Demonstration I1-11: Thermal Expansion is perennially one of mour most popular thermodynamics demonstrations. This simple setup has two primary pieces: a solid metal sphere, and a metal plate with a hole in it, each mounted on insulated handles. Under normal circumstances (i.e. at room temperature), the hole is just slightly too small for the ball to fit through it. See what happens in this video.

 All that changes, though, when the plate is heated.

A Puzzling Prediction

 Most students have learned already that metals tend to expand when heated. If using this in class, this demonstration is often best introduced after showing off one or two others that illustrate this (consider I1-15: Pin Breaker and I1-13: Bimetallic Strip for examples). But in this case, though, we have a puzzle: The metal plate has a hole in it. We expect the plate to expand outwards… but what happens to the hole? Does it get bigger, or smaller?

This is a good question to pose to your students before carrying out the demonstration. And don’t just ask the question – encourage them to discuss it amongst themselves, and to lay out not just what they predict will happen, but why. This is a good opportunity to encourage logical reasoning and communication skills, as well as an understanding of the physics involved.

The Hole Story

 Once the plate is heated, the ball moves freely through the hole, as seen in this video. As the plate expanded, the hole in it expanded as well!

What’s happening? The thermal expansion of the plate is linear – the distance between any two points on the plate expands by the same ratio. One way to approach understanding this: Imagine a plate with no hole in it, but with just a circle drawn on the plate. We’d of course expect the circle to expand as the plate does. The distance between the points along the circle expands at the same ratio as the distance between the outer edges of the plate. If we then cut a hole in the expanded plate, this distance wouldn’t change.* The same principle applies if the hole is there all along. The distance between any two points along the edge of the hole – essentially the “inner edge” of the plate – must expand at the same ratio as the rest of the plate. So the hole grows larger, and the plate cannot grow backwards into the hole. This constant ration is called the coefficient of expansion.

*OK, a technicality here: The distance wouldn’t change so long as our cutting tool didn’t bend the plate or make it hotter from friction as it cut!

This demonstration has been very popular in classes like PHYS121, PHYS171, and PHYS260 - anywhere that faculty want to introduce the idea of linear thermal expansion and the coefficient of expansion in an interesting way. It's also a fun and engaging way to drive students to think hard about a physical problem, and to practice discussing their reasoning and understanding the reasoning of others. This is perhaps the most valuable lesson of all.

 Figure 1

Atwood’s Machine is a device initially developed by George Atwood in the 18^th century as a way of testing Newton’s Laws mechanically. We have a three demonstrations of this device, C4-21: Atwood Machine, C4-22: Horizontal Atwood Machine, and C4-23: Atwood Machine with Heavy Pulleywhich can be used in the classroom to measure the acceleration of masses under gravity.

 At its simplest, Atwood’s Machine is a pulley with a string over it, with an object hanging on each end of the string, pulled down by gravity. According to Newton’s Laws, the force on each end of the string is dependent on the mass. In Atwood’s idealized, theoretical mathematical model of the experiment, the pulley and string themselves have no mass, so only the hanging objects’ mass we add affects the force.

So long as the masses are equal, there is the same force pulling down on each object – so there is no net force on the system. So if the two objects start out stationary, they remain stationary, by the principle of inertia. If they start out moving, they continue moving at the same speed (until they run out of string).

If one mass is greater than the other, then there are different forces acting on each end, so there is a net force on the system. This net force will cause the masses to accelerate. But this is where it gets complicated: the force on each end of the string from gravity is proportional to the mass on that end. But because the masses are tied together and transmit force to each other through the tension in the string, the acceleration is proportional to the net force and to the total mass of the system. But since each of the component forces is proportional to that mass, the final acceleration ends up being proportional to the ratio between the difference in masses and the total mass.

You can also use this device another way – by timing how fast the objects accelerate, if you know their masses, then you can measure the force of gravity!

In a real experiment, of course, the string and pulley very much do have mass. Additionally, there is another source of force on the system: the friction in the pulley. This frictional force acts to reduce the speed of the moving masses. Additionally, the force from gravity must move the mass of the rotating pulley as well as the mass of the hanging objects.

For one version of this demonstration (figure 1 above), we carefully chose a pulley with a very low mass, so it behaves very much like the theory predicts.

For the other (figure 2 below), we have a much heavier pulley. It has greater mass, and thus accelerating the hanging objects also means overcoming the pulley’s inertia as well.

 Figure 2

This is an important aspect of the difference between theory, simulation, and experiment, and valuable to talk about in class. In a theoretical model, we can propose things that don’t really exist in the real world – massless strings, frictionless pulleys, objects of constant mass but zero size. We use these theoretical experiments to test out theories mathematically, to see if our theories will give results that make sense, and to plan an experiment.

With a real experiment, there are more variables we have to take into account that can affect the results – friction, inertia, heat, objects that stretch and swing and wear down over time. But it’ is these real-world experiments that let us take that final step and ensure that our theories accurately predict the outcome of real-world physics.

 A simulation is somewhere in between the two. We can simulate a physics experiment inside a computer, as another way of testing our theories. This has its own limitations, since the computer only knows about the laws of physics we program into it – if our initial assumptions are wrong or if we set them up incorrectly, the simulation may give inaccurate results. But the simulation lets us try out many combinations of factors in succession, making tiny changes each time, in ways we might not be able to in the laboratory. It’s much faster to adjust a measurement in a simulation than it is to build a new device on the workbench.

 And, importantly, we can use a simulation to practice an experiment when we can’t get to the laboratory at all!

 This simulation, by Andrew Duffy of Boston University, lets us try out the Atwood Machine at home on our computers, or on the screen in the classroom http://physics.bu.edu/~duffy/HTML5/Atwoods_machine.html 

We can vary the mass of each of the two hanging objects, from 0.1 kilograms (100 grams, about the mass of a small bar of soap) to 2 kilograms (2,000 grams, close to the mass of a 5-pound bag of flour). You can set the masses and then start the simulation running, and it will measure the forces and acceleration as it runs. Try it out and see if you can replicate our demonstration, then experiment and test its limits!

One of our most popular electromagnetism demonstrations is J6-01: Electromagnet With A Bang! We discussed this demonstration in a previous highlight article. But now, you can see it in action in this new video with Landry Horimbere.

A massive block of steel is suspended by an electromagnet, courtesy of a single D-cell flashlight battery. When the switch is flipped to open the circuit, the electromagnet turns off, and the block falls dramatically to the table.

The operation of an electromagnet is based on the discovery that an electrical current generates a magnetic field as it flows through a conductor. By grouping many conductors together in a coil, arranged so that their fields align, we can sum their individual electromagnetic fields into a much stronger one. Thus, we can create a strong electromagnet even from a relatively weak current.

You can also try this out at home and in the classroom with this updated magnet simulator from the PhET collection at the University of Colorado.

The simulator has both permanent magnet and electromagnet options. Flip to the electromagnet tab; you should see, as in the screencap above, a battery connected to a coil, with many magnetic field indicators all around. Controls in the margin let you adjust the number of loops in the coil, and a slider lets you vary the voltage. Both the large magnetic compass and the magnetic field meter can be dragged around the screen to measure at different points. You can also swap the battery out for an AC power supply. Try it out!

One popular demonstration in our collection for introducing concepts of wave optics is M1-11: Laser Interference: Fixed Double Slits.

Collimated light waves come from the laser and pass through a pair of narrow slits in the slide; the light passes through and then projects on the distant screen. But light travels as an electromagnetic wave, so when the light comes out of the two slits, it forms two wavefronts, just like ripples from two stones dropped in a pond. These two wavefronts can interfere with each other, as we can model with this pair of overlapping concentric circles. Where two peaks or two valleys of the wave pattern line up, they add together, interfering constructively; when a peak and a valley overlap, they cancel out, interfering destructively. The same happens with light waves; the light from the two slits overlaps, and creates a pattern of bright spots (constructive interference) and dark spots (destructive interference).

When using this in class, we can adjust the slide to use different sets of slits, with different slit widths and different spacing between the slits. This is a good opportunity to challenge students to predict how changing these two variables will change the resulting interference pattern.

 The spacing between the bright and dark fringes ultimately depends on three things: the distance between the slits and the screen, the wavelength of the light, and the spacing between the two slits. The first is probably obvious from your everyday experience – if you step farther away from the point that light spreads out from, it spreads out more!

 Likewise, if you increase the wavelength, the space between the peaks of the waves gets larger, so it’s not a surprise that the spaces between their overlaps would get bigger, too. The effect of the slit spacing, though, takes a moment to think about. If the slits move closer together, the two wavefronts are more and more similar; so the differences between them, the points where they fully cancel out, are farther apart. So increasing the slit spacing decreases the spacing of the fringes, and decreasing the slit spacing increases the spacing of the fringes.

We can see this modeled in a ripple tank simulation here in the Physlet Physics collection at AAPT’s compadre.org: https://www.compadre.org/Physlets/optics/prob37_7.cfm Use your mouse to measure the positions of the peaks relative to the double slit at the base of the image.

To experiment with this at home, check out this PhET Simulation at the University of Colorado: https://phet.colorado.edu/sims/cheerpj/quantum-wave-interference/latest/quantum-wave-interference.html 

 Use the button on the right of the simulation screen to activate the double slit barrier. You will then be able to simulate precisely this experiment!

Light of a particular frequency is released from a source, passes through a double slit, and then is projected on a virtual “screen.” So you can see both the interference pattern as it is formed through space, and the final pattern that you can detect at a distance.

Carrying out the computations for this simulation is processor intensive, so it may run slowly; and as you can see in the image, the resolution is limited. There are limits to how well software can simulate reality!

But at the same time, we can use the simulator to try things we can’t easily do in the laboratory. The slider at the bottom will let you change the laser’s frequency, variable along the full visible spectrum and a bit beyond (fun challenge: is there anything remarkable about the spectrum selection here?).

The sliders at the right will let you change some of the physical parameters of the barrier. You can modify the width of the slits, or the separation distance between the two slits. The “Vertical position” slider lets you adjust the distance between the slits and the screen.

So here we can see the value of combining both real-world experiments and simulations – each alone is useful in learning, and each has its benefits and limitations, but the combination lets us see things we cannot with either one or the other on its own.

Earlier this summer, we introduced these demonstration highlight of the week articles with the Racing Balls demo. It’s been one of our most popular ever! This week we’re returning to it to check out some new resources.

Thanks to a UMD Teaching Innovation Grant, we’ve created new videos of many demonstrations, including this one, where Eliot Hammer presents the Racing Balls demo!

And Don Lynch has created a new animation of the demonstration as well, seen here:

This lets us see the balls in motion and compare their positions to an energy graph, showing the transition between kinetic and potential energy and how it affects the motion of the balls in the tracks. The balls each begin and end with the same potential and kinetic energy and the same velocity. But for a distance in the middle, the ball on the lowered track exchanges some of its potential energy for more kinetic energy, and a greater velocity. When the track returns to the original height, the ball is back to having the same potential and kinetic energy, and the same velocity it started with. But in the meantime, it has moved farther ahead in space during the time it had greater velocity, and so reaches the end of the track first.

This is a valuable way of showing the relationship between energy and velocity, and between velocity, position, and time. It’s also a good example of how the combination of real-world demonstrations and simulations can enhance our physics teaching.

 Check out additional videos, animations, and simulations in the Tools & Resources menu above!

Ready for a fun kinematics experiment you can experience and learn about from the comfort of home?

Demonstration C2-11: The Racing Balls is one of our most popular kinematics demonstrations, as it presents an interesting conundrum.

We have two long tracks, on which a pair of equal-mass billiard balls can roll with very little friction. At one end is a spring-loaded launcher that can send the balls rolling down the tracks with the same initial speed. One of the tracks remains straight and level all the way to the end. The other slopes down to a lower height, runs level at that height for a short distance, then slopes back up to the original level again.

 The challenge for you all is to predict what will happen when the balls reach the end of the track: will the ball on the straight track reach the end first, will the ball on the dipped track reach the end first, or will the get there at the same time?

 Think about this for a while before you scroll downto see what happens. 

Here’s what happens:When we release the spring, the balls start forward with the same speed. The ball on the rear track, which is straight, travels with this same speed the entire length of the track, until it reaches the end.

But when the ball on the bending track reaches the downward slope, it increases in speed as it goes downhill, as we see in this short video.

While that ball rolls along the flat lower section of track, it does so with this greater speed. So it covers this distance in less time than the ball on the higher track does. When the ball on the bent track rolls back uphill, it slows down to its original speed. Now this ball has undergone two periods of acceleration: a positive acceleration as it gained speed, and a negative acceleration as it lost that speed again. Let’s see that in slow motion in this video clip. 

So the two balls both have the same speed again when they reach the end of the track. But they don’t get there at the same time; the ball from the bent track gets there first!

 The ball on the bent track had a higher speed for a short time, while it was on the lowered section of track. This means that when you take the entire trip into account, that ball had a higher average speed, even though its initial and final speeds were the same.

 For further discussion, consider other ways you could analyze this problem. Can you separate out the force vectors acting on each ball? Can you calculate the energy of the system? Will this help predict the result, or not? This is an important question to look at when studying any problem – deciding which variables matter and need to be measured in order to predict the outcome.

.Welcome back! This week, we’re looking at one of our particularly popular and versatile demonstrations, the Ripple Tank.

 The Physics Demonstration Facility has two versions of this demo, so we can reach as many audiences as possible. We have a table top version for outreach events and local classrooms, and a portable version to reach out-of-building locations. These ripple tank demonstrations can both be used to highlight a variety of wave phenomena.

A properly set up ripple tank with its various accessories, can illustrate many different aspects of the physics of waves – single and double point source circular waves, plane waves, interference, diffraction through openings and around obstacles. With a movable mount and careful planning, it can even show the Doppler effect!

Some of this is hard to do at home, but fortunately, there are options. Simulators exist that can carry out at least some of the experiments you might usually use the ripple tank for. You can take screenshots of them to illustrate lectures, or send the link to students to experiment with at home.

There are several different ripple tank simulators, such as this versatile one from Paul Falstad: http://www.falstad.com/ripple/ 

• When first opened, the simulator defaults to emulating a tank of water with a single oscillating source in it. It is tinted a cerulean color for easy viewing, but can be switched to several different color schemes via a drop-down menu. If you prefer the traditional view, #4 on that menu is a greyscale view that closely approximates the familiar shadow projection of the tabletop ripple tank.

• A checkbox below this allows the simulation to be frozen and restarted; another lets you shift to an angled three-dimensional view that can be more difficult to see on small screens, but can be helpful in clarifying complex wave behaviour.

• Sliders adjacent to this let you vary the frequency of the oscillation, and turn on damping.

• Other sliders let you adjust aspects of the simulation process, changing the speed, brightness, and resolution of the simulation box; these are best left alone unless you are struggling with making it work on a slower computer or are having difficulty clarifying complex wave behaviour at an interface.

• In addition to the preprogrammed oscillators, you can excite the simulated ripple tank manually by clicking on it, just like dipping your finger into the water of a real ripple tank.

• A variety of oscillation sources and tank configurations can be selected from the Example dropdown menu. Some likely to be useful for our purposes include:

1. Single Source and Double Source for circular waves from point sources

2. Plane Wave (which does show edge effects at the sides of the “tank”)

3. Single Slit and Double Slit which show diffraction via a plane wave striking a barrier with one or two holes and producing the expected circular waves and interference pattern

4. Obstacle (with a single source circular wave and a small rectangular barrier)

5. Doppler Effect 1 (with a moving source of circular waves).

• For more complex uses, you can also modify the simulation. By right-clicking in the simulation box, you can place additional sources, barriers, and refracting elements. You can also right-click on existing elements and delete them, allowing you to clear the screen and produce an empty “tank” to create your own experiment in.

• You can also place “probes” that will display the wave pattern at that point in a movable oscilloscope-style box. This can be valuable as a challenge for students, to predict the pattern that a probe would read at a given point, or to construct a simulation to produce a particular result.

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.

Sometimes powerful things come in small packages, and this electromagnet is no exception! It features in two popular demonstrations in our collection, J6-01: Electromagnet with Bang and J6-04: Low-Power High-Force Electromagnet. These two demonstrations are frequently used, separately or together, in a variety of physics classes.They also featured in our popular Physics of Fantastic Worlds program!

This small electromagnet is powered by a single flashlight battery. But it is quite strong. In the first demonstration, we see a heavy block of steel being held up by the electromagnet. When we flip the switch to turn the electromagnet off, though, it falls to the table with a bang.

In the second, the electromagnet and a small steel plate are mounted on handles. If students grab the handles and touch the plate to the magnet, they cling together so tightly that even quite strong people cannot pull them apart. But flip the switch to turn off the electricity, and they fly apart!

But what is an electromagnet, and why does it work? Let’s find out.

The battery produces an electrical potential that causes a current to flow through the wire in the coil when the switch is closed. A current can only flow when the circuit is complete.

Maxwell’s Equations of Electromagnetism tell us that moving electrical charges, such as an electric current, create a magnetic field around it. This magnetic field acts just like the magnetic field of the permanent magnets we’re familiar with, like refrigerator magnets. The strength of the magnetic field is determined by the amount of current passing through an area.

(image credit: Wikimedia user Chetvornohttps://commons.wikimedia.org/wiki/File:Magnetic_field_of_wire_loop.svg)

Here we see a diagram of the magnetic field around a single loop of wire. We can see that the field wraps around the wire, so the direction of the force from the magnetic field will be different depending on where you are around the wire.We can see this field in motion in this animation from Penn State - click here!. See the animation "B Field Lines Due to a Current Loop."

The direction of the field also depends on which way the current flows; try this out in this simulator at JavaLab - click here! 

• Imagine the field around that single loop in the illustrations above turned on its side, lined up with more like it.If you flipped one of the wire loops around, its field would be oriented the other way, leaving a slightly weaker point in the field; but if you flipped all of them at once, the field of the entire coil flips directions. Try this out with the simulator!

• You can also flip the battery aroundin the simulator to change the direction of the currentflowing through the wire.

Compare for yourself: what happens if you change the direction of the wire, or change the direction of the current, or both at once?

The force from a single wire is not very strong, especially with only a small electric current. You could make a stronger electromagnet by having a power source with a higher electrical potential to make a stronger current;but that might not be very practical, and would certainly be more expensive.

But if we have many loops of wire, and line them up so that the fields all are aligned, then the small magnetic force from each wire will add up to a much stronger force. This is how a strong electromagnetlike the one in the photographs aboveis built.

 You can also try this at home; check out instructions to build your own small electromagnet on our outreach page, and try some experiments with it! Get the PDF here! 

Joseph Fourier and the Fourier Transform

Joseph Fourier was a French scientist in the late 18^th and early 19^th 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?

Make Your Own Waves

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!

Our Guitar and Oscilloscope demonstration is a fun hands-on way to show a visualization of a waveform while students also hear it. It illustrates how waves on a string become sound waves in air, and how the decay times of different components of a complex waveform affect its sound over time. You can see it in action in our new demonstration video, starring physics student Alana Dixon.

A guitar produces sound from the vibration of the strings. When you strum or pluck a string, the string vibrates. The frequency of this vibration is determined by the string’s length, its tension, and its weight. The body of the guitar, and the air chamber within, can couple and resonate with these vibrations. The energy therein is passed to the air, creating the sound waves we hear across the room. A pickup has been attached to the guitar; this pickup uses magnetic induction with the strings to detect the vibrations and transform them into an electrical signal. This is then amplified by the amplifier and displayed by the oscilloscope. An oscilloscope displays a changing electrical voltage as a moving point on a graph. It allows us to display visually how the signal changes over time.

When we pluck a string, we can see the resulting sound wave reflected as a waveform trace on the oscilloscope. There is a slight difference, though: the pickup is showing the vibrations from the string, and the vibration of the body as well since it is connected to the body, but this is not always exactly identical to the wave as transmitted through the air! The particular shape of the guitar body and sound hole can emphasize slightly different elements of the sound as they couple with the outside air. But for the purposes of our experiments, it’s close enough.

 The frequency produced by a vibrating string is determined by three factors: the length of the string, the tension in the string, and the linear density of the string. These define boundary conditions for the waveforms. Guitar strings have different linear densities (the weight per unit length) to help them produce a wider range of sounds. Then we adjust the tension in each string to tune the guitar to the exact frequencies we want. You’ll notice, though, that the waveforms we see here are not simple sine waves. The sound of a guitar is a very complex waveform that changes over time. The complexity of the waveform is in part due to the design of the instrument. There are several modes of coupling within the guitar, as energy passes from the vibrating strings to the surface of the guitar (the soundboard), from there to the air inside the body, and between the body and the outside air. And the complex shape of the instrument creates multiple possible resonances at various frequencies. All of these can reinforce certain harmonics of the fundamental wave, and these components add up to form the complex waveform we see on the oscilloscope screen. Additionally, the different components of the wave last different amounts of time. After the string is plucked, its vibration slowly dies down, but the vibrations it has set up within the instrument also last for different amounts of time – each component has its own decay time. As these components change in amplitude, the shape of the overall waveform changes, and that gives the guitar its complex sound and varying sound.

To understand this better, we can examine some simulations of how a string responds to being plucked, and its behaviour over time.

This simulation from Falstad http://www.falstad.com/loadedstring/ lets you pluck a string and see how the resulting wave in the string gradually decays. It will display graphs of both the amplitude and phases of various harmonics that make up the wave. Try adjusting the damping to see how that changes the decay over time!

If you’d like to learn more, check out this breakdown https://www.acs.psu.edu/drussell/Demos/Pluck-Fourier/Pluck-Fourier.html  by Dr. Daniel Russell of Penn State with both graphical and mathematical treatments of the initial conditions of a plucked string and its evolution over time.

Further Reading:

 Fred W. Inman. A Standing-Wave Experiment with a Guitar

The Physics Teacher 44, 465 (2006); https://doi.org/10.1119/1.2353595

Michael C. LoPresto. Experimenting with Guitar Strings

The Physics Teacher 44, 509 (2006); https://doi.org/10.1119/1.2362942

Polievkt Perov, Walter Johnson and Nataliia Perova-Mello. The physics of guitar string vibrations

American Journal of Physics 84, 38 (2016); https://doi.org/10.1119/1.4935088

Michael Sobel. Teaching Resonance and Harmonics with Guitar and Piano

The Physics Teacher 52, 80 (2014); https://doi.org/10.1119/1.4862108

Scott B. Whitfield and Kurt B. Flesch. An experimental analysis of a vibrating guitar string using high-speed photography

American Journal of Physics 82, 102 (2014); https://doi.org/10.1119/1.4832195

Welcome back! Today we’re taking a look at a popular demonstration related to the concept of relativity.

 We’re accustomed to thinking about the motion of a projectile from a perspective outside of its motion, the generally safer option in real life! The PhET collection of physics simulations has a lovely one for seeing how different parameters like mass, gravity, and air resistance affect the motion of a projectile; try it out here: https://phet.colorado.edu/en/simulations/projectile-motion

 When we observe and measure motion, we are inevitably making the measurement against some frame of reference. An inertial reference frame is the technical term for a frame of reference in which an object is observed to have no outside forces acting on it, so that it is moving freely in space. Sometimes we have to go to great lengths to determine what such a frame of reference might be – and in the case of this demonstration, it is literally a metal frame!

In demonstration P1-02 in our collection, two spring-powered cannon have been pointed so that if a projectile came out of either of them and moved in a straight line, the projectile would pass through a hole in a transparent barrier and then land in a sophisticated projectile catchment mechanism, also known as a sock. But of course, if we just launch a ball out of the cannon, that doesn’t happen! As soon as the ball leaves the cannon, it starts to fall due to the acceleration of gravity, following a parabolic path, so it slams into the transparent barrier far below the hole.

But, if we raise up the whole aluminum frame that holds the cannon, barrier, and catchment, and then drop it, we can fire the cannon while the aluminum frame is falling. Now, from the perspective of the frame, there’s no separate acceleration pulling the ball down, because the frame is falling at the same rate that the ball is! So the projectile moves “straight” across the frame, through the hole, and lands in the catchment. Meanwhile, from our own perspective outside the experiment, we see the ball following a parabolic path just like always, while the whole experiment falls down.

Read more here:

• https://lecdem.physics.umd.edu/p/p1/p1-02.html

• https://pubs.aip.org/aapt/ajp/article/57/12/1121/1053337/Nonequivalence-of-a-uniformly-accelerating

• https://pubs.aip.org/aapt/pte/article/16/2/103/267269/An-inertial-box

• https://pubs.aip.org/aapt/ajp/article/48/4/310/1051368/Local-inertial-frame-of-reference-demonstration

• https://pubs.aip.org/aapt/ajp/article/57/3/247/1040722/Motion-in-noninertial-systems-Theory-and

Dropped and Shot Masses: A Kinematics Experiment

Today we’re taking a look at a popular demonstration of how Newton’s Laws of Motion apply to falling objects. This device has a spring-loaded launcher on top of a tall stand. Two identical aluminum cubes are placed on it. The cube to your left is supported by the spring-loaded rod; the cube to your right is resting on the platform in front of the rod. When the spring is released, the rod will abruptly push to the right; it will release the cube on the left, allowing it to drop straight down, while it slams into the cube on the right, launching it out horizontally.

 Consider: Once the two cubes have left the launcher, what forces are acting on them? How will they accelerate?

 Try to predict which of the cubes will reach the ground first. The one the drops straight down, or the one launched out to the side? Or will they both reach the ground at once?

There are two key concepts to remember here: that a force is a vector that acts in a particular direction, and Newton’s principle of inertia: that a mass’s velocity (or lack of velocity!) remains the same until acted upon by an outside force.

 The cube on the right was given an initial velocity in the horizontal direction when it was struck by the rod, but not in the vertical direction. The cube on the left had no initial velocity in either the horizontal or vertical direction.

 The force of gravity is acting equally on the two identical cubes, pulling them down at the same rate. They will have the same vertical acceleration. This has no effect on the right-hand cube’s horizontal movement, and likewise its horizontal movement has no effect on gravity pulling it down.

So as a result, even though one has moved some distance away horizontally and the other has not, their vertical movement is identical, and they strike the ground at the same time! Check it out in this slow-motion video.

Today we’re looking at two demonstrations that are often used, individually or together, to discuss simple harmonic motion. Demonstration G1-11: Comparison of Simple Harmonic Motion and Uniform Circular Motion, is a simple mechanical model with a large rotating arm with a disc mounted on it. As the arm-mounted disc rotates around the center, we can see that its motion describes a circle in space. The arm is linked mechanically to a second disc mounted above, that slides back and forth as the arm rotates. The upper disc keeps pace with the lower disc, and as the arm rotates, the upper disc moves back and forth as though it were mounted on a spring.

Demonstration G1-12: Pendulum and Rotating Ball, lets us see that this is not just a coincidence of the model. A ball is mounted as the bob on a rigid pendulum, while an identical ball is mounted on a rotating platform below. The rotating platform is motorized so that it will spin at a constant speed; the pendulum is of an appropriate length so that the period of the swing is the same as the rotational period of the platform. If you start them moving from the same point at the same time, then you can see that the two balls move in sync. By positioning a bright light in front of the apparatus we can project the shadows of both balls on the wall behind, and we can see that the two balls are executing nearly the same motion.

A ball executing simple harmonic motion – the motion of a pendulum bob – is equivalent to the projection of a ball executing uniform circular motion. This is not just a coincidence of the apparatus, but a fundamental discovery about the mathematics behind repeating motion.

(diagram based on public domain work by Wikimedia user Yohai)

 If we make a graph of the linear position of a point on the rotating disc as a function of time, that graph traces out a repeating curve – a curve we can describe with the cosine function, Acos(ɷt),where A is the radius from the center of the circle to the point andt is time. For those of you who have studied thebehaviour of harmonic oscillators, that function should look familiar – it’s the same way we describe an object oscillating without damping, what’s called simple harmonic motion.ɷ(omega) isthe rate of rotation of the disc, and equivalent tothe angular frequency of the oscillation. And conversely, if you made a graph of the velocity of an oscillating mass against its position,rather than plotting the position or the velocity against time,that graph would also trace out a circle. It’s not just a coincidence, but reality – rotational motion and oscillating motion are fundamentally the same phenomenon from a mathematical perspective, just looked at in different dimensions.

 (PD Animation credits: Wikimedia users Chetvorno & Mazemaster)

Let's try this at home. This simulator, by Andrew Duffy of Boston University, lets us model this behaviour on the screen, and see what happens when we change parameters of the motion. Check it out at http://physics.bu.edu/~duffy/HTML5/SHM_circular_motion.html .

 This simulator lets us view this motion in real time. Press Play and see a point rotating on the disc, while two more masses oscillate on springs vertically and horizontally next to the disc. The graph plots out the vertical motion of both the point on the disc and the vertical oscillator over time. You can click the checkbox at the bottom of the screen to form virtual lines between the masses, to show they’re in sync.

Now try changing the experiment. There are two sliders at the bottom of the simulation. The slider on your left lets you change ɷ –try speeding it up and watch what happens! The slider on your right lets you change the radius of the disc, and thus the amplitude of the oscillation.

 Try it out for yourself! And think about where else you’ve seen graphs like that. There are many other physical phenomena that obey similar mathematics, including all types of waves. What examples can you think of?

The Pencil and Plywood demonstration is one of our most popular and dramatic illustrations of the principles of mechanics. We’ve all seeing experiments with collisions, but this one has some surprising results. 

In this demonstration, we have a long brass tube, and at the end is a chamber with a vise. We put a small piece of plywood in the vise. Then we put a pencil in the far end of the brass tube, and attach a fire extinguisher to it. When we release gas from the fire extinguisher, the force from the gas rapidly accelerates the pencil down the tube. The pencil flies through the tube at high speed, comes out the open end, and slams into the plywood.

When the pencil strikes the plywood, by Newton’s principal of inertia, it tends to keep moving unless an outside force (from the plywood) changes that motion. It’s moving very fast, though, and by the time the plywood has exerted enough force to stop it, the pencil has blasted partway through the wood! This is an exciting way to illustrate forces and inertia, though some have pointed out that it’s also a waste of a perfectly good #3 pencil.

Today, we’re going to take a closer look at the action, with some measurements you can make at home to complete the experiment.

 If you click here, you can see a close-up film of the collision in slow motion. 

This video was filmed at 600 frames per second – each frame of the video represents 1/600th of a second. So if we measure how far the pencil moves on the screen between two frames, we know it moved that far in 1/600th of a second. So if we know how big that distance is in real life, we can calculate how fast the pencil is moving, its velocity.

 Velocity is how fast something moves in a given direction. In this case, everything’s moving in a straight line, so we can find out the velocity by just measuring how far the pencil travels (the distance from the end of the tube to the plywood) and dividing that number by how long it takes to get there. If The distance were ten centimeters (it isn’t) and the pencil took two sixhundredths of a second (two frames of the video) to cross that distance, then the velocity would be (10 cm)/(2/600 sec), or about 3000 centimeters per second! (Of course it isn’t, because those aren’t the real measurements. You need to do those yourself!)

Now, obviously the easy way to know that distance would be to go measure it. But we’re all at home, not in the classroom! But you can estimate the distance be measuring other things. If you have wooden pencils of your own at home, you could measure their length and width, then compare that to the length and width of the pencil on the screen. If you don’t perhaps you can research pencils online and find out how long, wide, and heavy they usually are. Or you could measure how thick the wood is, look up online some of the widths plywood comes in, and use that to estimate how far apart things are. None of these will tell you exactly how big the pencil in the movie is, but you can make an estimate, and educated guess, of how big it is, and from that calculate an estimated velocity.

For advanced students, here’s a second challenge: If you have pencils around the house: weigh some pencils. Pencils are pretty light, and not all the same, but you can again get a good estimate of the mass of a pencil by measuring the mass of a group of several pencils, and dividing by how many pencils you have; this gives you their average mass. Again, if you don’t have the same kind of pencils at home, or a scale, you can do some research online and find out about them. Perhaps people who sell and ship pencils can tell you how many are in a box and how much the box weighs? 

Once you have estimates for both the velocity and the mass of the pencil in the video, now you can calculate two more things: the pencil’s momentum, which is the velocity times the mass of the pencil, and the pencil’s kinetic energy, which is one half of the mass of the pencil times its velocity squared. How much energy does that moving pencil have? Where does the energy go when the pencil stops? The law of conservation of energy says it doesn’t just stop existing, but it might move to somewhere else or take on a different form.

 And the final question of the day: Would you want to do all this math by hand using a pencil that’s sticking through a piece of plywood?

A few articles we ran across this week, two new and one old, have had us thinking about that ever-popular topic for the first week of the semester: Galileo, and the forces acting on objects in free fall.

 One of the basic concepts students struggle with in the early stages of introductory kinematics is the concept of free fall, and how different objects behave when falling. This question takes us back to the classic experiment Galileo may or may not have actually carried out, dropping objects of different masses from the top of a tower and observing that, barring drag, they fall in unison, regardless of their respective masses. Here at the Lecture-Demonstration Facility, we have a variety of demonstrations to help illustrate this concept, many of them quite popular in introductory classes... both because they are helpful illustrations of this important physics concept, and because falling objects make a satisfying bang. If you are teaching this topic here soon, be sure to explore sections C2 and C4 of the demonstrations index to see what we have to offer.

This was brought to mind recently when we came across an article from 2013 about efforts to rescue the endangered Leaning Tower of Pisa. Despite the legend, Galileo probably did not actually drop balls from the top of this tower; it does, however, make an excellent illustration for discussing the problem, and is popular in many textbooks for this reason regardless of historical relevance. The effort to save the tower from finally falling over entirely does itself lead to some interesting physics questions, discussed in the article, and could be interesting to students as an opportunity to talk about issues of force and torque, equilibrium and the center of mass. To explore other aspects of this problem in class, check out demonstrations B1-02 and B1-03.

In a recent paper in the European Journal of Physics Education, Balukovic & Slisko explore some of the potential causes of student confusion around weightlessness in free fall, and ways to address them. They recommend using multiple demonstrations and problem solving to help students engage in active learning around the topic. They also have suggestions on thinking about how we use language itself to talk about physical problems to improve clarity and understanding.

Leaving the Renaissance largely behind, this classic image of Galileo dropping his spheres is cited again in an article posted last week in Physics Today. Discussing recent research by Hebestreit, Novotny, et al., they report on the latest experiments in using optically-trapped nanoparticles as tiny force meters. When the optical trap is turned off, the nanoparticle “falls” or responds to other outside forces. By rapidly turning the trap off and on, they can measure the acceleration of the briefly free-falling particle to a high degree of precision, and can thus potentiallly use it as a measurement tool.

News items like this can be very useful in class to promote student engagement. Helping students see how the basic concepts we're teaching can be tied in to both cutting-edge research and real-world problems helps them both understand the concepts and better value what they learn.

Balukovic, J., & Slisko, J. (2018). Teaching and Learning the Concept of Weightlessness: An Additional Look at Physics Textbooks. European Journal of Physics Education, 9(1), 1-14. DOI: https://doi.org/10.20308/ejpe.v9i1.168

Hebestreit, E., Frimmer, M., Reimann, R., & Novotny, L. (2018). Sensing Static Forces with Free-Falling Nanoparticles. Physical Review Letters, 121(6), 063602. DOI: 10.1103/PhysRevLett.121.063602

Miller, J. (2018) Free-falling nanoparticles help to detect tiny forces. Physics Today.DOI:10.1063/PT.6.1.20180823a

Watt, S. (2013). Propping up the wall: How to rescue a leaning tower. Science in School, 26. 

This week marks the birthday of Émilie du Châtelet, French philosopher and scientist best remembered today for first developing the concepts of kinetic energy and the conservation of energy in physical systems.

Born Gabrielle Émilie Le Tonnelier de Breteuil in Paris in 1706, Émilie was the daughter of prominent couriters. She had an early talent for both languages and mathematics, and was fortunate to have parents who could provide her with tutors in a time when such topics were rarely available to women. At the age of 19 she married the older Marquis du Châtelet; his work kept him away on his travels, and she devoted much of her time to mathematics, philosophy, and the arts.

Émilie du Châtelet’s training in languages enabled to read Isaac Newton’s recently published Principia Mathematica, and she translated it into French (her translation remains the standard French version of the text to this day). In her studies and experiments on falling masses, she extended Newton’s concept of momentum to postulate a separate quantity that was not proportional to velocity, like momentum, but the square of the velocity – which we now know as kinetic energy. From this eventually developed the implication that energy is a constant quantity in a system that could be conserved, though the full mathematical understanding of this had to wait another two hundred years for the work of German mathematician Emmy Noether.

She also studied the physics and chemsitry of compusion, in part in collaboration with the philosopher (and her occasional partner) Voltaire, and published an essay on the topic in 1744.

Émilie du Châtelet was also famous in her day not only as a philosopher and scholar, but as a socialite and patron of the arts, fond of carousing, gambling, and drama. Perhaps some of her spirit lives on in our students today.

Today marks the birthday of Swiss physicist K. Alexander Müller, who shared the 1987 Nobel Prize in Physics with Georg Bednorz for their discovery of the first high temperature superconductor.

 Born on April 20^th, 1927 in Basel, Switzerland, Alex Müller attended the Eidgenössische Technische Hochschule Zürich, the Federal Institute of Technology at Zurich, where he received his PhD in 1957. He worked at a variety of institutions throughout Switzerland, studying various aspects of what we now term Condensed Matter Physics.

 In the 1980s, Müller and Bednorz were working together searching for high temperature superconductors. “High temperature,” in the context of superconductors, can be misleading to newcomers, as they are still very cold!

 Superconductors are materials whose resistance drops to zero at low temperatures. These materials have many fascinating properties – they can transmit electricity with no loss, and they repel all magnetic flux. Generally, a superconductor has a criticaltemperature below which it exhibits superconducting properties; above this temperature it does not, behaving as ordinary materials do. For many superconducting materials, and all of those discovered in the first seventy years of them being studied, this temperature is around ten to twenty Kelvin, a temperature very difficult to achieve, maintain, or work with.

 Müller and Bednorz, however, in 1986 discovered a ceramic compound material, lanthanum barium copper oxide, with a critical temperature of 35 Kelvin. Still very cold, but a definite improvement! More crucially, in addition to showing that higher critical temperatures were possible, they showed that superconductivity could be achieved in ceramics, driving other researchers to investigate similar compounds for this effect. Within a year, other such materials had been discovered, including the now popular yttrium barium copper oxide by Paul Chu of the University of Houston. This new material had a critical temperature of 92 Kelvin!

 92 Kelvin is still almost -300 degrees Fahrenheit below zero, obviously much colder than any temperature found naturally on Earth! But it is much warmer than the early metal superconductors. And crucially, it crosses an important line: 77 Kelvin is the temperature of liquid nitrogen, a refrigerant that is much cheaper and easier to manufacture than the liquid helium used in earlier studies, and vastly easier to work with. Since these newest materials can exhibit superconducting behaviour at liquid nitrogen temperatures, it means we can use them in practical technology and experiment with them more easily… including in classroom demonstrations!

We currently have two demonstrations that use high temperature superconductors, both taking advantage of their effect of excluding magnetic flux. Demonstration I7-21: Superconductor – Magnet Levitation uses a yttrium barium copper oxide (YBCO) disc bathed in a liquid nitrogen bath. When a small permanent magnet is placed on top of the disc, the strong magnetic field is repelled from the superconductor, so strongly that the magnet itself levitates above the disc!

Taking the opposite approach, demonstration I7-23: Magnetic Track and Superconductor, built by our own Don Lynch, consists of an array of powerful neodymium magnets. A puck of high temperature superconducting material wrapped in a Teflon sheath is soaked ahead of time in liquid nitrogen, cooling it down such that it will hold its temperature for a few minutes. The puck is cooled while resting above a small block of magnets. When taken out of its bath and placed on the track, it again holds itself at the same height above the magnets of the track.

Resources:

The Nobel Prize in Physics 1987 at nobelprize.org

J. G. Bednorz and K. A. Müller (1986). "Possible highT_c superconductivity in the Ba−La−Cu−O system". Z. Phys. B. 64 (1): 189–193.

One of the most popular and visually stunning illustrations of buoyancy and relationship between temperature and pressure is the hot air balloon. Some of you may have had a chance to see one recently at our Maryland STEM Festival event, FLIGHT!

 A hot air balloon rises in the air as a result of its buoyancy. As the air is heated, the increased average kinetic energy of the particles in the gas mean its average density is less, and so it rises through the air. In the outdoors, a modern hot air balloon carries its heat source with it, and can keep the air at a constant higher temperature, so the balloon will rise until it reaches equilibrium at an elevation where the density of the outer atmosphere is no longer sufficiently higher than that of the air in the balloon.

 Our demonstration balloon, however, more closely resembles the earliest experimental crewed hot air balloons, which heated the air with a heat source located on the ground (a bonfire then, an electric heat gun for us). So these balloons rise only until they have gone too far from the fixed heat source and the air begins to cool down again, reducing buoyancy until they settle back to the ground – or return to the heat source!

 The earliest records of the development of uncrewed hot air balloons, like ours, go back over 1,000 years in China, and are recorded some other parts of East and Southeast Asia as well. The were used for entertainment purposes and for signaling between distant points. The earliest known crewed hot air balloon experiments currently known date to the eighteenth century in Europe, though others may have occurred earlier elsewhere.

 Simple hot air balloons are easy to make, and are a fun home experiment. Larger demonstration models can be valuable in class to spur discussion of buoyancy and the behaviour of gases, and studying the history of both the technology and the theoretical understanding of their thermodyanmics can be a useful and interesting student project.

This is a simple demonstration, that students can easily try at home. It can also be valuable in class, and is recommended for introductory mechanics lectures. It is perhaps most suited to the second day of discussing center of mass and balance, after students have been introduced to the absic concept of what center of mass is and how it affects the equilibrium position of an object under gravity.

An empty soda can can sit upright on its bottom, or can be laid on its side, but cannot be at rest at any angle between these. However, this can be changed by adding a liquid to the system.

Pour approximately 150ml of water into the can, and then try carefully balancing the can at an angle, as seen in the photo above. (This may require experimenting to find the exact right amount of water for any given can; we recommend doing this in front of the class so they can see the process.)

Ask your students why this should happen? The mass has increased, but why does that change how it balances? The water moves when the can tilts, causing the center of mass to shift – with just the right amount of water, the new center of mass will be above the edge of the can, and so it will balance.

Some cans will tend towards a particular orientation and will roll along the edge to that point; invite students to hypothesize why this is. They may see that the location of the hole and tab in the top of the can affects the equilibrium position – consider how this can be used in class to relate to the concept of symmetry. The can’s behaviour as it reaches equilibrium is a damped harmonic oscillation, and is a good introduction to how fluid action can cause damping from within a system, as well as from an external source.

As you plan for your next class, check it out on our website at B1-18: Center of Mass - Soda Can and Water.