PIRATES OF THE CARIBBEAN: AT WORLD’S END
SNIPPET LESSON PLAN: HARMONIC MOTION
Subject: Science/Physics – Harmonic Motion, Waves;
Ages: 10+; Late Elementary to High School
Length: Film Clips: Pirates of the Caribbean – At World’s End: three minutes 30 seconds; Tacoma Bridge Collapse Video on YouTube: four minutes 13 seconds. Lesson: Playing the snippets and discussing the examples of harmonic motion described in the Helpful Background section will take about 40 minutes.
This film is available from Amazon.com.
Students will learn about resonance and be able to recognize it in everyday occurrences.
The sequence from the film Pirates of the Caribbean – At World’s End offers a spectacular fictional example of the use of resonance to tip over an entire pirate ship. This allows teachers to introduce or review this concept from physics and relate it to real-life examples, including the Tacoma Bridge Collapse of 1940. Resonance is also an interesting way to introduce harmonic motion and concepts such as amplitude and frequency.
DESCRIPTION OF SNIPPET
Jack Sparrow and the rest of the crew of the ship known as the Black Pearl try to follow a map with cryptic instructions. Jack Sparrow suddenly realizes that they need to tip the ship over. He gets his shipmates to join an effort to rock the ship by running back and forth from port to starboard at the exact regular intervals that amplify the oscillation until the ship capsizes.
A dramatic real-life instance of resonance is the famous Tacoma Bridge collapse of 1940. This film clip on YouTube shows footage from before and during the collapse of the bridge.
Resonance is a phenomenon that can occur in association with any movement that repeats itself at regular intervals. Examples are vibrations, oscillations, waves, and motion in closed trajectories, such as circles. These are all instances of harmonic motion.
Without knowing of the term “resonance” or its underlying physics, people have a sense of what constitutes harmonic motion. When children play on a swing (which is essentially a pendulum with a seat for a child), they instinctively know when to give the swing added momentum by moving their hanging legs forwards or backwards and tipping their bodies in the opposite direction. When a parent or a friend pushes a child on a swing, they also intuitively know that a slight push will have most effect if given immediately after the apogee of the swinging motion.
As the swing moves further, the extremes of its trajectory are further apart. The distance between the extremes of a pendulum is called its amplitude.
But, does a child in a swing or a weight at the end of a pendulum move back and forth at shorter time intervals when given an impulse at the top of the swing, just as the motion reverses and begins to go down? Try it out, either with a real swing in the schoolyard or using any simple “model to scale” pendulum made with a weight attached to the end of a string. Measure how often the child or the weight passes through a given position, for example one extreme of its trajectory. How often the weight at the end of the pendulum (or the child on the swing) passes through this point over a fixed period of time is called the frequency of the oscillation. Comparing the frequency of the swing/pendulum for a “small” oscillation (small amplitude) with that of a “big” oscillation (large amplitude), shows that regardless of the amplitude of the oscillation, the frequency of the pendulum is always the same for a given swing/child or string/weight combination. This is called the natural frequency of the pendulum’s oscillation.
Given a specific swing or length of string on a pendulum, frequency does not vary with changes in the distance covered by the oscillating motion. But if we increase the distance covered by the oscillating motion, the child on a swing or the weight on a pendulum has to travel a longer distance in the same time (the time it takes it to get back to one extreme of the oscillation). In other words, the child/weight will have a larger average speed in each longer oscillation. The speed is always zero at the extremes of the trajectory because the child/weight has to stop and reverse direction. The child/weight achieves the maximum speed in its motion between the extremes when it passes the lowest point of its trajectory. So, it is in this sense, and not in that of the frequency, that a swing, or any sort of pendulum, will go faster when pushed correctly.
To summarize, by giving a pendulum impulses at the right moments, the amplitude of its oscillation and the speed of the dangling object can be increased. We are not able, however, to influence the frequency of its oscillation. This frequency is fixed by other factors, namely by the length of the swing and its mass (weight). This can be shown by trying the experiment with shorter swings or heavier weights.
Now consider “the right moments” to apply the impulses, i.e. the pushes. Watch a parent pushing a child on a swing. The pushing of the swing is a periodic action to which a frequency can be ascribed (“how often” the pushing force is applied). Measure the frequency of the oscillation and also the frequency of the impulses. You will find that both frequencies are equal if in fact the impulses are contributing towards a larger oscillation, that is if pushing and swinging are in resonance.
If one looks closely, resonance is all around us.
1. The sound of the vibrating parts of musical instruments is amplified through resonance. See Resonance and Musical Instruments. Every time a sound is emitted, something vibrates. The oscillating motion of the vibrating objects is transferred to adjacent air molecules, which in turn bump their neighboring molecules into the oscillating motion. A wave is, therefore, a set of molecules oscillating in sequence about their position a rest. This can be simulated with a simple device. See Gummy Bear Wave Machine-2. Air molecules are represented by the Gummi Bears. It can be seen that each oscillating molecule (Gummy Bear) considered individually behaves like a harmonic oscillator. A wave is then formed when all adjacent molecules swing this way in an ordered sequence.
Humans have made use of controlled sound since very early times, developing what is called music. The vibrating part of a musical instrument is sometimes obvious, as in a guitar or a piano, and sometimes less so, as in most wind instruments. In fact, in most of the wind instruments, the vibrating parts are the players’ lips! In any case, if the only sound was made directly by the vibrating element, the sound could be heard only from very short distances. In an auditorium, only the first rows would be able to hear its sound. For the sound to be reasonably loud, instruments have what is known as a resonance chamber, which in fact makes up the main “body” of the instrument. It is so shaped that there is always one direction in which the wave from the sound that is being played gets “trapped” bouncing back and forth in a manner that waves traveling in one direction overlap with those traveling in the opposite direction, forming a resonating standing wave. Watch the following animation to see the resulting standing wave (top) from the combination of two equal waves traveling in opposite directions (middle and bottom) Standing Wave Demo 1.
Feedback frequently occurs when a microphone is placed near a loudspeaker to which it is connected. This is due to resonance. When the microphone picks up the sound, it sends a particular set of sound waves—encoded as electrical signals—to the electronic amplifier. The amplifier, in turn, feeds the loudspeakers with the same set of sound waves, except that their amplitude is larger. If this amplified sound reaches the microphone again while the original sound is still being emitted by its source, it will superimpose itself on the original sound, boosting the very same wavelengths, which will be amplified further. (Note that this usually does not happen with human speech, as a different sound is being pronounced by the time the amplified waves reach the microphone, and no boosting occurs. It is usually a tiny background hum or noise that sets off the feedback.) This becomes a loop in which loudness increases up to the speaker’s limits until someone breaks it by putting a hand over the microphone or disconnects it, or the loudspeakers are turned away from the microphone. See How to Eliminate Feedback.
2. Resonance is also the way microwave ovens work, as the water content of food is heated by making water molecules vibrate at their natural frequency. To understand this we need to bear in mind that the concept of “temperature” is nothing else than a measure of the kinetic energy of the molecules: the energy they have because of their motion. Hot stuff has faster molecules that have more energy than the slow ones of cold stuff. The energy of the kitchen fire when heating up food goes into the motion of the molecules. Molecules can move in many ways and vibrating is one of them. They can vibrate in various natural frequencies, also called modes of oscillation. So, why not push them into vibrating in one of those modes directly, instead of supplying heat and waiting for them to get moving? This is what microwave ovens do: food is irradiated with radio waves whose frequency corresponds to one natural frequency of water molecules. (Water is the main component of organic matter, such as food.) Resonance between those radio waves (microwaves) and the molecules sets them off into oscillation. The energy is transferred from the radiation to the molecules: the food is heated.
3. When an old car starts clattering when reaching a particular speed, this is because the vibrations of the engine get into resonance with the natural frequencies of loose parts.
1. Cue the DVD player on the location of the segment, which runs within chapter 9, from approximately 51:35 to 55:05.
2. Prepare the Tacoma Bridge Collapse video to be played.
3. Review the links referenced in this Guide, pick those that are appropriate for the class and the lesson, and decide when to present them to the class.
STEP BY STEP
1. Introduce vibrations and oscillations at the appropriate level of your students. The Helpful Background section offers a basic introductory explanation that can be adapted to the desired level.
2. In the course of the explanation, try to get students to give examples of harmonic motion and resonance from everyday life. Many of the examples mentioned in the Helpful Background section might come up.
3. Play the segment of Pirates of the Caribbean – At World’s End
4. Ask students if they think that loosening the cannons and other heavy objects inside the Black Pearl was really helpful in getting the ship to gain amplitude in its tipping motion. The correct answer is in the negative. The loose objects, by sliding always towards the lower side of the ship move in the opposite way from Jack Sparrow and the crew. These objects end up in the lower side of the ship precisely when that side is going to move up again, thus adding weight on the wrong side and actually slowing down the back and forth motion. The segment may need to be played again to make this clear.
5. Play the Tacoma Bridge video. Bridge builders always try to build with lighter materials and fewer of them, but in this case they inadvertently built a bridge with precisely a weight and size whose natural frequency matched that of typical wind gusts that often blew along the river crossed by the bridge. Note that the 42 mph wind speed is not a particularly high speed & the wind caused damage because it made the bridge oscillate at one of its natural frequencies.
6. Suggested project: Ask students to find and comment, individually or in groups, on real life instances of resonance, either spontaneous occurrences, like the Tacoma Bridge incident, or induced events, such as the apocryphal story that a soprano singer could break a glass with her voice. Is this really possible? There are numerous video clips showing alleged glass breaking through voice, but many, if not all, may be bogus. It is certainly possible with synthesized sound: watch it at: Breaking a Wine Glass Using Resonance. It is not necessary, though, to break a glass to see resonance in action. Watch a “glass harp” musician play beautifully on this YouTube channel: Glass harp-Toccata and fugue in D minor-Bach-BWV 565.