Vibrations, Waves and Sound Unit 7: Vibrations, Waves & Sound Chapter 18: Harmonic Motion 18.1 Harmonic Motion 18.2 Graphs of Harmonic Motion 18.3 Properties of Oscillators 18.1 Investigation: Harmonic Motion and the Pendulum Key Question: How do we describe the backand-forth motion of a pendulum? Objectives: Measure the amplitude and period of a pendulum. Describe any oscillator in terms of frequency, period, amplitude, and phase. Learn to read and represent frequency, period, amplitude, and phase on a graph. Harmonic motion A. Linear motion gets us from one place to another. B. Harmonic motion is motion that repeats over and over. Harmonic motion A pendulum is a device that swings back and force. A cycle is one unit of harmonic motion. Oscillators An oscillator is a physical system that has repeating cycles or harmonic motion. Systems that oscillate move back and forth around a center or equilibrium position. The term vibration is another word used for back and forth motion. Systems and oscillations Our solar system is a large oscillator with each planet in harmonic motion around the Sun. Earth is a part of several oscillating systems. Sound oscillations As a speaker cone moves back and forth, it pushes and pulls on air, creating oscillating changes in pressure that we can detect with our ears. Harmonic motion Harmonic motion can be fast or slow, but speed constantly changes during its cycle. We use period and frequency to describe how quickly cycles repeat themselves. The time for one cycle to occur is called a period. Harmonic motion The frequency is the number of complete cycles per second. Frequency and period are inversely related. One cycle per second is called a hertz, abbreviated (Hz). For a radio to play a specific station, the frequency of the oscillator in the radio must match the frequency of the oscillator signal. Calculating frequency The period of an oscillator is 15 minutes. What is the frequency of this oscillator in hertz? 1. Looking for: …the frequency in hertz. 2. Given: …the period (15 min). 3. Relationships: Use conversion factors and formula: ƒ = 1/T. 4. Solution: Convert min. to sec: 15 min x 60 s = 900 s ƒ = 1 cycle = 0.0011 Hz 1 min 900 s Amplitude Amplitude describes the “size” of a cycle. The amplitude is the maximum distance the oscillator moves away from its equilibrium position. Amplitude A pendulum with an amplitude of 20 degrees swings 20 degrees away from the center in either direction. We use the word damping to describe the gradual loss of amplitude of an oscillator such as a pendulum. Damping Friction slows a pendulum down, just as it slows all motion. If you wanted to make a clock with a pendulum, you would have to find a way to keep adding energy (through winding or electricity) to counteract the damping of friction. Unit 7: Vibrations, Waves & Sound Chapter 18: Harmonic Motion 18.1 Harmonic Motion 18.2 Graphs of Harmonic Motion 18.3 Properties of Oscillators 18.2 Investigation: Harmonic Motion Graphs Key Question: How do we make graphs of harmonic motion? Objectives: Construct graphs of harmonic motion. Interpret graphs of harmonic motion to determine phase, amplitude, and period. Use the concept of phase to describe the relationship between two examples of harmonic motion. Graphs of harmonic motion A graph is a good way to show harmonic motion because you can quickly recognize cycles. Graphs of linear motion do not show cycles. Determining period from a graph To find the period from a graph, start by identifying one complete cycle. The cycle must begin and end in the same place in the pattern. Subtract the beginning time from the ending time. Determining amplitude from a graph On a graph of harmonic motion, the amplitude is half the distance between the highest and lowest points on the graph. The amplitude is 20 cm. Phase The phase tells you exactly where an oscillator is in its cycle. Circular motion is a kind of harmonic motion because rotation is a pattern of repeating cycles. Because circular motion always has cycles of 360 degrees, we use degrees to measure phase. Phase and harmonic motion graphs Imagine a peg on a rotating turntable. A bright light casts a shadow of the peg on the wall. As the turntable rotates, the shadow goes back and forth on the wall. If we make a graph of the position of the shadow, we get a harmonic motion graph. Pendulums in phase The concept of phase is most important when comparing two or more oscillators. We say these pendulums are in phase because their cycles are aligned. Pendulums out of phase by 90o Although, they have the same cycle, the first pendulum is always a little bit ahead in its cycle compared to the second pendulum. The pendulums are out of phase by 90 degrees. Pendulums out of phase by 180o Two oscillators that are 180 degrees out of phase are one-half cycle apart. When pendulum number 1 is all the way to the left, pendulum number 2 is all the way to the right. Unit 7: Vibrations, Waves & Sound Chapter 18: Harmonic Motion 18.1 Harmonic Motion 18.2 Graphs of Harmonic Motion 18.3 Properties of Oscillators 18.3 Investigation: Natural Frequency Key Question: What kinds of systems oscillate? Objectives: Build a mechanical oscillator and find its period and natural frequency. Change the natural frequency of an oscillator. Systems and equilibrium Systems that have harmonic motion always move back and forth around a central, or equilibrium position. Restoring force A restoring force is any force that always acts to pull a system back toward equilibrium. Restoring force and inertia A restoring force keeps a pendulum (or child) swinging. The restoring force is related to the weight of the mass and the lift force (or tension) of the string supporting the mass. Harmonic motion and machines Mechanical systems usually do not depend on a restoring force or inertia. The piston in a car engine is harmonic motion, but the motion is caused by the rotation of the crankshaft and the attachment of the connecting rod. Natural frequency and resonance The natural frequency is the frequency (or period) at which a system naturally oscillates. Every system that oscillates has a natural frequency. Natural frequency Every oscillating system has a natural frequency. Microwave ovens, musical instruments, and cell phones are common devices that use the natural frequency of an oscillator. The strings of a guitar are tuned by adjusting the natural frequency of vibrating strings to match musical notes. Natural frequency and resonance You can get a swing moving by pushing it at the right time every cycle. A force that is repeated over and over is called a periodic force. Natural frequency and resonance Resonance happens when a periodic force has the same frequency as the natural frequency. When each push adds to the next one, the amplitude of the motion grows. Jump rope A jump rope depends on resonance. If you shake it at the right frequency, it makes a wave. If the frequency is too fast or too slow, the rope will not make the wave pattern at all. Simple oscillators A mass on a spring is an oscillating system. When the spring is compressed, it pushes the mass back to equilibrium. When the spring is extended, it pulls the mass back toward equilibrium. Simple oscillators A vibrating string oscillator is a rubber band stretched between two rods. If the middle of the rubber band is pulled to the side, it will move back toward equilibrium when it is released. Stretching the rubber band to the side creates a restoring force. When the rubber band is released, inertia carries it past equilibrium and it vibrates. Skyscrapers and Harmonic Motion The John Hancock Tower is one of the tallest skyscrapers in New England. While this skyscraper was being built in 1972 and 1973, a disaster struck— windowpanes started falling out all over the building and crashing to the ground.