energy conservation experiments physics

4 Conservation of Energy

4–1 What is energy?

In this chapter, we begin our more detailed study of the different aspects of physics, having finished our description of things in general. To illustrate the ideas and the kind of reasoning that might be used in theoretical physics, we shall now examine one of the most basic laws of physics, the conservation of energy.

There is a fact, or if you wish, a law , governing all natural phenomena that are known to date. There is no known exception to this law—it is exact so far as we know. The law is called the conservation of energy . It states that there is a certain quantity, which we call energy, that does not change in the manifold changes which nature undergoes. That is a most abstract idea, because it is a mathematical principle; it says that there is a numerical quantity which does not change when something happens. It is not a description of a mechanism, or anything concrete; it is just a strange fact that we can calculate some number and when we finish watching nature go through her tricks and calculate the number again, it is the same. (Something like the bishop on a red square, and after a number of moves—details unknown—it is still on some red square. It is a law of this nature.) Since it is an abstract idea, we shall illustrate the meaning of it by an analogy.

Imagine a child, perhaps “Dennis the Menace,” who has blocks which are absolutely indestructible, and cannot be divided into pieces. Each is the same as the other. Let us suppose that he has $28$ blocks. His mother puts him with his $28$ blocks into a room at the beginning of the day. At the end of the day, being curious, she counts the blocks very carefully, and discovers a phenomenal law—no matter what he does with the blocks, there are always $28$ remaining! This continues for a number of days, until one day there are only $27$ blocks, but a little investigating shows that there is one under the rug—she must look everywhere to be sure that the number of blocks has not changed. One day, however, the number appears to change—there are only $26$ blocks. Careful investigation indicates that the window was open, and upon looking outside, the other two blocks are found. Another day, careful count indicates that there are $30$ blocks! This causes considerable consternation, until it is realized that Bruce came to visit, bringing his blocks with him, and he left a few at Dennis’ house. After she has disposed of the extra blocks, she closes the window, does not let Bruce in, and then everything is going along all right, until one time she counts and finds only $25$ blocks. However, there is a box in the room, a toy box, and the mother goes to open the toy box, but the boy says “No, do not open my toy box,” and screams. Mother is not allowed to open the toy box. Being extremely curious, and somewhat ingenious, she invents a scheme! She knows that a block weighs three ounces, so she weighs the box at a time when she sees $28$ blocks, and it weighs $16$ ounces. The next time she wishes to check, she weighs the box again, subtracts sixteen ounces and divides by three. She discovers the following: \begin{equation} \label{Eq:I:4:1} \begin{pmatrix} \text{number of}\\ \text{blocks seen} \end{pmatrix}+ \frac{(\text{weight of box})-\text{$16$ ounces}}{\text{$3$ ounces}}= \text{constant}. \end{equation} \begin{align} \begin{pmatrix} \text{number of}\\ \text{blocks seen} \end{pmatrix}&+ \frac{(\text{weight of box})-\text{$16$ ounces}}{\text{$3$ ounces}}\notag\\[1ex] \label{Eq:I:4:1} &=\text{constant}. \end{align} There then appear to be some new deviations, but careful study indicates that the dirty water in the bathtub is changing its level. The child is throwing blocks into the water, and she cannot see them because it is so dirty, but she can find out how many blocks are in the water by adding another term to her formula. Since the original height of the water was $6$ inches and each block raises the water a quarter of an inch, this new formula would be: \begin{align} \begin{pmatrix} \text{number of}\\ \text{blocks seen} \end{pmatrix}&+ \frac{(\text{weight of box})-\text{$16$ ounces}} {\text{$3$ ounces}}\notag\\[1ex] \label{Eq:I:4:2} &+\frac{(\text{height of water})-\text{$6$ inches}} {\text{$1/4$ inch}}= \text{constant}. \end{align} \begin{align} \begin{pmatrix} \text{number of}\\ \text{blocks seen} \end{pmatrix}&+ \frac{(\text{weight of box})-\text{$16$ ounces}} {\text{$3$ ounces}}\notag\\[1ex] &+\frac{(\text{height of water})-\text{$6$ inches}} {\text{$1/4$ inch}}\notag\\[2ex] \label{Eq:I:4:2} &=\text{constant}. \end{align} In the gradual increase in the complexity of her world, she finds a whole series of terms representing ways of calculating how many blocks are in places where she is not allowed to look. As a result, she finds a complex formula, a quantity which has to be computed , which always stays the same in her situation.

What is the analogy of this to the conservation of energy? The most remarkable aspect that must be abstracted from this picture is that there are no blocks . Take away the first terms in ( 4.1 ) and ( 4.2 ) and we find ourselves calculating more or less abstract things. The analogy has the following points. First, when we are calculating the energy, sometimes some of it leaves the system and goes away, or sometimes some comes in. In order to verify the conservation of energy, we must be careful that we have not put any in or taken any out. Second, the energy has a large number of different forms , and there is a formula for each one. These are: gravitational energy, kinetic energy, heat energy, elastic energy, electrical energy, chemical energy, radiant energy, nuclear energy, mass energy. If we total up the formulas for each of these contributions, it will not change except for energy going in and out.

It is important to realize that in physics today, we have no knowledge of what energy is . We do not have a picture that energy comes in little blobs of a definite amount. It is not that way. However, there are formulas for calculating some numerical quantity, and when we add it all together it gives “$28$”—always the same number. It is an abstract thing in that it does not tell us the mechanism or the reasons for the various formulas.

4–2 Gravitational potential energy

Conservation of energy can be understood only if we have the formula for all of its forms. I wish to discuss the formula for gravitational energy near the surface of the Earth, and I wish to derive this formula in a way which has nothing to do with history but is simply a line of reasoning invented for this particular lecture to give you an illustration of the remarkable fact that a great deal about nature can be extracted from a few facts and close reasoning. It is an illustration of the kind of work theoretical physicists become involved in. It is patterned after a most excellent argument by Mr. Carnot on the efficiency of steam engines. 1

Consider weight-lifting machines—machines which have the property that they lift one weight by lowering another. Let us also make a hypothesis: that there is no such thing as perpetual motion with these weight-lifting machines. (In fact, that there is no perpetual motion at all is a general statement of the law of conservation of energy.) We must be careful to define perpetual motion. First, let us do it for weight-lifting machines. If, when we have lifted and lowered a lot of weights and restored the machine to the original condition, we find that the net result is to have lifted a weight , then we have a perpetual motion machine because we can use that lifted weight to run something else. That is, provided the machine which lifted the weight is brought back to its exact original condition , and furthermore that it is completely self-contained —that it has not received the energy to lift that weight from some external source—like Bruce’s blocks.

A very simple weight-lifting machine is shown in Fig.  4–1 . This machine lifts weights three units “strong.” We place three units on one balance pan, and one unit on the other. However, in order to get it actually to work, we must lift a little weight off the left pan. On the other hand, we could lift a one-unit weight by lowering the three-unit weight, if we cheat a little by lifting a little weight off the other pan. Of course, we realize that with any actual lifting machine, we must add a little extra to get it to run. This we disregard, temporarily . Ideal machines, although they do not exist, do not require anything extra. A machine that we actually use can be, in a sense, almost reversible: that is, if it will lift the weight of three by lowering a weight of one, then it will also lift nearly the weight of one the same amount by lowering the weight of three.

We imagine that there are two classes of machines, those that are not reversible, which includes all real machines, and those that are reversible, which of course are actually not attainable no matter how careful we may be in our design of bearings, levers, etc. We suppose, however, that there is such a thing—a reversible machine—which lowers one unit of weight (a pound or any other unit) by one unit of distance, and at the same time lifts a three-unit weight. Call this reversible machine, Machine $A$. Suppose this particular reversible machine lifts the three-unit weight a distance $X$. Then suppose we have another machine, Machine $B$, which is not necessarily reversible, which also lowers a unit weight a unit distance, but which lifts three units a distance $Y$. We can now prove that $Y$ is not higher than $X$; that is, it is impossible to build a machine that will lift a weight any higher than it will be lifted by a reversible machine. Let us see why. Let us suppose that $Y$ were higher than $X$. We take a one-unit weight and lower it one unit height with Machine $B$, and that lifts the three-unit weight up a distance $Y$. Then we could lower the weight from $Y$ to $X$, obtaining free power , and use the reversible Machine $A$, running backwards, to lower the three-unit weight a distance $X$ and lift the one-unit weight by one unit height. This will put the one-unit weight back where it was before, and leave both machines ready to be used again! We would therefore have perpetual motion if $Y$ were higher than $X$, which we assumed was impossible. With those assumptions, we thus deduce that $Y$ is not higher than  $X$, so that of all machines that can be designed, the reversible machine is the best.

We can also see that all reversible machines must lift to exactly the same height . Suppose that $B$ were really reversible also. The argument that $Y$ is not higher than $X$ is, of course, just as good as it was before, but we can also make our argument the other way around, using the machines in the opposite order, and prove that $X$ is not higher than  $Y$. This, then, is a very remarkable observation because it permits us to analyze the height to which different machines are going to lift something without looking at the interior mechanism . We know at once that if somebody makes an enormously elaborate series of levers that lift three units a certain distance by lowering one unit by one unit distance, and we compare it with a simple lever which does the same thing and is fundamentally reversible, his machine will lift it no higher, but perhaps less high. If his machine is reversible, we also know exactly how high it will lift. To summarize: every reversible machine, no matter how it operates, which drops one pound one foot and lifts a three-pound weight always lifts it the same distance, $X$. This is clearly a universal law of great utility. The next question is, of course, what is $X$?

Suppose we have a reversible machine which is going to lift this distance $X$, three for one. We set up three balls in a rack which does not move, as shown in Fig.  4–2 . One ball is held on a stage at a distance one foot above the ground. The machine can lift three balls, lowering one by a distance $1$. Now, we have arranged that the platform which holds three balls has a floor and two shelves, exactly spaced at distance $X$, and further, that the rack which holds the balls is spaced at distance $X$, (a). First we roll the balls horizontally from the rack to the shelves, (b), and we suppose that this takes no energy because we do not change the height. The reversible machine then operates: it lowers the single ball to the floor, and it lifts the rack a distance $X$, (c). Now we have ingeniously arranged the rack so that these balls are again even with the platforms. Thus we unload the balls onto the rack, (d); having unloaded the balls, we can restore the machine to its original condition. Now we have three balls on the upper three shelves and one at the bottom. But the strange thing is that, in a certain way of speaking, we have not lifted two of them at all because, after all, there were balls on shelves $2$ and $3$ before. The resulting effect has been to lift one ball a distance $3X$. Now, if $3X$ exceeds one foot, then we can lower the ball to return the machine to the initial condition, (f), and we can run the apparatus again. Therefore $3X$ cannot exceed one foot, for if $3X$ exceeds one foot we can make perpetual motion. Likewise, we can prove that one foot cannot exceed $3X$ , by making the whole machine run the opposite way, since it is a reversible machine. Therefore $3X$ is neither greater nor less than a foot , and we discover then, by argument alone, the law that $X=\tfrac{1}{3}$ foot. The generalization is clear: one pound falls a certain distance in operating a reversible machine; then the machine can lift $p$ pounds this distance divided by $p$. Another way of putting the result is that three pounds times the height lifted, which in our problem was $X$, is equal to one pound times the distance lowered, which is one foot in this case. If we take all the weights and multiply them by the heights at which they are now, above the floor, let the machine operate, and then multiply all the weights by all the heights again, there will be no change . (We have to generalize the example where we moved only one weight to the case where when we lower one we lift several different ones—but that is easy.)

We call the sum of the weights times the heights gravitational potential energy —the energy which an object has because of its relationship in space, relative to the earth. The formula for gravitational energy, then, so long as we are not too far from the earth (the force weakens as we go higher) is \begin{equation} \label{Eq:I:4:3} \begin{pmatrix} \text{gravitational}\\ \text{potential energy}\\ \text{for one object} \end{pmatrix}= (\text{weight})\times(\text{height}). \end{equation} It is a very beautiful line of reasoning. The only problem is that perhaps it is not true. (After all, nature does not have to go along with our reasoning.) For example, perhaps perpetual motion is, in fact, possible. Some of the assumptions may be wrong, or we may have made a mistake in reasoning, so it is always necessary to check. It turns out experimentally , in fact, to be true.

The general name of energy which has to do with location relative to something else is called potential energy. In this particular case, of course, we call it gravitational potential energy . If it is a question of electrical forces against which we are working, instead of gravitational forces, if we are “lifting” charges away from other charges with a lot of levers, then the energy content is called electrical potential energy . The general principle is that the change in the energy is the force times the distance that the force is pushed, and that this is a change in energy in general: \begin{equation} \label{Eq:I:4:4} \begin{pmatrix} \text{change in}\\ \text{energy} \end{pmatrix}= (\text{force})\times \begin{pmatrix} \text{distance force}\\ \text{acts through} \end{pmatrix}. \end{equation} We will return to many of these other kinds of energy as we continue the course.

The principle of the conservation of energy is very useful for deducing what will happen in a number of circumstances. In high school we learned a lot of laws about pulleys and levers used in different ways. We can now see that these “laws” are all the same thing , and that we did not have to memorize $75$ rules to figure it out. A simple example is a smooth inclined plane which is, happily, a three-four-five triangle (Fig.  4–3 ). We hang a one-pound weight on the inclined plane with a pulley, and on the other side of the pulley, a weight $W$. We want to know how heavy $W$ must be to balance the one pound on the plane. How can we figure that out? If we say it is just balanced, it is reversible and so can move up and down, and we can consider the following situation. In the initial circumstance, (a), the one pound weight is at the bottom and weight $W$ is at the top. When $W$ has slipped down in a reversible way, (b), we have a one-pound weight at the top and the weight $W$ the slant distance, or five feet, from the plane in which it was before. We lifted the one-pound weight only three feet and we lowered $W$ pounds by five feet. Therefore $W=\tfrac{3}{5}$ of a pound. Note that we deduced this from the conservation of energy , and not from force components. Cleverness, however, is relative. It can be deduced in a way which is even more brilliant, discovered by Stevinus and inscribed on his tombstone. 2 Figure  4–4 explains that it has to be $\tfrac{3}{5}$ of a pound, because the chain does not go around. It is evident that the lower part of the chain is balanced by itself, so that the pull of the five weights on one side must balance the pull of three weights on the other, or whatever the ratio of the legs. You see, by looking at this diagram, that $W$ must be $\tfrac{3}{5}$ of a pound. (If you get an epitaph like that on your gravestone, you are doing fine.)

Let us now illustrate the energy principle with a more complicated problem, the screw jack shown in Fig.  4–5 . A handle $20$ inches long is used to turn the screw, which has $10$ threads to the inch. We would like to know how much force would be needed at the handle to lift one ton ($2000$ pounds). If we want to lift the ton one inch, say, then we must turn the handle around ten times. When it goes around once it goes approximately $126$ inches. The handle must thus travel $1260$ inches, and if we used various pulleys, etc., we would be lifting our one ton with an unknown smaller weight $W$ applied to the end of the handle. So we find out that $W$ is about $1.6$ pounds. This is a result of the conservation of energy.

Take now the somewhat more complicated example shown in Fig.  4–6 . A rod or bar, $8$ feet long, is supported at one end. In the middle of the bar is a weight of $60$ pounds, and at a distance of two feet from the support there is a weight of $100$ pounds. How hard do we have to lift the end of the bar in order to keep it balanced, disregarding the weight of the bar? Suppose we put a pulley at one end and hang a weight on the pulley. How big would the weight $W$ have to be in order for it to balance? We imagine that the weight falls any arbitrary distance—to make it easy for ourselves suppose it goes down $4$ inches—how high would the two load weights rise? The center rises $2$ inches, and the point a quarter of the way from the fixed end lifts $1$ inch. Therefore, the principle that the sum of the heights times the weights does not change tells us that the weight $W$ times $4$ inches down, plus $60$ pounds times $2$ inches up, plus $100$ pounds times $1$ inch has to add up to nothing: \begin{equation} \label{Eq:I:4:5} -4W+(2)(60)+(1)(100)=0,\quad W=\text{$55$ lb}. \end{equation} \begin{equation} \begin{gathered} -4W+(2)(60)+(1)(100)=0,\\[.5ex] W=\text{$55$ lb}. \end{gathered} \label{Eq:I:4:5} \end{equation} Thus we must have a $55$-pound weight to balance the bar. In this way we can work out the laws of “balance”—the statics of complicated bridge arrangements, and so on. This approach is called the principle of virtual work , because in order to apply this argument we had to imagine that the structure moves a little—even though it is not really moving or even movable . We use the very small imagined motion to apply the principle of conservation of energy.

4–3 Kinetic energy

To illustrate another type of energy we consider a pendulum (Fig.  4–7 ). If we pull the mass aside and release it, it swings back and forth. In its motion, it loses height in going from either end to the center. Where does the potential energy go? Gravitational energy disappears when it is down at the bottom; nevertheless, it will climb up again. The gravitational energy must have gone into another form. Evidently it is by virtue of its motion that it is able to climb up again, so we have the conversion of gravitational energy into some other form when it reaches the bottom.

We must get a formula for the energy of motion. Now, recalling our arguments about reversible machines, we can easily see that in the motion at the bottom must be a quantity of energy which permits it to rise a certain height, and which has nothing to do with the machinery by which it comes up or the path by which it comes up. So we have an equivalence formula something like the one we wrote for the child’s blocks. We have another form to represent the energy. It is easy to say what it is. The kinetic energy at the bottom equals the weight times the height that it could go, corresponding to its velocity: $\text{K.E.}= WH$. What we need is the formula which tells us the height by some rule that has to do with the motion of objects. If we start something out with a certain velocity, say straight up, it will reach a certain height; we do not know what it is yet, but it depends on the velocity—there is a formula for that. Then to find the formula for kinetic energy for an object moving with velocity $V$, we must calculate the height that it could reach, and multiply by the weight. We shall soon find that we can write it this way: \begin{equation} \label{Eq:I:4:6} \text{K.E.}=WV^2/2g. \end{equation} Of course, the fact that motion has energy has nothing to do with the fact that we are in a gravitational field. It makes no difference where the motion came from. This is a general formula for various velocities. Both ( 4.3 ) and ( 4.6 ) are approximate formulas, the first because it is incorrect when the heights are great, i.e., when the heights are so high that gravity is weakening; the second, because of the relativistic correction at high speeds. However, when we do finally get the exact formula for the energy, then the law of conservation of energy is correct.

4–4 Other forms of energy

We can continue in this way to illustrate the existence of energy in other forms. First, consider elastic energy. If we pull down on a spring, we must do some work, for when we have it down, we can lift weights with it. Therefore in its stretched condition it has a possibility of doing some work. If we were to evaluate the sums of weights times heights, it would not check out—we must add something else to account for the fact that the spring is under tension. Elastic energy is the formula for a spring when it is stretched. How much energy is it? If we let go, the elastic energy, as the spring passes through the equilibrium point, is converted to kinetic energy and it goes back and forth between compressing or stretching the spring and kinetic energy of motion. (There is also some gravitational energy going in and out, but we can do this experiment “sideways” if we like.) It keeps going until the losses—Aha! We have cheated all the way through by putting on little weights to move things or saying that the machines are reversible, or that they go on forever, but we can see that things do stop, eventually. Where is the energy when the spring has finished moving up and down? This brings in another form of energy: heat energy .

Inside a spring or a lever there are crystals which are made up of lots of atoms, and with great care and delicacy in the arrangement of the parts one can try to adjust things so that as something rolls on something else, none of the atoms do any jiggling at all. But one must be very careful. Ordinarily when things roll, there is bumping and jiggling because of the irregularities of the material, and the atoms start to wiggle inside. So we lose track of that energy; we find the atoms are wiggling inside in a random and confused manner after the motion slows down. There is still kinetic energy, all right, but it is not associated with visible motion. What a dream! How do we know there is still kinetic energy? It turns out that with thermometers you can find out that, in fact, the spring or the lever is warmer , and that there is really an increase of kinetic energy by a definite amount. We call this form of energy heat energy , but we know that it is not really a new form, it is just kinetic energy—internal motion. (One of the difficulties with all these experiments with matter that we do on a large scale is that we cannot really demonstrate the conservation of energy and we cannot really make our reversible machines, because every time we move a large clump of stuff, the atoms do not remain absolutely undisturbed, and so a certain amount of random motion goes into the atomic system. We cannot see it, but we can measure it with thermometers, etc.)

There are many other forms of energy, and of course we cannot describe them in any more detail just now. There is electrical energy, which has to do with pushing and pulling by electric charges. There is radiant energy, the energy of light, which we know is a form of electrical energy because light can be represented as wigglings in the electromagnetic field. There is chemical energy, the energy which is released in chemical reactions. Actually, elastic energy is, to a certain extent, like chemical energy, because chemical energy is the energy of the attraction of the atoms, one for the other, and so is elastic energy. Our modern understanding is the following: chemical energy has two parts, kinetic energy of the electrons inside the atoms, so part of it is kinetic, and electrical energy of interaction of the electrons and the protons—the rest of it, therefore, is electrical. Next we come to nuclear energy, the energy which is involved with the arrangement of particles inside the nucleus, and we have formulas for that, but we do not have the fundamental laws. We know that it is not electrical, not gravitational, and not purely kinetic, but we do not know what it is. It seems to be an additional form of energy. Finally, associated with the relativity theory, there is a modification of the laws of kinetic energy, or whatever you wish to call it, so that kinetic energy is combined with another thing called mass energy . An object has energy from its sheer existence . If I have a positron and an electron, standing still doing nothing—never mind gravity, never mind anything—and they come together and disappear, radiant energy will be liberated, in a definite amount, and the amount can be calculated. All we need know is the mass of the object. It does not depend on what it is—we make two things disappear, and we get a certain amount of energy. The formula was first found by Einstein; it is $E=mc^2$.

It is obvious from our discussion that the law of conservation of energy is enormously useful in making analyses, as we have illustrated in a few examples without knowing all the formulas. If we had all the formulas for all kinds of energy, we could analyze how many processes should work without having to go into the details. Therefore conservation laws are very interesting. The question naturally arises as to what other conservation laws there are in physics. There are two other conservation laws which are analogous to the conservation of energy. One is called the conservation of linear momentum. The other is called the conservation of angular momentum. We will find out more about these later. In the last analysis, we do not understand the conservation laws deeply. We do not understand the conservation of energy. We do not understand energy as a certain number of little blobs. You may have heard that photons come out in blobs and that the energy of a photon is Planck’s constant times the frequency. That is true, but since the frequency of light can be anything, there is no law that says that energy has to be a certain definite amount. Unlike Dennis’ blocks, there can be any amount of energy, at least as presently understood. So we do not understand this energy as counting something at the moment, but just as a mathematical quantity, which is an abstract and rather peculiar circumstance. In quantum mechanics it turns out that the conservation of energy is very closely related to another important property of the world, things do not depend on the absolute time . We can set up an experiment at a given moment and try it out, and then do the same experiment at a later moment, and it will behave in exactly the same way. Whether this is strictly true or not, we do not know. If we assume that it is true, and add the principles of quantum mechanics, then we can deduce the principle of the conservation of energy. It is a rather subtle and interesting thing, and it is not easy to explain. The other conservation laws are also linked together. The conservation of momentum is associated in quantum mechanics with the proposition that it makes no difference where you do the experiment, the results will always be the same. As independence in space has to do with the conservation of momentum, independence of time has to do with the conservation of energy, and finally, if we turn our apparatus, this too makes no difference, and so the invariance of the world to angular orientation is related to the conservation of angular momentum . Besides these, there are three other conservation laws, that are exact so far as we can tell today, which are much simpler to understand because they are in the nature of counting blocks.

The first of the three is the conservation of charge , and that merely means that you count how many positive, minus how many negative electrical charges you have, and the number is never changed. You may get rid of a positive with a negative, but you do not create any net excess of positives over negatives. Two other laws are analogous to this one—one is called the conservation of baryons . There are a number of strange particles, a neutron and a proton are examples, which are called baryons. In any reaction whatever in nature, if we count how many baryons are coming into a process, the number of baryons 3 which come out will be exactly the same. There is another law, the conservation of leptons . We can say that the group of particles called leptons are: electron, muon, and neutrino. There is an antielectron which is a positron, that is, a $-1$ lepton. Counting the total number of leptons in a reaction reveals that the number in and out never changes, at least so far as we know at present.

These are the six conservation laws, three of them subtle, involving space and time, and three of them simple, in the sense of counting something.

With regard to the conservation of energy, we should note that available energy is another matter—there is a lot of jiggling around in the atoms of the water of the sea, because the sea has a certain temperature, but it is impossible to get them herded into a definite motion without taking energy from somewhere else. That is, although we know for a fact that energy is conserved, the energy available for human utility is not conserved so easily. The laws which govern how much energy is available are called the laws of thermodynamics and involve a concept called entropy for irreversible thermodynamic processes.

Finally, we remark on the question of where we can get our supplies of energy today. Our supplies of energy are from the sun, rain, coal, uranium, and hydrogen. The sun makes the rain, and the coal also, so that all these are from the sun. Although energy is conserved, nature does not seem to be interested in it; she liberates a lot of energy from the sun, but only one part in two billion falls on the earth. Nature has conservation of energy, but does not really care; she spends a lot of it in all directions. We have already obtained energy from uranium; we can also get energy from hydrogen, but at present only in an explosive and dangerous condition. If it can be controlled in thermonuclear reactions, it turns out that the energy that can be obtained from $10$ quarts of water per second is equal to all of the electrical power generated in the United States. With $150$ gallons of running water a minute, you have enough fuel to supply all the energy which is used in the United States today! Therefore it is up to the physicist to figure out how to liberate us from the need for having energy. It can be done.

• Our point here is not so much the result, ( 4.3 ), which in fact you may already know, as the possibility of arriving at it by theoretical reasoning. ↩
• Stevinus' tombstone has never been found. He used a similar diagram as his trademark. ↩
• Counting antibaryons as $-1$ baryon. ↩

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Table of contents.

• Experiment 1 - Heart Rate Meter
• Experiment 2 - Kinematics
• Experiment 3 - Newton's Second Law

Experiment 4 - Conservation of Energy

• Experiment 5 - Momentum and Impulse
• Experiment 6 - Biceps Muscle Model
• Experiment 7 - Rotation and Gyroscopic Precession

Click Here for Experiment 4 -- Conservation of Energy

Shown in the diagram and picture below (both with a top-down view):

• Air track, springs and bracket, thread, glider
• Smart pulley and mount
• Dumb pulley (on the same mount)

Not shown in the images above:

• Computer and Pasco interface
• Scale and weight set
• Mass hanger

INTRODUCTION

In this experiment, you will test the Law of Energy Conservation by monitoring an oscillating air track glider connected to springs by a thread which passes over a smart pulley.

Consider a glider of mass \(M\) on a nearly frictionless air track. As the glider oscillates back and forth, there is a continuous exchange of mechanical energy between two forms: kinetic energy contained in the moving glider, and potential energy stored in the stretched or compressed springs. The Law of Energy Conservation tells us that the total mechanical energy of the system (i.e., the sum of the kinetic and potential terms) remains constant in time. In reality, however, a small amount of mechanical energy may be lost to friction.

INITIAL SETUP

Turn on the air track and level it by adjusting the leveling screw such that a glider on the track has no apparent tendency to move in either direction. Since air tracks are often bowed in the middle, place the glider at several different positions on the track to verify that it is level.

Weigh the glider to obtain its mass, and record this value in the “Data” section.

Assemble the springs, glider, smart pulley, and thread as shown in the figure below. One spring is attached to a bracket at the end of air track and to one end of the glider. The other end of the glider is attached to a second spring by a thread which passes over the smart pulley. The second spring is then attached to the bracket. The springs should be tensioned so that you can get an end-to-end glider motion over a distance of at least 40 cm without either spring being completely compressed.

Mount a dumb pulley (which has low friction, but is not connected to the computer) on the same fixture that holds the smart pulley, albeit in a vertical plane. Using a long piece of thread, connect the mass hanger to the glider. Make sure the thread passes over the dumb pulley so that weights added to the hanger will displace the glider.

PROCEDURE PART 1: MEASURING THE SPRING CONSTANT \(k\)

Our first task is to measure \(k\), the force constant in \(F = -kx\) of a Hooke's Law spring. Call up a blank Excel worksheet and prepare three columns to record the total mass in the hanger (including the mass of the hanger itself) in grams, its weight in newtons, and the position of the glider in meters. In the mass column, type “0” and “10” in the first two cells. Select these two cells, position the cursor at the lower left corner of the bottom cell until it turns into a lopsided square, and drag down the column to fill it with a series up to 60 grams. The running yellow box shows how far the series continues.

Fill down the next column with the force values. Remember how to do this? Type “ =A3*9.8/1000 ” in the cell next to the mass value of zero, being sure to use the correct cell designation corresponding to your spreadsheet (where we typed “ A3 ” above). Then fill down the forces next to the masses.

Now make the measurements. Turn on the air blower, and help the glider come to equilibrium. Add mases to the mass hanger, one at a time, and read the distance (in meters!) from the scale on the air track aligned with one corner of the glider. On your spreadsheet, record the distance corresponding to the addition of each mass in the mass hanger. Be sure that you use values corresponding to the entries in the mass column (including the mass of the hanger itself). If you need to use different masses, change the mass entries; the forces will be recalculated instantly.

When you have filled in the distance column next to the force column, chart these variables against each other in Excel, and find the slope. Here is a reminder of how this is done:

• Select the cells with numbers in the force and distance columns.
• In the “Insert” tab menu, select the “Scatter” option without connecting lines. After clicking on this line-less scatter option, the chart should appear.
• Within the “Design” tab of the “Chart Tools”, in the “Chart Layouts” area, click on the left-most (and upper-most) layout option. This will place titles on the chart that you can edit or delete.
• Create your own title and labels for the axes.
• In the “Layout” tab of the “Chart Tools”, click on “Trendline” in the “Analysis” area. Select “More Trendline Options...”. In the pop-up window, make sure “Linear” is selected, and check the box beside “Display Equation on chart”. Click on “Close”.

Convince yourself that the slope is \(1/k\) and not \(k\). Record the value of \(k\) in the “Data” section.

PROCEDURE PART 2: PLOTTING ENERGIES

Unhook the mass hanger string. Hook up the smart pulley physically and virtually, in Data Studio. Double-click on the smart pulley icon, and check “Position”, “Velocity”, and “Acceleration”. Arrange a graph to plot the position \(x\), the velocity \(v\), and the acceleration \(a\) as a stack of three plots on a single graph window. With the air blower on, pull the glider out, click “Start”, let the glider oscillate several times, and click “Stop”. The velocity and acceleration graphs resemble the sinusoidal oscillations of a simple harmonic oscillator, but the position graph consists of a series of S-shaped curves increasing in \(y\) value. This shape results because the smart pulley does not distinguish between the forward and reverse directions of motion; it merely counts the number of times the spokes block the photosensor and records the result as positive distance. Thus, each S-shaped curve on the position graph is produced as the oscillator moves from one endpoint of its motion to the other.

Now prepare Data Studio to calculate the kinetic energy in real time. Click the calculator button in the graph tool bar (or in the top tool bar). In the equation area of the pop-up window, type “ y = 0.5*m*v^2 ” (for \((1/2)mv^2\)). You will be asked to define the variables \(m\) and \(v\) below. Here \(m\) is a constant (the measured mass of the glider in kilograms), but \(v\) is a data measurement variable (the velocity, Ch. 1 (m/s)). When you have finished defining \(m\) and \(v\), click “Accept” again.

The kinetic energy calculation now appears in the “Data” column. Drag it to a graph, and perform a run with the glider oscillating to check that you are obtaining reasonable-looking kinetic energy curves. (The kinetic energy never quite goes to zero at the end points.)

Calculating the potential energy with the smart pulley is trickier. First, as demonstrated above, the pulley does not distinguish between forward and backward motion, so we can look at only the first half-oscillation. Second, when we pull the glider out and “Start” the distance measurement, the software assigns zero to the first distance measurement. However, we want to assign zero to the equilibrium position. In other words, we want to calculate \(PE = (1/2)k(x-x_0)^2\), with \(x_0\) the equilibrium position.

To accomplish this, record one full S-shaped curve of position: Move the glider away from its equilibrium position to a point where the smart-pulley LED just turns OFF. Wiggle the glider back and forth to locate the ON/OFF transition point as precisely as you can. If the LED is ON, then move away from equilibrium until the light turns OFF. The photogate is now unblocked. Within the first millimeter of motion, it will be blocked by a spoke, and the timing will begin.

Have your partner click “Start”; then release the glider. Click “Stop” just after the glider has reached its maximum position on the other side of equilibrium. Check your position graph, and repeat the experiment until you have a nice S-shaped curve containing 25 – 30 data points with a few points from the next S-curve.

Layer a graph with position and velocity from your best run. The velocity first swings through a minimum (near zero when the glider reaches the endpoint opposite where it was released. Select the position values from \(t=0\) through this endpoint, and have Data Studio calculate the mean value of the position. This is your \(x_0\) in \(PE = (1/2)k(x-x_0)^2\); record it in the “Data” section.

Now click “New” in the calculator window again, and type in the equation “ y2 = 0.5*k*(x-x0)^2 ”, and define its variables \(k\), \(x\), and \(x0\) appropriately. (You can change the name of the variable \(y2\) if you wish.) Click “Accept”.

Finally, click “New” in the calculator window again, and type in the equation “ y3 = P + K ”. This is to be the total energy, so define \(P\) as the data measurement variable \(y2\) (the potential energy calculation) and \(K\) as the data measurement variable \(y\) (the kinetic energy calculation).

These data calculations all appear in the “Data” column. Get a new layered graph with your variables \(y\), \(y2\), and \(y3\) plotted on the \(y\)-axis, and time on the \(x\)-axis. Title your graph and show it to your TA before printing it out. Be prepared to comment to your TA on whether your graphs show reasonable data for the kinetic, potential, and total energies.

(Initial Setup, step 2) Mass of glider =                                              

(Procedure Part 1, step 4) Spring constant \(k\) =                                              

(Procedure Part 2, step 5) Equilibrium position \(x_0\) =                                              

ADDITIONAL CREDIT PART 1 (5 mills)

This is a harder experiment to do well, requiring more careful experimental technique. After you started, you may have found that you had not tensioned the springs enough to get 25 points on the S-shaped curve without compressing one of the springs. Changing the spring tension may require remeasuring \(k\). (Should \(k\) change significantly?)

Repeat the experiment more carefully. If necessary, retension the springs so you can get at least 25 points on the S-shaped curve for the half-oscillation, and remeasure \(k\) carefully.

Averaging the distance values to obtain \(x_0\) may not be the best method of determining the true \(x_0\). From studying your various measurements, what would be another method to obtain a value for \(x_0\)? Does this agree with the averaging method? Include this method in you repeat of the experiment, if you think it would improve the measurement of \(x_0\). Print out the final results.

Full additional credit for this assignment requires obtaining good \(PE\), \(KE\), and total \(E\) curves without significant assistance from your TA. The total energy curve should be a nearly-level line, decreasing slightly through the motion owing to frictional losses. Label your curves as in the illustration below. Include an explanation of any improvements you made in the measuring techniques.

ADDITIONAL CREDIT PART 2 (2 mills)

The glider-spring system is a simple harmonic oscillator. You will study the physics of such systems later in the course. The frequency of oscillation is given by

\begin{eqnarray} f &=& (1/2\pi)\sqrt{k/m}. \end{eqnarray}

Devise a way to obtain the frequency of oscillation from your Data Studio measurements (or make new measurements), and use your values of \(k\) and \(m\) to check this formula. Report the results with experimental error.

The Law of Conservation of Energy and Examples

• The Albert Team
• Last Updated On: February 28, 2023

We’ve now introduced three values that all have the same unit – kinetic energy , potential energy , and work . Since they have the same unit of measurement, we should be able to use them all together. We know there’s some correlation between these three values, but we don’t know how to use them to describe how an object changes after an action has been completed. This is where the law of conservation of energy comes in. In the following blog post, we’ll explain the law of conservation of energy, the formula that governs it, and common conservation of energy examples.

What We Review

What is The Law of Conservation of Energy?

The Law of Conservation of Energy states that the energy in a system before an action plus the work done to complete the action will be equal to the energy in the system after the action. This doesn’t always play out perfectly in real life as some energy can be lost to the creation of noise and heat. For the time being, this law will play out perfectly in the problems you’re given in your physics courses.

Principles of the Law of Conservation of Energy

The catchier definition of this law you may have heard before is that energy cannot be created or destroyed. While that is true, it’s missing something. Energy can be transferred from one object to another. This is what allows one pool ball to set another into motion. It effectively tells us that an object or system needs energy from somewhere else in order to change or complete an action. This is also why a perpetual motion machine could never truly exist.

Conservation of Energy Formula and Its Application

The conservation of energy formula can be written as:

Recall that E_{K} represents kinetic energy , E_{P} represents potential energy , and W represents work. The reason we’ve added numbers to the subscripts is to show that the left side of the equation represents values from before an action was taken while the right side was after the action was taken. So, before and after something falls or before and after two things collide or something like that. The \sum might be new, though.

Conservation of Energy Examples

Kinetic and potential energy and work.

We discussed kinetic energy, potential energy, and work in depth previously, but let’s check our understanding before we jump into some conservation of energy examples.

First, kinetic energy is energy that comes from motion. It is determined by an object’s mass and velocity and is given by the equation E_{K}=\frac{1}{2}mv^{2} . 

So far, we’ve covered two types of potential energy – gravitational potential energy and elastic potential energy. Gravitational potential energy is determined by an object’s mass, gravitational field strength, and height above ground. It is given by the equation E_{P}=mgh . Elastic potential energy is determined by the spring constant of the elastic object and the distance it has been stretched or compressed. It is given by the equation E_{P}=\frac{1}{2}kx^{2} .

Lastly, work is a measure of the effort needed from an external force to cause an action to occur and it is commonly found with the equation W=Fd\cos\theta . It’s also important to keep in mind that the work put into a system is equal to the change in energy of that system. This is what allows for the law of conservation of energy equation to work.

Example: A Simple Lab Experiment

Let’s consider a fairly simple experiment that can be used to test the law of conservation of energy. You only need a scale, a stopwatch, and a few things you can drop. The idea of this experiment is simply to take the mass of a few different objects with notably different masses. You could use a pen, a notebook, and an empty water bottle – anything that will fall without gathering too much air resistance. We’ll be using some sample data below, but feel free to recreate this experiment to find your own data or try a simulation .

Step 1: Calculate Potential Energy Values

Firstly, you’ll want to take the mass of each object and calculate the potential energy.

Step 2: Record Drop Times

Secondly, drop each object from the height you used above and time how long it takes each item to drop. Then, we’ll use this to figure out the average velocity of each object as it falls. You’ll want to drop each item a few times to make sure you have a good average to decrease any potential errors in the data.

Step 3: Calculate the Kinetic Energy

Thirdly, we’ll use our known mass and our calculated average velocities to find the final kinetic energy for each object.

Step 4: Check the Conservation of Energy

Once we’ve collected the above data, we can apply the law of the conservation of energy by comparing the potential and kinetic energy values that we calculated. It is important to realize that we can make this comparison directly from our conservation of energy formula. If we follow our standard problem-solving steps a bit, you’ll soon see why it works.

• E_{K1}=0\text{ J}
• E_{P1}=\text{known}
• W=0\text{ J}
• E_{K2}=\text{known}
• E_{P2}=0\text{ J}

So, we can see that the potential energy we calculated for each object should be equal to the kinetic energy we calculated. Keep in mind that there was some error in our measured times and calculations. Let’s check the law of conservation of energy.

As you can see, we have some error. Specifically, our kinetic energy values aren’t a perfect match to our potential energy values. Given our percent error values are around 3% to 5%, we can therefore assume they were created by inaccurate measurements, rounding errors through the problem, or other human factors. So, we have successfully conducted a lab that proves the Law of the Conservation of Energy.

How Does the Conservation of Energy Apply to Roller Coasters?

Let’s take another example and look at how the conservation of energy applies to roller coasters. In this case, we’ll assume a roller coaster car filled with people has a mass of 700\text{ kg} . On this particular coaster, the car is brought up to its first hill, which is 75\text{ m} tall. Assuming the vehicle is at rest at the top of the hill and no work is done on the car as it falls, what will be its velocity at the bottom of the hill?

• h_{1}=75\text{ m}
• g=9.81\text{ m/s}^{2}
• v_{1}=0\text{ m/s}
• h_{2}=0\text{ m}
• v_{2}=\text{?}

At first, you may be feeling a little lost on how to solve this one. You probably know you need the formula for the conservation of energy given the context of this article. As for how to use it, we’ll need to break it down by substituting equations for our values.

That probably looks quite long and overwhelming because, well, it is. We can simplify it immediately, though. We were told to assume no work was done and no elastic potential energy was ever mentioned, so we can take out both of those values.

You can see that this looks a bit better and a bit more manageable. Now, we have a variable matching all of our knowns from our problem statement and the only variable we don’t know is the one we were told to find. You could jump in and rearrange to solve for the final velocity now, or you can simplify a little more. We have a couple of known values of zero, so let’s plug just those in and see what goes away as a result.

This is now as simple as we can get. Let’s finish following our usual problem-solving steps to find the answer to this question.

The Law of Conservation of Energy is an important one in Physics. It gives engineers the knowledge they need to create things like roller coasters and rocket ships. It also allows us to combine seemingly unrelated ideas like the height of an object and the stretch of spring. Working with this equation can seem overwhelming at first, but it gets easier over time. The more problems you apply it to, the more patterns you’ll see in it, and before no time it will be more helpful than stressful.

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7.6 Conservation of Energy

Learning objectives.

By the end of this section, you will be able to:

• Explain the law of the conservation of energy.
• Describe some of the many forms of energy.
• Define efficiency of an energy conversion process as the fraction left as useful energy or work, rather than being transformed, for example, into thermal energy.

Law of Conservation of Energy

Energy, as we have noted, is conserved, making it one of the most important physical quantities in nature. The law of conservation of energy can be stated as follows:

Total energy is constant in any process. It may change in form or be transferred from one system to another, but the total remains the same.

We have explored some forms of energy and some ways it can be transferred from one system to another. This exploration led to the definition of two major types of energy—mechanical energy KE + PE KE + PE and energy transferred via work done by nonconservative forces ( W nc ) ( W nc ) . But energy takes many other forms, manifesting itself in many different ways, and we need to be able to deal with all of these before we can write an equation for the above general statement of the conservation of energy.

Other Forms of Energy than Mechanical Energy

At this point, we deal with all other forms of energy by lumping them into a single group called other energy ( OE OE ). Then we can state the conservation of energy in equation form as

All types of energy and work can be included in this very general statement of conservation of energy. Kinetic energy is KE KE , work done by a conservative force is represented by PE PE , work done by nonconservative forces is W nc W nc , and all other energies are included as OE OE . This equation applies to all previous examples; in those situations OE OE was constant, and so it subtracted out and was not directly considered.

Making Connections: Usefulness of the Energy Conservation Principle

The fact that energy is conserved and has many forms makes it very important. You will find that energy is discussed in many contexts, because it is involved in all processes. It will also become apparent that many situations are best understood in terms of energy and that problems are often most easily conceptualized and solved by considering energy.

When does OE OE play a role? One example occurs when a person eats. Food is oxidized with the release of carbon dioxide, water, and energy. Some of this chemical energy is converted to kinetic energy when the person moves, to potential energy when the person changes altitude, and to thermal energy (another form of OE OE ).

Some of the Many Forms of Energy

What are some other forms of energy? You can probably name a number of forms of energy not yet discussed. Many of these will be covered in later chapters, but let us detail a few here. Electrical energy is a common form that is converted to many other forms and does work in a wide range of practical situations. Fuels, such as gasoline and food, carry chemical energy that can be transferred to a system through oxidation. Chemical fuel can also produce electrical energy, such as in batteries. Batteries can in turn produce light, which is a very pure form of energy. Most energy sources on Earth are in fact stored energy from the energy we receive from the Sun. We sometimes refer to this as radiant energy , or electromagnetic radiation, which includes visible light, infrared, and ultraviolet radiation. Nuclear energy comes from processes that convert measurable amounts of mass into energy. Nuclear energy is transformed into the energy of sunlight, into electrical energy in power plants, and into the energy of the heat transfer and blast in weapons. Atoms and molecules inside all objects are in random motion. This internal mechanical energy from the random motions is called thermal energy , because it is related to the temperature of the object. These and all other forms of energy can be converted into one another and can do work.

Table 7.1 gives the amount of energy stored, used, or released from various objects and in various phenomena. The range of energies and the variety of types and situations is impressive.

Problem-Solving Strategies for Energy

You will find the following problem-solving strategies useful whenever you deal with energy. The strategies help in organizing and reinforcing energy concepts. In fact, they are used in the examples presented in this chapter. The familiar general problem-solving strategies presented earlier—involving identifying physical principles, knowns, and unknowns, checking units, and so on—continue to be relevant here.

Step 1. Determine the system of interest and identify what information is given and what quantity is to be calculated. A sketch will help.

Step 2. Examine all the forces involved and determine whether you know or are given the potential energy from the work done by the forces. Then use step 3 or step 4.

Step 3. If you know the potential energies for the forces that enter into the problem, then forces are all conservative, and you can apply conservation of mechanical energy simply in terms of potential and kinetic energy. The equation expressing conservation of energy is

Step 4. If you know the potential energy for only some of the forces, possibly because some of them are nonconservative and do not have a potential energy, or if there are other energies that are not easily treated in terms of force and work, then the conservation of energy law in its most general form must be used.

In most problems, one or more of the terms is zero, simplifying its solution. Do not calculate W c W c , the work done by conservative forces; it is already incorporated in the PE PE terms.

Step 5. You have already identified the types of work and energy involved (in step 2). Before solving for the unknown, eliminate terms wherever possible to simplify the algebra. For example, choose h = 0 h = 0 at either the initial or final point, so that PE g PE g is zero there. Then solve for the unknown in the customary manner.

Step 6. Check the answer to see if it is reasonable . Once you have solved a problem, reexamine the forms of work and energy to see if you have set up the conservation of energy equation correctly. For example, work done against friction should be negative, potential energy at the bottom of a hill should be less than that at the top, and so on. Also check to see that the numerical value obtained is reasonable. For example, the final speed of a skateboarder who coasts down a 3-m-high ramp could reasonably be 20 km/h, but not 80 km/h.

Transformation of Energy

The transformation of energy from one form into others is happening all the time. The chemical energy in food is converted into thermal energy through metabolism; light energy is converted into chemical energy through photosynthesis. In a larger example, the chemical energy contained in coal is converted into thermal energy as it burns to turn water into steam in a boiler. This thermal energy in the steam in turn is converted to mechanical energy as it spins a turbine, which is connected to a generator to produce electrical energy. (In all of these examples, not all of the initial energy is converted into the forms mentioned. This important point is discussed later in this section.)

Another example of energy conversion occurs in a solar cell. Sunlight impinging on a solar cell (see Figure 7.19 ) produces electricity, which in turn can be used to run an electric motor. Energy is converted from the primary source of solar energy into electrical energy and then into mechanical energy.

Even though energy is conserved in an energy conversion process, the output of useful energy or work will be less than the energy input. The efficiency Eff Eff of an energy conversion process is defined as

Table 7.2 lists some efficiencies of mechanical devices and human activities. In a coal-fired power plant, for example, about 40% of the chemical energy in the coal becomes useful electrical energy. The other 60% transforms into other (perhaps less useful) energy forms, such as thermal energy, which is then released to the environment through combustion gases and cooling towers.

PhET Explorations

Masses and springs.

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energies for each spring.

• 1 Representative values

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Access for free at https://openstax.org/books/college-physics-2e/pages/1-introduction-to-science-and-the-realm-of-physics-physical-quantities-and-units

• Authors: Paul Peter Urone, Roger Hinrichs
• Publisher/website: OpenStax
• Book title: College Physics 2e
• Publication date: Jul 13, 2022
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-physics-2e/pages/1-introduction-to-science-and-the-realm-of-physics-physical-quantities-and-units
• Section URL: https://openstax.org/books/college-physics-2e/pages/7-6-conservation-of-energy

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5 Ideas for Experiments on Conserving Energy

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We're surrounded by vampires . Only they live on energy, not blood. That big-screen TV, your computer, Xbox , furnace, refrigerator, stove, microwave -- they all suck energy. Some, like your cable box or DVD player, are truly ghoulish: They drink energy even when not in use.

These power-hungry monsters can do some serious damage. About 70 percent of our energy comes from non-renewable sources like oil and natural gas. When they're gone, they're gone for good. Also, some energy sources, such as the coal-burning power plants that generate nearly half of U.S. electricity , pour out carbon dioxide, which threatens to change Earth's climate in dangerous ways [source: Dosomething.org ].

Energy vampires drain pocketbooks as well. In 2011, each American spent an average of $4,410 a year on energy [source: Clayton ]. Putting a stake through the heart of energy waste can put dollars in your pocket.

There are three simple ways to conserve energy:

• Use energy efficiently. Modern refrigerators use much less energy than older ones. Compact fluorescent and LED lights burn less juice than incandescent bulbs.
• Embrace alternative forms of energy. A passive solar room or photovoltaic outdoor lighting takes advantage of free energy from the sun.
• Stop wasting energy . Fix dripping faucets. Plug leaks where cold air can seep in. Turn off appliances when you're not using them.

To slay the energy vampires, you first have to understand how each of these energy conservation methods works. The experiments that follow can help to make people of all ages more energy savvy -- some may even improve your bottom line.

• Shower or bath?
• Which is the best insulator?
• The Energy Effects of Shade and House Paint
• Make a Solar Water Heater
• Wind Power to Electricity

Author's Note: 5 Ideas for Experiments on Conserving Energy

5: shower or bath.

You need energy to heat water. Therefore, if you can get clean without using so much hot water, you'll save. But which uses more hot water, a bath or a shower? This experiment will help you determine that.

Ask everybody in your household to take a bath instead of a shower when they need to wash. Make sure each person leaves the water in the tub when he or she is finished. Then check the height of the water. You can mark it with a bath crayon or with colored tape. Or you can measure it with a ruler and record the level.

Next, have everybody take a shower. (This works best if you have a shower/tub combo.) Ask them to plug the drain before they start so the water doesn't run out. Measure the amount of water left in the tub when each person is finished and mark or record it.

Now compare the average amount of water used for each method. The one that required less water is the one that will save more energy -- and the one you should recommend for use in your household.

Think of ways you could save even more energy. What if you turned off the shower when you're soaping? Using a low-flow shower head, which can put out as much as 72 percent less water, can make a difference, too [sources: Energy Information Administration: Saving, titanheater.com ].

4: Which is the best insulator?

Insulating your house holds heat in, so you run your furnace less often when it's cold outside. However, insulation can also keep things cold -- your refrigerator, for example, is insulated to keep the heat out. Either way, insulation saves energy by keeping your appliances from having to work as hard to maintain the temperature. This experiment helps you find out which materials make the best insulators.

Start with two cardboard shoe boxes. Tape black paper to the outside of the bottom of each. Place the top of each box upside down in a sunny location. Lay a thermometer on each top.

Now, fit the bottom of one box over the top so that the black paper faces up toward the sun. This uninsulated box is your control. It shows what happens without insulation.

Next, take the material you want to test, shape it to completely fill the bottom of the second box and tape it to make sure it stays in place. You can choose almost any material to test, including Styrofoam, a towel, layers of newspaper, a piece of bubble wrap or some artificial fleece. Fit the box over the other top. Make sure the two boxes receive the same amount of sunlight.

After 15 minutes, open the boxes and record the temperature in each. The inside of the insulated box should be cooler than the control box. The greater the temperature difference, the better the material works as an insulator. Repeat the experiment with different materials. Finally, think about why some materials insulate better. For example, maybe they trap dead air spaces that don't allow heat to circulate [source: Energy Information Administration: Insulation].

3: The Energy Effects of Shade and House Paint

Can something as simple as shade from a tree or paint help save on home energy bills? This experiment will help answer that question. The "house" in this case will be a small cardboard box. A 100-watt bulb in a reflector lamp will play the role of the sun.

First, set up the box. Make sure it has a lid that shuts tightly. Put a thermometer inside to measure the temperature. Then, arrange the light so that it shines directly on the box. After 20 minutes, record the temperature inside the box.

Next, set a house plant between the lamp and the box so that its shadow rests on the "house." You can also use branches broken from trees or bushes for this, instead of a house plant -- just stand them in pots of sand to provide stable shade. Check the temperature again after 20 minutes. Did the shade keep the box cooler? Try different plants with different types of leaves. Which makes the most difference in temperature?

To test paint's energy effect, take two identical boxes and paint one white and one black or a dark color. Put each an equal distance from the light and record the temperature inside after 20 minutes. Which box gets hotter? How would this affect energy usage if it were an actual house?

Basically, choosing the right shade trees or exterior paint color could help save energy if they keep a house cooler in summer and cut the need for air conditioning. In winter, however, it might be better not to have shade, so that the house could absorb heat from the sun. In that case, trees that lose their leaves in winter might be the best energy savers [source: Energy Information Administration: Sun].

2: Make a Solar Water Heater

You can make your own hot water using only the power of the sun . You'll start with a flat box that's about 3 feet (90 centimeters) square and 2 or 3 inches (5 or 7.5 centimeters) deep. Paint the inside black or line it with black paper.

Next, take 20 feet (6 meters) or more of flexible black tubing. Make two holes in the sides of the box, one near a corner and the other near the opposite corner. Insert the end of the tube through one hole. Shape the tubing into flat loops that don't overlap but run up and down the bottom of the box. Keep looping until you've used up most of the length. Pass the end out through the other hole.

Place a glass cover on the box. Tape the edges to hold it down. You now have a solar collector like the ones you might sometimes see on rooftops. Put the box in the sun. Immerse one end of the tubing in a bucket of water. Suck on the other end to fill the tube and start a siphon action. Keep the end you sucked on lower than the other and let the water trickle into another container. Use a clothespin to crimp the tube and limit the flow to a trickle.

Place a thermometer in each container of water to track the temperature change. Does the angle of the box toward the sun make a difference? What if you use a longer piece of tubing and make more loops? The warmer the water coming out, the more heat it's absorbing from the sun. If you were using water heated this way in your home, you would take the load off your electric, gas or oil water heater , saving energy. [source: Benrey]

1: Wind Power to Electricity

You might have seen giant wind turbines spinning on hilltops. Well, with this experiment, you can make your own model turbine to try generating electricity from wind .

Start with a 1.5 volt direct current motor. These battery-powered motors are available from hobby shops or online sources. In your experiment, instead of using electricity to turn the motor, you'll be turning the motor shaft with "wind" in order to generate electricity. Mount the motor horizontally on a board with the shaft extending over the edge.

Take a model airplane propeller and fix it to the motor's shaft, perpendicular to the board. Make sure the propeller fits tightly on the shaft or screws down so that when the propeller turns, it turns the shaft. Your "wind turbine" is ready to generate power.

Attach the two wires coming from the motor to a voltmeter that measures up to 5 volts. Between the motor and the voltmeter, connect these two wires using a 100-ohm resistor. You'll need to remove an inch of insulation from each wire and solder the resistor leads (the wires coming from each end of the resistor) to the bare wires. This setup is needed to allow the voltmeter to accurately measure the electricity.

Place the motor and propeller in front of a fan, which will produce the "wind." When the propeller starts spinning, check the voltmeter. It will show you how much electricity is being produced. If you see no movement, or if the voltmeter needle tries to go to the left instead of the right, remove the wires from the voltmeter and reattach them, reversing the sides. See how the voltage changes as the fan speed increases or decreases.

Experiment with different types of propellers. Do larger or smaller propellers affect the amount of electricity produced? Try designing your own propeller. Experiment with multiple blades and blades at different angles. Which produces the most electricity at a given wind speed? Efficient turbines can harness wind energy and conserve energy from other sources [source: National Renewable Energy Laboratory ].

These experiments are intended to teach us about energy conservation principles, but they can suggest practical applications as well. For example, my neighbor has a swimming pool. In the spring, he takes a long length of garden hose and lays loops of it on the roof of his south-facing pool house. It's a lot like the solar hot water experiment. During the day, he uses a pump to run water from the pool through the hose and back in. Solar energy lets him keep his pool a comfortable temperature and he gets into the water earlier than anybody else.

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• Benrey, Ronald M. and Schultz, Robert T. "Alternative Energy Sources," Edison Electric Institute. (April 5, 2012) http://www.charlesedisonfund.org/experiments/edison-pdf/edison_ch2.pdf
• Clayton, Mark. "American anger at gas prices fueled by rising household energy costs," Christian Science Monitor, May 18, 2011. (April 5, 2012) http://www.csmonitor.com/USA/2011/0518/American-anger-at-gas-prices-fueled-by-rising-household-energy-costs
• Dosomething.org. "11 Facts About Energy." (April 5, 2012) http://www.dosomething.org/actnow/tipsandtools/11-facts-about-energy
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• U.S. Energy Information Administration. "Insulation." (April 5, 2012) http://www.need.org/needpdf/IntInsulation.pdf
• U.S. Energy Information Administration. "Saving Hot Water." (April 5, 2012) http://www.need.org/needpdf/PriSavingHotWater.pdf
• U.S. Energy Information Administration. "Sun or Shade." (April 5, 2012) http://www.need.org/needpdf/PriSunorShade.pdf

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conservation of energy , principle of physics according to which the energy of interacting bodies or particles in a closed system remains constant. The first kind of energy to be recognized was kinetic energy , or energy of motion. In certain particle collisions , called elastic , the sum of the kinetic energy of the particles before collision is equal to the sum of the kinetic energy of the particles after collision. The notion of energy was progressively widened to include other forms. The kinetic energy lost by a body slowing down as it travels upward against the force of gravity was regarded as being converted into potential energy , or stored energy, which in turn is converted back into kinetic energy as the body speeds up during its return to Earth . For example, when a pendulum swings upward, kinetic energy is converted to potential energy. When the pendulum stops briefly at the top of its swing, the kinetic energy is zero, and all the energy of the system is in potential energy. When the pendulum swings back down, the potential energy is converted back into kinetic energy. At all times, the sum of potential and kinetic energy is constant. Friction , however, slows down the most carefully constructed mechanisms, thereby dissipating their energy gradually. During the 1840s it was conclusively shown that the notion of energy could be extended to include the heat that friction generates. The truly conserved quantity is the sum of kinetic, potential, and thermal energy. For example, when a block slides down a slope, potential energy is converted into kinetic energy. When friction slows the block to a stop, the kinetic energy is converted into thermal energy . Energy is not created or destroyed but merely changes forms, going from potential to kinetic to thermal energy. This version of the conservation-of-energy principle, expressed in its most general form, is the first law of thermodynamics . The conception of energy continued to expand to include energy of an electric current , energy stored in an electric or a magnetic field , and energy in fuels and other chemicals. For example, a car moves when the chemical energy in its gasoline is converted into kinetic energy of motion.

With the advent of relativity physics (1905), mass was first recognized as equivalent to energy. The total energy of a system of high-speed particles includes not only their rest mass but also the very significant increase in their mass as a consequence of their high speed. After the discovery of relativity, the energy-conservation principle has alternatively been named the conservation of mass-energy or the conservation of total energy.

When the principle seemed to fail, as it did when applied to the type of radioactivity called beta decay (spontaneous electron ejection from atomic nuclei), physicists accepted the existence of a new subatomic particle , the neutrino , that was supposed to carry off the missing energy rather than reject the conservation principle. Later, the neutrino was experimentally detected.

Energy conservation, however, is more than a general rule that persists in its validity. It can be shown to follow mathematically from the uniformity of time . If one moment of time were peculiarly different from any other moment, identical physical phenomena occurring at different moments would require different amounts of energy, so that energy would not be conserved.

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• Published: 13 April 2020

Kinematically complete experimental study of Compton scattering at helium atoms near the threshold

• Max Kircher   ORCID: orcid.org/0000-0002-1592-5706 1 ,
• Florian Trinter 2 , 3 ,
• Sven Grundmann   ORCID: orcid.org/0000-0001-6512-6921 1 ,
• Isabel Vela-Perez 1 ,
• Simon Brennecke 4 ,
• Nicolas Eicke 4 ,
• Jonas Rist 1 ,
• Sebastian Eckart   ORCID: orcid.org/0000-0002-9245-9059 1 ,
• Salim Houamer   ORCID: orcid.org/0000-0002-5085-6638 5 ,
• Ochbadrakh Chuluunbaatar 6 , 7 , 8 ,
• Yuri V. Popov 6 , 9 ,
• Igor P. Volobuev 9 ,
• Kai Bagschik 2 ,
• M. Novella Piancastelli   ORCID: orcid.org/0000-0003-3303-7494 10 , 11 ,
• Manfred Lein   ORCID: orcid.org/0000-0003-1489-8715 4 ,
• Till Jahnke 1 ,
• Markus S. Schöffler 1 &
• Reinhard Dörner   ORCID: orcid.org/0000-0002-3728-4268 1  

Nature Physics volume  16 ,  pages 756–760 ( 2020 ) Cite this article

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• Atomic and molecular interactions with photons
• Quantum mechanics

Compton scattering is one of the fundamental interaction processes of light with matter. When discovered 1 , it was described as a billiard-type collision of a photon ‘kicking’ a quasi-free electron. With decreasing photon energy, the maximum possible momentum transfer becomes so small that the corresponding energy falls below the binding energy of the electron. In this regime, ionization by Compton scattering becomes an intriguing quantum phenomenon. Here, we report on a kinematically complete experiment studying Compton scattering off helium atoms in that regime. We determine the momentum correlations of the electron, the recoiling ion and the scattered photon in a coincidence experiment based on cold target recoil ion momentum spectroscopy, finding that electrons are not only emitted in the direction of the momentum transfer, but that there is a second peak of ejection to the backward direction. This finding links Compton scattering to processes such as ionization by ultrashort optical pulses 2 , electron impact ionization 3 , 4 , ion impact ionization 5 , 6 and neutron scattering 7 , where similar momentum patterns occur.

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Data availability.

The data that support the plots within this Letter are available from the corresponding authors upon reasonable request.

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Acknowledgements

This work was supported by DFG and BMBF. O.C. acknowledges support from the Hulubei-Meshcheryakov programme JINR-Romania and the RUDN University Program 5-100. Y.V.P. is grateful to the Russian Foundation of Basic Research (RFBR) for financial support under grant no. 19-02-00014a. S.H. thanks the Direction Generale de la Recherche Scientifique et du Developpement Technologique (DGRSDT-Algeria) for financial support. We are grateful to the staff of PETRA III for excellent support during the beam time. Calculations were performed on the Central Information and Computer Complex and heterogeneous computing platform HybriLIT through supercomputer ‘Govorun’ of JINR.

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Institut für Kernphysik, J. W. Goethe Universität, Frankfurt, Germany

Max Kircher, Sven Grundmann, Isabel Vela-Perez, Jonas Rist, Sebastian Eckart, Till Jahnke, Markus S. Schöffler & Reinhard Dörner

FS-PETRA-S, Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany

Florian Trinter & Kai Bagschik

Molecular Physics, Fritz-Haber-Institut der Max-Planck-Gesellschaft, Berlin, Germany

Florian Trinter

Institut für Theoretische Physik, Leibniz Universität Hannover, Hannover, Germany

Simon Brennecke, Nicolas Eicke & Manfred Lein

LPQSD, Department of Physics, Faculty of Science, University Sétif-1, Setif, Algeria

Salim Houamer

Joint Institute for Nuclear Research, Dubna, Moscow, Russia

Ochbadrakh Chuluunbaatar & Yuri V. Popov

Institute of Mathematics and Digital Technologies, Mongolian Academy of Sciences, Ulaanbaatar, Mongolia

Ochbadrakh Chuluunbaatar

Peoples’ Friendship University of Russia (RUDN University), Moscow, Russia

Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University, Moscow, Russia

Yuri V. Popov & Igor P. Volobuev

Sorbonne Universités, CNRS, UMR 7614, Laboratoire de Chimie Physique Matière et Rayonnement, Paris, France

M. Novella Piancastelli

Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden

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Contributions

M.K., F.T., S.G., I.V.-P., J.R., S.E., K.B., M.N.P., T.J., M.S.S. and R.D. contributed to the experimental work. S.B., N.E., S.H., O.C., Y.V.P., I.P.V. and M.L. contributed to theory and numerical simulations. All authors contributed to the manuscript.

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Supplementary video 1.

Electron and ion momentum distributions for different scattering angles. Here, the blue arrow indicates the direction of the incoming photon, the light green arrow the direction of the scattered photon and the dark purple arrow the direction of the momentum transfer.

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Kircher, M., Trinter, F., Grundmann, S. et al. Kinematically complete experimental study of Compton scattering at helium atoms near the threshold. Nat. Phys. 16 , 756–760 (2020). https://doi.org/10.1038/s41567-020-0880-2

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Experimental proof for Conservation of Total Energy

In physics, the law conservation of energy states that the total energy of an isolated system cannot change—it is said to be conserved over time. (Source : wikipedia(conservation of energy)

Truly isolated systems cannot exist in nature, other than allegedly the universe itself, and they are thus hypothetical concepts only. (Source : wikipedia (isolated system)

Clearly depict a loophole in the law of conservation of energy as the law holds in case of isolated systems and apparently these do not exist.

I thought of various practical experiments to inspect the actuality of the law, but in reality I could not think of a single experiment which does not send off energy in form of heat/sound/light out of the system. Even the experiments performed in UCLA here states that some energy will be uncaccounted for in the experiment !

Even a simple experiment of a ball falling down creates heat from friction with air. Clearly the law must hold if we conduct the entire experiment in an isolated system , but these simply do not exist.

Magnetic levitation to experiment with conservation of energy experiment is not an option as the levitation requires energy and going back to source, we come up with the same heat release problem which makes this experiment only a complex version of the first one.

Note : I am not talking about approximations

Since we can not make any sort of isolated system without it leaking/absorbing energy into the surroundings. The discussion of the conservation of energy boils down to the conservation of energy of the entire universe.

Through various texts I came to the conclusion that the total energy of Universe is not a constant. Those texts being ( 1 , 2 , 3 )

Even the ArxiV paper listed as 3 states only

Despite remaining hesitations about where additional mass could be coming from, on the overall balance of consistency, we conclude that the energy conservation law is better obeyed by means of increasing mass of the universe with its radius Conserves only a type of energy as it suggests increment in mass which is also a type of energy.

I believe this looks extremely like challenging the law of conservation of energy which is supported by neother's theorem among many others. But I just wish to determine whether practically this law is actually valid or not !

I do not expect to get answers of 99.99% conservation with the negligible heat being lost neglected.

I understand that GR does not say energy is conserved but the scales of day to day life and that of GR are different which makes energy conservation at earthly scales even if 99.99% a boon. But this just means that energy conservation depends on the working scale and experiment and ranges from nearly 100% in some to not conserved in various aspects of GR.

Finally the bottom line of question is, can we experimentally prove that energy, total energy is actually conserved ? (a Yes answer requires a detailed experiment with complete conservation and no loopholes)

• experimental-physics
• energy-conservation

• 1 $\begingroup$ A theory can't be proved by experiment, it can only be shown to not contradict given experiment, or disproved. Moreover, the history of science suggests that physical theories are just approximations of laws of nature, and approximations break sooner or later. $\endgroup$ –  Ruslan Commented Feb 6, 2014 at 14:53
• $\begingroup$ @Ruslan : Do you mean to say that Conservation of Energy is ultimately just an approximation, and under various circunstances be circumvent to produce machines such as even a perpetual motion machine ! $\endgroup$ –  Rijul Gupta Commented Feb 6, 2014 at 14:56
• $\begingroup$ @rijulgupta I believe what he's trying to say is that the law fits everything we know today, but it might be possible that we find out that in some specific cases this interpretation might fail. For instance, we know that energy fluctuations do temporarly occur (see here en.wikipedia.org/wiki/Quantum_fluctuation ) $\endgroup$ –  cinico Commented Feb 6, 2014 at 15:03
• $\begingroup$ I mean to say that it's currently known law, but no one can guarantee that it won't ever broken. In quantum mechanics you already have something which makes this law fuzzy: Heisenberg uncertainty principle. Something might be discovered which would indeed allow making perpetual motion machine, but according to current understanding of nature, it's impossible. $\endgroup$ –  Ruslan Commented Feb 6, 2014 at 15:03
• 1 $\begingroup$ This question and its answers might be useful. $\endgroup$ –  Ruslan Commented Feb 6, 2014 at 15:17

3 Answers 3

Elementary particle physicists have been doing this for more than sixty years. Conservation of energy is one of the main constraints that built up the standard model of particle physics.

We have first to agree about "proof". In my above assertion I consider it a "proof" that the law has not been falsified by any of the data used to discover the great symmetries of nature in the standard model. And the number of experiments and events in the experiments are way over the five sigma deviation for statistical proof.

We also have to agree that for any measurement in the real world there will be errors, and all the values measured are accompanied by a +/- of the error of the measurement.

Energy conservation is, within errors, always true in decays. Look at these decays in the bubble chamber of a lambda and an antilambda:

Caption: Bubble chamber photo of the production & decay of a lambda particle & its antimatter equivalent, an antilambda. The particles are produced from the annihilation of an antiproton which enters the picture at bottom. Being neutral, these lambda & antilambda leave no tracks, but they reveal their presence by decaying into charged particles which form V-shaped pairs of tracks. The two "vees" near the bottom of the picture are produced by the antilambda (left) & the lambda. The antilambda decays into an antiproton (left) & a positive pion, the lambda into a proton (left) & a negative pion.

We can measure the momenta of the decay products, with the experimental errors, we can identify the proton(anti) and the pi-(+) by the ionization they leave in the bubble chamber . We have done this in a huge number of experiments and measured the mass of the lamda as

1115.683±0.006 MeV/c**2

It all comes because in elementary particle interactions four momentum is conserved . Fitting interactions with the constraint of four momenta conservation has built up for us the particles and resonances that fit so beautifully into the group representations of the standard model.

• $\begingroup$ My apologies if my comment is based on wrong understanding of your answer, but haven't energy conservation most doubtful in the realm of sub atomic particles ? I have read many times that due to the necessity of energy conservation we keep on finding new particles to balance the equations, I suppose their might be many particles still not discovered that might be in use to satisfy equations of energy conservation at the scale. If what I am saying is in fact true then woudnt it be wrong to chose these experiments to prove energy conservation ? $\endgroup$ –  Rijul Gupta Commented Mar 9, 2014 at 3:15
• 1 $\begingroup$ Well, you can stop using your eyes too if you see too many new sights. I have shown the lamdas above because they cannot fall into your argument of "keep finding new particles" . We see them, they are there, we can measure their four vectors, and a bubble chamber is a hands on instrument even for non physicists. Experimental physicists believe the evidence of the highly enginneered detectors at the LHC ( for example) as much as their eyes looking and measuring the bubble chamber pictures by ruler and angular meter. You are free to believe in angels on the tip of a pin, but that is not physics $\endgroup$ –  anna v Commented Mar 9, 2014 at 4:34
• $\begingroup$ I see that the above experiments is very very loosely like a collision of two balls and then measuring their moments before and after to check energy conservation or other things. Is there in no way anything we are not accounting ? Like in macroscopic balls we leave the heat changes. Since these particles must not be releasing heat, are we accounting for everything ? Maybe some loose photons ? $\endgroup$ –  Rijul Gupta Commented Mar 9, 2014 at 4:59
• $\begingroup$ Yes we are leaving out little angels which force what we call the decay products of the lambda to always have the mass of the lambda with great accuracy. $\endgroup$ –  anna v Commented Mar 9, 2014 at 5:12
• 1 $\begingroup$ You are wrong. The scales of this lamda detection are of the order of weak interactions hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html look at that 10^-18m . Heat has no meaning in elementary interactions. Heat is an emergent quantity in large numbers of statistics. If there were photons missing the mass would not come accurate. I am no longer going to respond because it is evident that your mind is made up, and not by physics arguments. $\endgroup$ –  anna v Commented Mar 9, 2014 at 9:31

No, we cannot prove that energy is perfectly conserved experimentally.

The first problem is that you would have to run an uncountably infinite number of experiments to "prove" that conservation of energy is true. In some hypothetical universe, you might run one trillion experiments where you calculate the energy of, say, a falling ball as well as the energy it imparts to the air on its fall. In this wonderful universe, these two things add up to exactly 100% with no decimals. What if when you hold your tongue out 45 degrees to the left, the ball falls faster or some other ridiculous thing? So, from a philosophical stance, it's not possible to prove any scientific hypothesis/theory/law perfectly. You can "prove" it beyond reasonable doubt, but not perfectly.

A second problem is that of accuracy and precision. Where should we get a machine that measures energy (or some value from which energy can be calculated) to an infinite number of decimal places? You're saying that accounting for 99.99% of the energy is insufficient. I assume that accounting for 99.99999999999% of the energy would also be insufficient? In that case we would need some kind of perfect instrument that could never exist in our universe (so far as we know).

A third problem: ∆E∆t >= hbar/2. Our current understanding of the universe indicates that even if we somehow had perfect equipment, energy can never be measured perfectly. Essentially, the uncertainty in energy multiplied by the uncertainty in time can never be less than hbar/2.

So, what does that all mean? Ultimately, all of science (and therefore physics) is our current best guess. To say that something is absolutely proven beyond any possible doubt is to say that we are no longer open to new information. This goes against everything that science stands for. But does this mean that we're shooting in the dark? Did some crazy person just guess that energy is conserved and we all went with it?

There have been probably hundreds of thousands or millions of measurements of the conservation of energy. Of those many measurements, no set of measurements that disagrees with conservation of energy has ever stood up to serious scrutiny. In science, when we make a measurement we need to also look at the uncertainty in the measurement. Perhaps we have a system where we can measure both the before and after energy of an object, as well as the heat and sound generated by it. We add up all the before energies and all the after energies to find the before is 10 J, and the after is 9.9 J. There. Conservation disproven? The experiment, and any instruments used in it will have some uncertainty. What we should really write is before = 10.0 ± 0.2 J and after = 9.9 ± 0.1 J. Since they agree to within experimental error, all we can say is that conservation does not appear to be violated.

Unfortunately, that's the reality. No measurement can ever be exact. If you think that only accounting for 99.99999% of the energy in a system means that energy isn't actually conserved... there's a small (so small as to be virtually indistinguishable from zero) chance you could be right! In science though, we have to eventually accept certain things as true in order to move on. In a perfect world, every scientist would verify every principle for themselves from first principles. But if that were the case, we would all still be tossing rocks in the air and timing how long it takes them to fall. Perhaps the best argument that conservation of energy works is that planes stay in the air, phones connect to the internet, and the sun rises every morning. It would be amazing if one of the core tenets of physics was found to be extremely incorrect and yet all these things that we've based our physics on actually work.

No physicist can ever impart absolute truth to you. We leave that up to the mathematicians.

1) Homogeneity of time throughout the universe (electronic transitions of atoms and molecules, then telescopes and spectroscopy). 2) Noether's theorems. 3) Mass-energy is conserved.

However, (1) is not exactly true locally. We live at the bottom of a gravitational potential well. GPS demonstrates that time passes faster when residing higher in that well. General Relativity lacks conservation laws. Magnitude matters! Given the uncontested loophole in Noether, exploit it.

• $\begingroup$ GR still has conservation of energy-momentum tensor $\endgroup$ –  Jim Commented Feb 6, 2014 at 18:13

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Charged Lepton Flavor Violation Experiments

• Published: 09 September 2022
• Volume 77 , pages 136–140, ( 2022 )

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We briefly summarize the current status and plans of a sample of charged lepton flavor violation experimental efforts, emphasizing muon-based experiments.

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Charged lepton flavour violation: An experimental and theoretical introduction

A comparative study of z′ mediated charged lepton flavor violation at future lepton colliders.

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ACKNOWLEDGMENTS

I would like to thank Fermilab for their support, and my colleagues on the Mu2e experiment for their support and many informative discussions.

This work is supported under a grant from the US Department of Energy, Office of High Energy Physics, award DESC0013895.

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Miller, J.P. Charged Lepton Flavor Violation Experiments. Moscow Univ. Phys. 77 , 136–140 (2022). https://doi.org/10.3103/S0027134922020692

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Physicists solve the mystery of vanishing particles in graphene

Researchers have explained the puzzling phenomenon of particle-antiparticle annihilation in graphene, recognized by specialists as Auger recombination

Moscow Institute of Physics and Technology

image: Two scenarios of electron-hole recombination in graphene. In radiative recombination (left), the mutual annihilation of an electron and a hole, shown as blue and red spheres respectively, frees energy in the form of a photon, a portion of light. In Auger recombination (right), this energy is picked up by an electron passing by. The Auger process is harmful for semiconductor lasers, because it consumes the energy that could be used to produce laser light. For a long time, the Auger process was considered to be impossible in graphene due to the energy and momentum conservation laws. view more 

Credit: Elena Khavina/MIPT Press Office

Researchers from the Moscow Institute of Physics and Technology and Tohoku University (Japan) have explained the puzzling phenomenon of particle-antiparticle annihilation in graphene, recognized by specialists as Auger recombination. Although persistently observed in experiments, it was for a long time thought to be prohibited by the fundamental physical laws of energy and momentum conservation. The theoretical explanation of this process has until recently remained one of the greatest puzzles of solid-state physics. The theory explaining the phenomenon was published in Physical Review B .

In 1928, Paul Dirac predicted that an electron has a twin particle, which is identical in all respects but for its opposite electric charge. This particle, called the positron, was soon discovered experimentally. Several years later, scientists realized that the charge carriers in semiconductors -- silicon, germanium, gallium arsenide, etc. -- behave like electrons and positrons. These two kinds of charge carriers in semiconductors were called electrons and holes. Their respective charges are negative and positive, and they can recombine, or annihilate each other, releasing energy. Electron-hole recombination accompanied by the emission of light provides the operating principle of semiconductor lasers, which are devices crucial for optoelectronics.

The emission of light is not the only possible outcome of an electron coming in contact with a hole in a semiconductor. The liberated energy is often lost to thermal vibrations of the neighboring atoms or picked up by other electrons (figure 1). The latter process is referred to as Auger recombination and is the main "killer" of active electron-hole pairs in lasers. It bears the name of French physicist Pierre Auger, who studied these processes. Laser engineers strive to maximize the probability of light emission upon electron-hole recombination and to suppress all the other processes.

That is why the optoelectronics community greeted with enthusiasm the proposal for graphene-based semiconductor lasers formulated by MIPT graduate Victor Ryzhii. The initial theoretical concept said that Auger recombination in graphene should be prohibited by the energy and momentum conservation laws. These laws are mathematically similar for electron-hole pairs in graphene and for electron-positron pairs in Dirac's original theory, and the impossibility of electron-positron recombination with energy transfer to a third particle has been known for a long time.

However, experiments with hot charge carriers in graphene consistently returned the unfavorable result: Electrons and holes in graphene do recombine with a relatively high rate, and the phenomenon appeared attributable to the Auger effect. Moreover, it took an electron-hole pair less than a picosecond, or one-trillionth of a second, to disappear, which is hundreds of times faster than in contemporary optoelectronic materials. The experiments suggested a tough obstacle for the implementation of a graphene-based laser.

The researchers from MIPT and Tohoku University found that the recombination of electrons and holes in graphene, prohibited by the classical conservation laws, is made possible in the quantum world by the energy-time uncertainty principle. It states that conservation laws may be violated to the extent inversely proportional to the mean free time of the particle. The mean free time of an electron in graphene is quite short, as the dense carriers form a strongly-interacting "mash." To systematically account for the uncertainty of particle energy, the so-called nonequilibrium Green's functions technique was developed in modern quantum mechanics. This approach was employed by the authors of the paper to calculate Auger recombination probability in graphene. The obtained predictions are in good agreement with the experimental data.

"At first, it looked like a mathematical brain teaser, rather than an ordinary physical problem," says Dmitry Svintsov, the head of the Laboratory of 2D Materials for Optoelectronics at MIPT. "The commonly accepted conservation laws permit recombination only if all three particles involved are moving precisely in the same direction. The probability of this event is like the ratio between the volume of a point and the volume of a cube -- it approaches zero. Luckily, we soon decided to reject abstract mathematics in favor of quantum physics, which says that a particle cannot have a well-defined energy. This means that the probability in question is finite, and even sufficiently high to be experimentally observed.

The study does not merely offer an explanation for why the "prohibited" Auger process is actually possible. Importantly, it specifies the conditions when this probability is low enough to make graphene-based lasers viable. As particles and antiparticles rapidly vanish in experiments with hot carriers in graphene, the lasers can exploit the low-energy carriers, which should have longer lifetimes, according to the calculations. Meanwhile, the first experimental evidence of laser generation in graphene has been obtained at Tohoku University in Japan.

The study was supported by the Russian Science Foundation.

Physical Review B

10.1103/PhysRevB.97.205411

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