This is a collection of simple, real-world analogies that can be used to teach K-12 students about concepts such as convection, center of mass,gravity, reflection etc.


connection between temperature and densityEdit

  • Activity to do with children. Mark off an area 10 feet x 10 feet. Ask a whole bunch of children to fill the space. Ask the class to count the number of "molecules". There should be enough children in the space that they can't move. Draw attention to this fact. Now start removing molecules and ask the remaining molecules to run around in the space trying to cover all the area. Keep removing molecules. Draw attention to the fact that as the number of molecules decreases, the remaining have to run faster to cover the same area. As they run faster, they get hotter and hotter.

Contributed by Sayre Peralta and Tara Chklovski

  • Analogy for why hot air rises & cold air falls: Think of air as a fluid (like water) and think of objects that will float and sink in water, like a beach ball and a dime. In this analogy, the hot air is like a beach ball, and cold air is like a dime. The beach ball floats on water because it is less dense than water, and the dime sinks because it is more dense than water. Both objects are approximately the same mass, but the density difference is what makes one float and one sink.

Contributed by Eloy Ivan Balderas and Minh Dan Vuong


Buoyancy as a density difference, and gravityEdit

Gravity always pulls objects down. So why, then, do some objects like wood float on top of water? Or if you release wood while submerged in water, why will it float upwards? Is gravity pushing the wood up?

The wood is sitting in a fluid, namely water. We know that gravity pulls everything down, including the fluid. If we dump water out of a container, it falls on the floor because gravity pulls it downward. Now, when we have a piece of wood sitting in water, gravity is trying to pull down both the piece of wood and the water. So what happens? If the force of gravity is stronger on the wood, then the wood sinks through the water. If the force of gravity is stronger on the water, then the water gets pulled downward under the wood, and as a consequence, the wood floats upward. So the wood only goes up because gravity is pulling water downward.

So, what make the force of gravity stronger in wood or water? This is a property known as density, or how closely packed the molecules are inside an object. When molecules are tightly packed together, we say that the object is dense. When molecules are spread far apart, we say that the object is light. Since gravity acts on mass (F=mg) when something has more mass per volume (or density) than something else, the force of gravity is greater on that object.

Now, onto a concept known as buoyancy. Buoyancy is simply the density of a solid object relative to the fluid it is sitting in. For this reason, buoyancy is a property of an object that can change, depending on its surroundings. For example, wood is negatively buoyant in air (i.e. more dense than air), while wood is positively buoyant in water (i.e. less dense than water). Air is less dense than water. If you trap an air bubble in a cup and release it under water, the air will float to the surface. This also explains why water will sit nicely in a tank, and not float up and out of it. The water is heavier than the air which fills up the rest of the room that the tank is sitting in.

In summary, object float upwards because gravity pulls the fluid that the objects are sitting in downward, which describes what we commonly call positive buoyancy.

Contributed by Kevin Miklasz


explaining low Reynolds number and high viscosityEdit

Viscosity is friction in fluids. Imagine you are very small (as small as a molecule) and wading through water. As you move, you have to fight these other molecules that are blocking your movement. If you were Gulliver, you could easily move through a crowd of Lilliputians, but if you were a Lilliputian yourself, then it would take you much longer to get through the crowd.

Also, think of the play pits that are filled with plastic balls. It is hard to move though this pit of balls, because the balls you are moving through are nearly the same size as you. Now imagine walking through air. Air molecules are a lot smaller than you, so you barely notice them.

Contributed by Kevin Miklasz

File:Teaching about Reynolds Number

Difference between skin drag and pressure dragEdit

At low Reynolds numbers, or in extra sticky fluids, skin drag dominates. The fluid sticks to the surfaces of objects, or their skin, thus the term skin drag. This is just like the friction between an object sliding on a surface, it is the contact between the fluid and the object which creates friction and slows something down. The total exposed surface area is all that matters- a tiny long string has a lot more drag than a big fat cylinder of equal volume.

On the other hand, pressure drag dominates at large Re. Pressure is basically a description of how "annoyed" a fluid is with its current situation, and how much it would rather be somewhere else. Fluids only get annoyed at big Reynolds number, at lower Reynolds numbers, the fluid sticks together and can't really move around much at all, therefore its ability to bounce around and get annoyed is reduced. Pressure drag results from a fluid's annoyance at having to go around an obstacle. A tiny string, no matter how long, is no problem to go around, but a big fat cylinder pushes a lot of fluid to the side and is quite a pain. A high pressure zone builds up in front of the cylinder, where everything is really annoyed at having to move around this object. And a low pressure zone builds up behind the object, which is where everything wants to be.

The bigger the object, the bigger the pressure drag because the fluid has to move even farther around the object and gets even more annoyed. Think of driving in a traffic jam, with an accident ahead. If the accident is only in one lane, traffic slows down and you are slightly annoyed, but everyone moves out of that one lane and keeps going. On the other hand, if you have a three lane accident on a four lane highway, traffic has to really get redirected, resulting in a lot of annoyance and a buildup of pressure.

Contributed by Kevin Miklasz


What is an airfoil?Edit

An airfoil is a 2-dimensional cross-section of a wing, taken at some span-wise location. This image of a plane with airfoil high-lighted helps to orient the audience to the overall idea.[1] An airfoil can be described as having a leading edge and a trailing edge, which are the two points on the airfoil which are furthest apart. Typically the trailing edge is the apex of an acute angle. The chord line can be defined as the straight line that connects the leading and trailing edges. The upper surface and the lower surface are defined as the two surfaces(actually lines, but called surfaces typically to relate to wing) that connect the leading and trailing edges. The upper surface is conventionally the one that has a longer path, or more area enclosed by it and the chord line. The two surfaces are occasionally referred to as the upper and lower camber lines. The mean camber line is created by taking the locus of points which are created by systematically moving along the chord line, and at each point drawing a line perpendicular to the chord line and extending to the lower and upper surfaces. Taking the midpoint of each of these lines will form the mean camber line. This image helps explain the leading and trailing edges, the chord line, the upper and lower surfaces, and the mean camber line.[2] The thickness of an airfoil changes along the chord, and at any given chord-wise location, the thickness is the distance between the upper and lower surface, taken perpendicular to the chord line. The angle of attack for an airfoil is defined as the angle between the on-coming wind and the chord line. The concept of on-coming wind might be confusing for kids, so explain that while the plane is moving through still air, the air looks like its moving for people in the plane. The direction the plane is moving is also the direction of the on-coming wind.[3]

Contributed by John McArthur

How does an airfoil generate lift?Edit

This question is difficult, and is still somewhat debated. In my experience, I've come across two dominant theories:

1) Lift is generated by the pressure difference between the upper and lower surfaces. The upper surface has lower pressure than the lower surface, therefore the net force is upward. The pressure difference comes from the Bernoulli principle that faster air has lower pressure. The air above the wing is moving faster because the upper surface is curved up, while the lower surface is usually flat, or pretty close to flat. I recall one person trying to describe it as the upper surface having a longer path than the lower surface. His logic was that two particles that start together at the leading edge, but one takes the upper surface path and the other takes the lower surface path, then the one going around the upper surface must go faster if they both arrive at the trailing edge together. This could be good for kids to visualize, but its NOT correct. The flaw is that they don't necessarily have to arrive at the trailing edge together.

The correct way to describe why air going over the upper surface is faster is that it is "pinched" more than the lower surface. This is a result of the continuity principle that says that mass is neither created nor destroyed (unless you're dealing with atomic physics). If you take a straw and blow with a certain amount of pressure, then the air will come out at a certain speed. If you pinch the end of the straw, then the air will come out faster. The same is true (and fun) about a water hose, and that's why you put you're finger over the end of the hose to get higher speed water. (Well, that's partly true, its also to increase the water pressure in the hose...but this is a good visualization).

In summary, this pressure difference explanation goes like this: Upper surface is pinching the flow more than the lower surface, therefore it has higher speeds, and therefore it has lower pressures.

A great demonstration of this explanation is to take a piece of 8.5"x11" paper and hold it under your lips so it bends down under as it goes away from your lips. Then blow lightly, and the paper should lift up slightly. This is because the higher speed air has lower pressure, and so the high pressure air underneath the paper is lifting it up. [Note - this is not true, as a vertically held paper will not move to on side when you blow down it. See]

2) Lift is generated because the air flow is being curved and pushed downwards by the airfoil. Using Newton's laws, if the air is pushed down by the airfoil, then the airfoil is pushed up by the air. This is known as "for every action, there is an equal and opposite reaction". In other words, every force exerted has an equal and opposite force exerted back. This idea can be demonstrated by having a skate board and a basketball. If you stand on the skate board with the basketball, then throw the basketball in the direction of the skate board wheels, then you and the skate board will move in the opposite direction. Equal and opposite. The faster you throw the ball, the faster you'll move. Also, the heavier the ball, the faster you'll move.

So, which of these is correct? One of my professors told me that these two ways of generating lift are connected. In other words, you can't get one without the other. If you want to have lower pressures on the top surface, then you must turn the flow. If you want to turn the flow, then you must have lower pressure on top. However, I don't fully agree. There is a set of equations called the Navier-Stokes equations, and they describe all of fluid motion. These equations can include what's called a "body force", which in this case would be the force exerted on the airfoil. There are three other forces described by these equations: the pressure force, the inertial force, and the viscous force. To me, it makes sense that the force exerted on the airfoil is the sum of these three forces. I have asked 3 different professors about this, and 1 agreed, while the other 2 danced around the issue. So, is my way of viewing it correct? I'm not sure, and there doesn't seem to be much consensus on the internet either.

For further reading, I'll direct you to[4]. This site basically describes what I talk about above, but includes one other method of generating lift, which is by circulation. I would say that the combination technique that I describe above is very similar to the circulation method, but his description of the circulation method is rather lacking of details, so its hard to say. However, you can form the Navier-Stokes equations in a vorticity form, and taking the integral of vorticity in a flow gives the circulation. So, I think that what he's trying to describe is basically using the Navier-Stokes equations, which include the pressure, inertial, and viscous forces.

Finally, I leave you with a historical/political note. You'll see that the two methods above stem from to different and very prominent names in mathematics and science. One derives mostly from the Bernoulli principle, the other from Newton's "Laws" (if we're calling them laws). Newton was English, was prominent in the late 1600's and early 1700's, and never worked extensively in mainland Europe. Bernoulli was Swiss, was prominent in the mid 1700's, and worked throughout mainland Europe (including Russia) but never in England. During the 1700's and since, England had many wars with various powers from the mainland, from French, Italian, Spanish and German. Generally there is quite a strong division between all countries in Europe. So, is the lack of agreement and bitter dispute about how lift is generated more scientific, or more political? Certainly, the English version of the invention of the Calculus (Newton developed it) was very different from the German version (Leibniz invented it) for many years. Just food for thought. There is always a debate over what is correct, no matter what field you're in, and I think its important to convey that to kids so that they feel okay in disagreeing about something that they learn. Sometimes that debate can be motivated by real concern over what is the truth, and other times it can be motivated by personal ego or national pride.

Contributed by John McArthur


temporal and spatial frequency Edit

  • Frequency is the rate, or how frequent (how often) , an event occurs. The two most common forms of frequency are temporal (in time) and spatial (in space). If something occurs once every second, it has a temporal frequency of 1 Hertz (1 Hz = 1/1 second). For example, if you tap your finger twice every second on the table, the tapping sound you make has a frequency of 2 Hz. By comparison, waves that transmit between the radio station and the radio antenna in a car have temporal frequencies on the order of mega-Hertz (mega=million). In contrast to time, there is also spatial frequency, that is, how often something occurs for a given distance. For example, if you are driving down the highway and notice that there is a light pole for every mile. The spatial frequency of the light pole is then 1/1 mile. If there were two light poles for every mile, then the spatial frequency would be 1/2 mile. The concept of frequency is used in so many scientific and engineering fields. Note that temporal frequency has units of inverse time; spatial frequency has units of inverse distance.

Trivia -

Q: the average rate a human breathes while resting is about 10 breaths per minute. What is the breathing frequency?

A: 10 breaths / minute = 10 breaths / 60 seconds = 1 breaths / 6 seconds = 1/6 Hz.

Q: The human heart beats on average 120 a minute during strenuous activity. What is the frequency of heart beat?

A: 2 Hz.

Q: A farmer plants rows of corn stalks. Every six rows of stalks occupy three meters. What is the spatial frequency of the rows?

A: 6 row of stalks / 3 meters = 2 rows of stalks / 1 meter. Spatial frequency is 2 inverse meters.

Contributed by Houchun Harry Hu


Contributed by Houchun Harry Hu

Linear, one-to-one mapping Edit

In all forms of science and engineering, a linear relationship is often used to relate one variable or parameter to another.

space vs. frequencyEdit

Let's consider a piano. A typical modern piano has 88 keys. Each key, when struck, hits a string that resonates at a specific frequency, thereby generating a musical chord. The frequency of the chord is a form of temporal frequency. A high pitched chord resonates (vibrates) much faster than a lower pitched chord. A one-to-one relationship exists between each of the keys and their corresponding resonant frequencies. Thus, an expert piano player, blind-folded, can identify which keys are being played solely by listening to the chords that are generated, because he/she memorized the one-to-one mapping between the location of the piano keys and their resonant frequencies.

Trivia -

Q: Can you think of any other examples where space and frequency are related?

A: Ever heard of Roy G. Biv? The letters list in order from lowest to highest frequencies, the principle colors that are visible to the human eye. R=Red, O=Orange, Y=Yellow, G=Green, B=Blue, I=Indigo, V=Violet. The principle colors are characterized by different frequencies of light. Next time you see a rainbow, remember Roy G. Biv ... the order never changes. (Search Roy G. Biv on Google and Wikipedia and you'll find a wealth of examples.)

speed vs. distance vs. timeEdit

We're all familiar with the concept of speed, distance, and time. The faster a car drives, the greater the distance it covers. If a car travels at 60 miles per hour, it will have covered a distance of 60 miles in 1 hour. The relationship between speed and distance is a classic example one-to-one mapping example, and the variable that relates speed to distance is time.

Trivia -

Q: What's the difference between speed and velocity?

A: Consider car A driving eastwards, and car B driving northbound. If both cars are traveling at 60 miles per hour, both car A and car B have the same speed (namely, 60 miles per hour). However, the velocity of car A is 60 miles per hour eastwards; the velocity of the car B is 60 miles per hour northbound. Simply put, velocity is speed with direction. Depending on the location of an observer, the speed of both cars A and B will always be 60 miles per hour, but the velocity of cars A and B will be different.

Contributed by Houchun Harry Hu


Nearly all of us has had a chest X-ray scan at the local clinic or hospital. In the past, films were used to capture the X-ray image. In modern medical imaging, the film has become nearly obsolete (although some doctors still prefer film), and the digital era of computers has taken over. Have you ever wondered how an X-ray image is formed. It's quite simple. Most of us are familiar with the X-ray image of a human chest, where the ribs (bones) are white, against a dark background. Why is this? A new piece of film that has not yet been exposed to X-rays and processed is completely white. It's only upon exposure to X-rays that the film turns black. So hint hint, now do you have a feeling of why bone is white on an X-ray image? Curious, let's take a closer look.

In X-ray imaging, X-rays, a form of electromagnetic radiation, with a frequency of 10^15 to 10^18 Hz, are passed from the X-ray source and through a person's body. The film is essentially the detector. Since a human body is made up of different materials, each with a different density (tissue, bone, air, water), the amount of X-rays that are absorbed (or attenuated) by each material will vary. As a result, although the amount of X-ray entering the person at each position across the body is the same, different tissues and solids in the body will cause different amounts of X-rays to exit the person and strike the film. One term that refers to this effect is attenuation. Some materials attenuate (absorb) more X-rays then others.

Analogy - Think of the scenario where you hold up a flashlight and shine it through a thin piece of paper. As expected, much of the light will pass through the paper. The thin piece of paper has low attenuation. Now try it with papers of different thickness. How about a piece of wood, a brick, or a piece of metal? As the material gets thicker and denser (and more solid, like wood, and metal), less light passes through. Thus, their attenuation is higher.

In X-ray based imaging, a material's ability to attenuate (or block) the passage of X-ray is called the Hounsfield Unit (HU) , named after Sir Godfrey Hounsfield (shared winner of the Nobel Prize in 1979 for Physiology or Medicine with Allan Cormack). The lower the Hounsfield unit, the less the attenuation. The higher the Hounsefield unit, the greater the attenuation.

Bone, since it's a solid, attenuates a great level of X-ray (HU = +400). Thus not a whole lot of X-rays pass through bone, and that's why bone is white on X-ray films. Air has a HU of -1000. Thus nearly all X-rays pass through air (e.g. lung space). That's why the lungs appear black on X-ray films, because exposure to X-rays turns film from white to black.

So here's the bottom line - A X-ray image is essentially a map of HU. At every anatomical location, we get a linear mapping of the X-ray attenuation capability of the body.

Contributed by Houchun Harry Hu


There are two kingdoms connected by a bridge. One kingdom has very little people called centimeters and the other has giant people called kilometers. The bridge is the great equalizer! If little people want to go over to the giant kingdom, 100,000 of them have to go over and become 1 giant kilometer. Contributed by Nan Wang

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