## mathematics

• ### Demo Highlight: Vector Addition

A vector is a mathematical construct that has two traits: a magnitude and a direction. Many common quantities in physics, like velocity and force, are vectors. Adding together two vectors is not as simple as just adding the magnitudes together; because a vector is pointing in a particular direction, you have to add together the components of the vectors in any given direction to find out the final vector’s total magnitude and final direction. For example, if you tell someone to walk three meters east and then four meters north, they are not actually seven meters away from where they started! In physics, we often need to add vector quantities, and we have developed several demonstrations to help model this.

Demonstrations A2-22: Magnetic Vectors and A2-24: Vector Algebra are a popular way to provide visible, manipulable vector models in the classroom. Magnetic vectors of several lengths can be attached to the lecture hall chalkboards, and a projected grid can both serve as length measurement and provide axes. If we rotate the grid, we see that the vectors themselves, and their sum, stay the same even if we’re measuring them on different axes. Demonstration C2-41 presents a physical example of adding vectors together. Two hammers are mounted 90 degrees apart above a ball. If we drop one hammer, it hits the ball and sends it in one direction. If we drop the other hammer, it hits them ball and sends the ball in a direction 90 degrees off from the first. If both hammers strike the ball at the same time and with the same force, the ball moves off faster, and at an angle 45 degrees between the two. One force vector produces an acceleration in the same direction as the force; adding two force vectors gives an acceleration in the direction of the sum of the two forces. You can try this out at home, if you happen to have some balls and mallets and a lot of patience. But if you don’t, or if your family gets upset when you break things, you can try out vector addition with a simulator instead.

This simulator (linked here), developed by Dr. Andrew Duffy of the Boston University physics department, allows you to add vectors together at home without the risk of breaking any windows. The simulator is set up to add direction vectors together, but as we have seen with the model vectors in the classroom, the addition is the same no matter what the units are.

Two sliders let you adjust the length, or magnitude, of each vector. Two more let you adjust the angle each vector makes with the horizontal axis. If you want to add two vectors at right angles, like our demonstration with the hammers does, set one to 0 degrees and one to 90 degrees, then set the two magnitudes equal. You should see a new sum vector that connects the two. On this graph, the vectors are added up tip-to-tail, rather than all starting from the same point like the velocity of a ball does. But as we saw in the photos of the demonstrations above, the addition is the same no matter how we slide them around! Changing the axes doesn't change how the underlying mathematics works.

Now try experimenting – change the magnitude of one vector and see how that change affects the sum. Try changing the angle. See if you can do it in reverse – note what the sum of two vectors is, change one of the vectors to that magnitude, then change the angle to see what angle you need to get the original vector’s magnitude out.

• ### Demonstration Highlight: Diameter and Circumference

Today, Third Month Fourteenth, we celebrate Pi Day. π=3.14…. is a well known fact of science, from our grade school mathematics classes and many dessert-related puns, but what does that actually mean?

Let’s take a look at demonstration A2-11. We see a large metal cylinder mounted on a stand. A bead chain can be wrapped around the cylinder, or pulled off and stretched out straight. When you stretch it out, π is the ratio between its length and the diameter of the cylinder. Technically, π= ½ (c/r), where c is the circumference and r is the radius, radius being the distance from the center to the edge or half the diameter.

Why don’t we define the ratio without the ½? Or why do we use radius rather than diameter? The answer is that using radius makes it easier to generalize to other calculations. If we want to calculate the area, rather than the circumference, we use the square of the radius – which is less annoying to calculate than the square of half the diameter. And even more so when we generalize to three or more dimensions. Ultimately, the factor of 2 falls out from the process of taking derivatives and integrals, just like in elementary calculus.

To see this effect virtually, check out this animation from wikipedia: Pi unrolled. As you can see, if you  have a cylinder 1 unit in diameter, its circumference “unrolls” to be approximately 3.14 units long.

Now, check out the Pi(e) Day events at the Maryland Science Center

• ### Demonstration Highlight: Diffusion Distribution Models

Today we’re taking a look at some models of diffusion: Demonstrations I6-21 and I6-25. These both use the behaviour of ping-pong balls to model the behaviour of molecules in a gas. Each of these models uses ping-pong balls of different colors to represent different molecules in a gas. In I6-21, we have a mechanically shaken chamber divided by a plastic barrier. We can put balls of one color on one side and balls of another color on the other side. When the chamber vibrates, the balls bounce around like the molecules in a gas. When the barrier is removed, the balls begin to drift onto each other’s sides, and soon there is no distinction between the two.

This is also a good example of the principle of entropy – while it is very easy and probable to disorder this system, as the two sets of balls mix together, it is highly improbable (though not impossible, given a small enough number of balls) that all of the balls of each color will suddenly sort themselves out again! Thus, the system tends towards the more disordered state.

In I6-25, we start with columns of balls at the top of an array of pegs. The balls are held in place by a small plastic baffle. When the baffle is removed, the balls fall down through the array, scattering as they go. By the time they reach the bottom, they have spread out into a curve, roughly approximating a proability graph. The columns at the bottom with more balls are the areas more probable for balls to scatter into, and those with few or no balls are less probable. As with I6-21, we can use different colors of balls to show how gases diffuse together over time.

Now, you can try this in class or at home with this simulation from the PhET Collection at the University of Colorado. You can let a small or large number of particles of two different gases diffuse through each other, and watch their behaviour. How do the simulated particles here resemble the model “particles” of our demonstrations? What’s different? How can we explore the differences when talking about the behaviour or real gases?

And explore more such experiments in our Directory of Simulations!

• ### Demonstration Highlight: Fourier Analysis

Joseph Fourier and the Fourier Transform

Joseph Fourier was a French scientist in the late 18th and early 19th centuries. He made important contributions to subjects ranging from algebra to thermodynamics, including early studies on the greenhouse effect on Earth’s climate, but today is best remembered for his discovery that many mathematical functions can be approximated more simply as a sum of basic trigonometric functions (sines and cosines).

This process is particularly useful to us because of the realization that you can analyze the structure of any waveform by breaking it down into a series of sine waves. By doing this, we can represent the wave as a list of simple sines and cosines, and their relative amplitudes and phases. We can build up a complex waveform by taking a single sine wave, then adding harmonics of it (sine waves whose frequency is an integral multiple of the fundamental sine wave) in different amplitudes and different phases.

We can then work with these sine and cosine waves mathematically in order to manipulate the original waveform. This is used in modern technology for many things, from audio equalizers on music players, to cleaning up errors in digital photographs, to analyzing the complex interference patterns from spectroscopy and crystallography used to identify substances in the laboratory.

This all sounds very complex; but the fundamentals of it are quite simple, and you can try it for yourself!

Each of these pairs of images represents a single waveform. In the first picture, we see the full wave. In the second, we see the Fourier Transform of that wave – the spread of sine waves of different frequencies that can be assembled to build that waveform. Each spike in the Fourier Transform graph represents a sine wave; the height of each spike is how large the amplitude of that sine wave should be to make the full wave. When the waveform we put in is just a sine wave itself, of course the Fourier Transform of it is a single line – it’s just that same sine wave again! This more complicated sawtooth wave is made up of many Fourier components – multiple sine waves. As the frequency goes up, the amplitude goes down.

Each of these sine waves is a harmonic of the first one; the frequency of each is two, or three, or four, etc times the frequency of the first, or fundamental, sine wave. That fundamentalhas the same frequency as the original sawtooth wave.

These graphs were all created with an oscilloscope and waveform generator in our facility; check one out here! Match the Wave!

Now try it for yourself! Here are some more waveforms: and some Fourier transforms. Can you guess which Fourier transform came from which wave? Even without a complex electronic synthesizer, you can try this at home with a simulator.

This interactive simulatorin the PhET collection lets you build up waveforms by adding Fourier components: https://phet.colorado.edu/en/simulation/legacy/fourier

And the Falstad collection has another interactive simulator to discover the Fourier components of many different wave forms, and see how the breakdown of components changes when the wave does. You can also turn on the sound generator and compare how different waveforms sound to your ear. Try it out, and see what you can change in a wave to change what you hear – and what you can change and have the wave still sound the same. Can you hear a chance in frequency? A change in phase? http://www.falstad.com/fourier/

Try out both, and see what waves you can build and explore!

• ### STEM News Tip: The Mathematics of a Pandemic

This month in Physics Today, Prof. Alison Hill of Johns Hopkins explores the mathematical underpinnings of the science of epidemics.

Hill discusses the variables that characterize a particular disease, and how these are used to model and predict disease behaviour and determine the right approaches to control and treatment.

• ### Teatime in Physics

A question that came in via Twitter recently is one that comes up a lot this time of year, as we tend to want to spend more and more time curled up with a warm beverage. How does my little round teapot fill up so many cups? And why is the tea in the pot still warm when the tea in my cup has gone cold? The answer comes down to geometry!

Here’s a pretty ordinary sized teapot from the cabinet, and an official UMD Physics mug. We’ve tested it twice today and confirmed: This teapot can fill this mug six times. Sure, the pot is bigger than the mug, but it doesn’t look that much bigger, right?  The teapot can even fill this bigger UMD Physics travel mug four times! How?  The answer is related to what biologists call the Square-Cube Law. As an object grows in size, its volume increases faster than its surface area. If you take a cubical container and double its length, width, and height, multiplying by 2 in each direction, then its surface area is multiplied by 4, the square of 2. But its volume is multiplied by 8, the cube of 2. The exact numbers will change, though, depending on the shape of the container. Every shape has its own relationship among liner size, area, and volume. As it turns out, the most efficient shape, with the highest ratio of volume to surface, is a sphere.

This sounds like just abstruse math, but it actually explains a lot about things we deal with every day, from teapots and fuel tanks to giraffes and polar bears. (OK, maybe not all of us deal with polar bears every day, but it’s good to know about them anyway.)

Here’s an example from the demonstration collection. This round flask and this tall cylinder each hold the same volume of water, 500 milliliters. The cylinder is much longer and narrower than the sphere, so it looks bigger, but it has the same volume!  One thing that makes this interesting is that, having a larger surface area, the cylinder is also heavier. It takes more glass to make a 500mL cylinder than to make a 500mL sphere. That might not matter much for our purposes, when we just want one container to sit on the table, but it can make a big difference in large storage containers, or in places where weight is important, like spacecraft.

This is also why fluids in free fall, like raindrops, form into spheres. The surface tension of the liquid is pulling inwards, compressing the surface to the smallest area for that volume of water: a sphere. On a larger scale, this even happens to big accumulations of rock, pulled in by gravity over a long period of time. We call them planets – and luck for us, they do tend to end up round!

That’s all interesting, but isn’t my tea getting cold after all this?

No, and here’s why: The total amount of heat in the container is proportional to its volume. But the radiation of heat away from the container is proportional to its surface area. So my nearly spherical teapot loses heat a lot more slowly than that tall cylinder does. Plus, because there’s less surface area for the same volume, we can make the walls thicker for the same weight, giving it better insulation.

And that’s where the polar bears come in. (Not literally, polar bears should not drink tea.) Ever wonder why so many animals in warm climates evolved long, lanky builds, while arctic animals tend to be rounder? A lot of it comes down to heat. A round polar bear loses heat a lot more slowly, so they can burn fewer calories to stay warm. That can be important in the long winters when there’s not much to eat. In a hot climate where the bigger problem is staying cool, many animals tend to be thinner. Others find other ways to increase their surface area, like the big ears on an elephant, to radiate heat away faster. There are lots of other factors at play in evolution as well, of course, but heat is always an important one.

This relates to why animals only come in certain sizes, too. If you scale up an ant 100 times in each direction, its mass increases by one million - but the surface area of its legs doesn't, so it can't stand up!

So sit back, make a pot of tea, and curl up with a good book about somewhere warmer. And spring will be here before you know it!

(Note: No tea was harmed in the creation of this blog post. But quite a lot of it was consumed.)