(\Delta E\) has the usual interpretation: the final amount of energy in a certain form minus the initial amount that was present in that form. Power has units of J/s, which are abbreviated as watts, W (rhymes with “lots”).

If the rate of energy transfer is not constant, the power at any instant can be defined as the slope of the tangent line on a graph of \(E\) versus \(t\). Likewise \(\Delta E\) can be extracted from the area under the \(P\)-versus-\(t\) curve.

Example 8: Converting kilowatt-hours to joules

\(\triangleright\) The electric company bills you for energy in units of kilowatt-hours (kilowatts multiplied by hours) rather than in SI units of joules. How many joules is a kilowatt-hour?

\(\triangleright\) 1 kilowatt-hour = (1 kW)(1 hour) = (1000 J/s)(3600 s) = 3.6 MJ.

Example 9: Human wattage

\(\triangleright\) A typical person consumes 2000 kcal of food in a day, and converts nearly all of that directly to heat. Compare the person's heat output to the rate of energy consumption of a 100-watt lightbulb.

\(\triangleright\) Looking up the conversion factor from calories to joules, we find

\[\begin{equation*} \Delta E=2000\ \text{kcal}\times\frac{1000\ \text{cal}}{1\ \text{kcal}}\times\frac{4.18\ \text{J}}{1\ \text{cal}}=8\times10^6\ \text{J} \end{equation*}\]

for our daily energy consumption. Converting the time interval likewise into mks,

\[\begin{equation*} \Delta t=1\ \text{day}\times\frac{24\ \text{hours}}{1\ \text{day}}\times\frac{60\ \text{min}}{1\ \text{hour}}\times\frac{60\ \text{s}}{1\ \text{min}}=9\times10^4\ \text{s} . \end{equation*}\]

Dividing, we find that our power dissipated as heat is 90 J/s = 90 W, about the same as a lightbulb.

It is easy to confuse the concepts of force, energy, and power, especially since they are synonyms in ordinary speech. The table on the following page may help to clear this up:

\begin{minipagefullpagewidth}





force

energy

power





conceptual definition & A force is an interaction between two objects that causes a push or a pull. A force can be defined as anything that is capable of changing an object's state of motion. & Heating an object, making it move faster, or increasing its distance from another object that is attracting it are all examples of things that would require fuel or physical effort. All these things can be quantified using a single scale of measurement, and we describe them all as forms of energy. & Power is the rate at which energy is transformed from one form to another or transferred from one object to another.
\hline

operational definition & A spring scale can be used to measure force. & If we define a unit of energy as the amount required to heat a certain amount of water by a \(1°\text{C}\), then we can measure any other quantity of energy by transferring it into heat in water and measuring the temperature increase. & Measure the change in the amount of some form of energy possessed by an object, and divide by the amount of time required for the change to occur.
\hline

scalar or
[4] vector?
& vector --- has a direction in space which is the direction in which it pulls or pushes & scalar --- has no direction in space & scalar --- has no direction in space
\hline

unit & newtons (N) & joules (J) & watts (W) = joules/s
\hline

Can it run out? Does it cost money? & No. I don't have to pay a monthly bill for the meganewtons of force required to hold up my house. & Yes. We pay money for gasoline, electrical energy, batteries, etc., because they contain energy. & More power means you are paying money at a higher rate. A 100-W lightbulb costs a certain number of cents per hour.
\hline

Can it be a property of an object? & No. A force is a relationship between two interacting objects. A home-run baseball doesn't “have” force. & Yes. What a home-run baseball has is kinetic energy, not force. & Not really. A 100-W lightbulb doesn't “have” 100 W. 100 J/s is the rate at which it converts electrical energy into light.
\hline

\end{minipagefullpagewidth}

Summary

Vocabulary

energy — A numerical scale used to measure the heat, motion, or other properties that would require fuel or physical effort to put into an object; a scalar quantity with units of joules (J).

power — The rate of transferring energy; a scalar quantity with units of watts (W).

kinetic energy — The energy an object possesses because of its motion.

heat — A form of energy that relates to temperature. Heat is different from temperature because an object with twice as much mass requires twice as much heat to increase its temperature by the same amount. Heat is measured in joules, temperature in degrees. (In standard terminology, there is another, finer distinction between heat and thermal energy, which is discussed below. In this book, I informally refer to both as heat.)

temperature — What a thermometer measures. Objects left in contact with each other tend to reach the same temperature. Cf. heat. As discussed in more detail in chapter 2, temperature is essentially a measure of the average kinetic energy per molecule.

Notation

\(E\) — energy

J — joules, the SI unit of energy

\(KE\) — kinetic energy

\(P\) — power

W — watts, the SI unit of power; equivalent to J/s

Other Notation

\(Q\) or \(\Delta Q\) — the amount of heat transferred into or out of an object

\(K\) or \(T\) — alternative symbols for kinetic energy, used in the scientific literature and in most advanced textbooks

thermal energy — Careful writers make a distinction between heat and thermal energy, but the distinction is often ignored in casual speech, even among physicists. Properly, thermal energy is used to mean the total amount of energy possessed by an object, while heat indicates the amount of thermal energy transferred in or out. The term heat is used in this book to include both meanings.

Summary

{}

Heating an object, making it move faster, or increasing its distance from another object that is attracting it are all examples of things that would require fuel or physical effort. All these things can be quantified using a single scale of measurement, and we describe them all as forms of energy. The SI unit of energy is the Joule. The reason why energy is a useful and important quantity is that it is always conserved. That is, it cannot be created or destroyed but only transferred between objects or changed from one form to another. Conservation of energy is the most important and broadly applicable of all the laws of physics, more fundamental and general even than Newton's laws of motion.

Heating an object requires a certain amount of energy per degree of temperature and per unit mass, which depends on the substance of which the object consists. Heat and temperature are completely different things. Heat is a form of energy, and its SI unit is the joule (J). Temperature is not a measure of energy. Heating twice as much of something requires twice as much heat, but double the amount of a substance does not have double the temperature.

The energy that an object possesses because of its motion is called kinetic energy. Kinetic energy is related to the mass of the object and the magnitude of its velocity vector by the equation

\[\begin{equation*} KE = \frac{1}{2}mv^2 . \end{equation*}\]

Power is the rate at which energy is transformed from one form to another or transferred from one object to another,

\[\begin{equation*} P = \frac{\Delta E}{\Delta t} . \shoveright{\text{[only for constant power]}} \end{equation*}\]

The SI unit of power is the watt (W).

Homework Problems

\begin{homeworkforcelabel}{sweatold}{1}{}{1} This problem is now problem 14 in chapter 12, on page 315. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{negativeke}{1}{}{2}Can kinetic energy ever be less than zero? Explain. [Based on a problem by Serway and Faughn.] \end{homeworkforcelabel}

\begin{homeworkforcelabel}{sprinter}{1}{}{3}Estimate the kinetic energy of an Olympic sprinter. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{crash}{1}{}{4}(answer check available at lightandmatter.com) You are driving your car, and you hit a brick wall head on, at full speed. The car has a mass of 1500 kg. The kinetic energy released is a measure of how much destruction will be done to the car and to your body. Calculate the energy released if you are traveling at (a) 40 mi/hr, and again (b) if you're going 80 mi/hr. What is counterintuitive about this, and what implication does this have for driving at high speeds? \end{homeworkforcelabel}

\begin{homeworkforcelabel}{astronaut}{1}{}{5} A closed system can be a bad thing --- for an astronaut sealed inside a space suit, getting rid of body heat can be difficult. Suppose a 60-kg astronaut is performing vigorous physical activity, expending 200 W of power. If none of the heat can escape from her space suit, how long will it take before her body temperature rises by 6°{}C (11°{}F), an amount sufficient to kill her? Assume that the amount of heat required to raise her body temperature by 1°{}C is the same as it would be for an equal mass of water. Express your answer in units of minutes. (answer check available at lightandmatter.com) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{melt-antarctic}{1}{}{6}All stars, including our sun, show variations in their light output to some degree. Some stars vary their brightness by a factor of two or even more, but our sun has remained relatively steady during the hundred years or so that accurate data have been collected. Nevertheless, it is possible that climate variations such as ice ages are related to long-term irregularities in the sun's light output. If the sun was to increase its light output even slightly, it could melt enough Antarctic ice to flood all the world's coastal cities. The total sunlight that falls on Antarctica amounts to about \(1\times10^{16}\) watts. In the absence of natural or human-caused climate change, this heat input to the poles is balanced by the loss of heat via winds, ocean currents, and emission of infrared light, so that there is no net melting or freezing of ice at the poles from year to year. Suppose that the sun changes its light output by some small percentage, but there is no change in the rate of heat loss by the polar caps. Estimate the percentage by which the sun's light output would have to increase in order to melt enough ice to raise the level of the oceans by 10 meters over a period of 10 years. (This would be enough to flood New York, London, and many other cities.) Melting 1 kg of ice requires \(3\times10^3\) J. \end{homeworkforcelabel}

\begin{homeworkforcelabel}{bulletthroughbook}{1}{}{7} A bullet flies through the air, passes through a paperback book, and then continues to fly through the air beyond the book. When is there a force? When is there energy? (solution in the pdf version of the book) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{boatengine}{1}{}{8}Experiments show that the power consumed by a boat's engine is approximately proportional to third power of its speed. (We assume that it is moving at constant speed.) (a) When a boat is crusing at constant speed, what type of energy transformation do you think is being performed? (b) If you upgrade to a motor with double the power, by what factor is your boat's crusing speed increased? [Based on a problem by Arnold Arons.] (solution in the pdf version of the book) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{aronskeratio}{1}{}{9}Object A has a kinetic energy of 13.4 J. Object B has a mass that is greater by a factor of 3.77, but is moving more slowly by a factor of 2.34. What is object B's kinetic energy? [Based on a problem by Arnold Arons.] (solution in the pdf version of the book) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{earthmoonke}{1}{}{10}The moon doesn't really just orbit the Earth. By Newton's third law, the moon's gravitational force on the earth is the same as the earth's force on the moon, and the earth must respond to the moon's force by accelerating. If we consider the earth and moon in isolation and ignore outside forces, then Newton's first law says their common center of mass doesn't accelerate, i.e., the earth wobbles around the center of mass of the earth-moon system once per month, and the moon also orbits around this point. The moon's mass is 81 times smaller than the earth's. Compare the kinetic energies of the earth and moon. (We know that the center of mass is a kind of balance point, so it must be closer to the earth than to the moon. In fact, the distance from the earth to the center of mass is 1/81 of the distance from the moon to the center of mass, which makes sense intuitively, and can be proved rigorously using the equation on page 360.) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{microwave-waste}{1}{}{11}My 1.25 kW microwave oven takes 126 seconds to bring 250 g of water from room temperature to a boil. What percentage of the power is being wasted? Where might the rest of the energy be going? (solution in the pdf version of the book) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{colliding-balls}{1}{}{12} The multiflash photograph shows a collision between two pool balls. The ball that was initially at rest shows up as a dark image in its initial position, because its image was exposed several times before it was struck and began moving. By making measurements on the figure, determine numerically whether or not energy appears to have been conserved in the collision. What systematic effects would limit the accuracy of your test? [From an example in PSSC Physics.] \end{homeworkforcelabel}

hw-colliding-balls

g / Problem 12.

\begin{homeworkforcelabel}{balls}{1}{}{13}This problem is a numerical example of the imaginary experiment discussed on p. 292 regarding the relationship between energy and relative motion. Let's say that the pool balls both have masses of 1.00 kg. Suppose that in the frame of reference of the pool table, the cue ball moves at a speed of 1.00 m/s toward the eight ball, which is initially at rest. The collision is head-on, and as you can verify for yourself the next time you're playing pool, the result of such a collision is that the incoming ball stops dead and the ball that was struck takes off with the same speed originally possessed by the incoming ball. (This is actually a bit of an idealization. To keep things simple, we're ignoring the spin of the balls, and we assume that no energy is liberated by the collision as heat or sound.) (a) Calculate the total initial kinetic energy and the total final kinetic energy, and verify that they are equal. (b) Now carry out the whole calculation again in the frame of reference that is moving in the same direction that the cue ball was initially moving, but at a speed of 0.50 m/s. In this frame of reference, both balls have nonzero initial and final velocities, which are different from what they were in the table's frame. [See also problem 15 on p. 376.] \end{homeworkforcelabel}

\begin{homeworkforcelabel}{columbia}{1}{}{14}One theory about the destruction of the space shuttle Columbia in 2003 is that one of its wings had been damaged on liftoff by a chunk of foam insulation that fell off of one of its external fuel tanks. The New York Times reported on June 5, 2003, that NASA engineers had recreated the impact to see if it would damage a mock-up of the shuttle's wing. “Before last week's test, many engineers at NASA said they thought lightweight foam could not harm the seemingly tough composite panels, and privately predicted that the foam would bounce off harmlessly, like a Nerf ball.” In fact, the 1.7-pound piece of foam, moving at 531 miles per hour, did serious damage. A member of the board investigating the disaster said this demonstrated that “people's intuitive sense of physics is sometimes way off.” (a) Compute the kinetic energy of the foam, and (b) compare with the energy of a 170-pound boulder moving at 5.3 miles per hour (the speed it would have if you dropped it from about knee-level).(answer check available at lightandmatter.com)
(c) The boulder is a hundred times more massive, but its speed is a hundred times smaller, so what's counterintuitive about your results? \end{homeworkforcelabel}

../../../share/mechanics/figs/hydraulic-ram \begin{homeworkforcelabel}{hydraulic-ram}{1}{}{15}The figure above is from a classic 1920 physics textbook by Millikan and Gale. It represents a method for raising the water from the pond up to the water tower, at a higher level, without using a pump. Water is allowed into the drive pipe, and once it is flowing fast enough, it forces the valve at the bottom closed. Explain how this works in terms of conservation of mass and energy. (Cf. example 1 on page 283.) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{melt-and-boil}{1}{}{16}The following table gives the amount of energy required in order to heat, melt, or boil a gram of water.

heat 1 g of ice by1degunittextupC 2.05 J
melt 1 g of ice 333 J
heat 1 g of liquid by1degunittextupC4.19 J
boil 1 g of water 2500 J
heat 1 g of steam by1degunittextupC2.01 J


(a) How much energy is required in order to convert 1.00 g of ice at -20 \(°\text{C}\) into steam at 137 \(°\text{C}\)? (answer check available at lightandmatter.com)
(b) What is the minimum amount of hot water that could melt 1.00 g of ice? (answer check available at lightandmatter.com) \end{homeworkforcelabel}

\begin{homeworkforcelabel}{fly-wing-ke}{1}{}{17} Estimate the kinetic energy of a buzzing fly's wing. \end{homeworkforcelabel}

(c) 1998-2013 Benjamin Crowell, licensed under the Creative Commons Attribution-ShareAlike license. Photo credits are given at the end of the Adobe Acrobat version.

Footnotes
[1] In standard, formal terminology, there is another, finer distinction. The word “heat” is used only to indicate an amount of energy that is transferred, whereas “thermal energy” indicates an amount of energy contained in an object. I'm informal on this point, and refer to both as heat, but you should be aware of the distinction.