Transcript a notes
The Frequency and Period of an Oscillator Objectives • Convert from frequency to period, or period to frequency. • Create graphs of position vs. time for an oscillator. • Determine amplitude, period, and frequency from a graph of oscillatory motion. • Investigate the factors that determine the period of a pendulum and a spring/mass system. • Describe the restoring forces of oscillatory motion in various types of media. • Describe the energy transformations of oscillatory motion in various types of media. Physics terms • oscillation • phase • equilibrium • oscillator • amplitude • damping • period • frequency Equations The period of an oscillator is the time to complete one cycle. The frequency of an oscillator is the inverse of its period. NOTES Oscillators An oscillator is a system with motion that repeats in cycles. NOTES Oscillators The graph shows repeated cycles. Period A full cycle is one complete back and forth motion. The period is the time it takes to complete one full cycle. Period T is measured in seconds. NOTES Period What is the period of the following oscillators? 1. Earth in its rotation 86,400 seconds, 24 hours, or 1 day 2. your heartbeat 3. the minute hand on a clock 4. a classroom pendulum Frequency Frequency is how many cycles are completed each second. Frequency f is measured in hertz, or Hz. 1. 100 – 800 Hz NOTES Frequency Frequency is how many cycles are completed in one second. What is the frequency of these oscillators? 1.your heartbeat 2.a fan that rotates 360 times a minute 1.0.5 – 1 per second of 0.5 – 1 Hz 3.the vibration of a guitar string 1.100 – 800 Hz Frequency and period The period of an oscillator is one over its frequency. The frequency of an oscillator is one over its period. Engaging with the concepts Javier is on a swing. His feet brush the ground every 3.0 seconds. What is Javier’s frequency? Frequency Engaging with the concepts Marie has a spring-mass system with a frequency of 4 Hz. What is the system’s period? 4 Period What causes oscillations? Oscillations occur in systems with stable equilibrium. Stable systems have restoring forces that act to return them to the equilibrium position if they are displaced. NOTES What causes oscillations? What provides the restoring force for a simple pendulum? The force of gravity What provides the restoring force for a mass on a spring? The spring force NOTES Amplitude NOTES Amplitude is the maximum displacement from the average. A = 4 meters Period NOTES Period is the time per cycle. T = 9 seconds Frequency Frequency is the number of cycles in 1 second. f = 0.11 Hz NOTES Assessment 1. Determine the amplitude, period, and frequency from the graph. Assessment 2. An object has a frequency of 50 Hz. What is the period? 3. A spring mass system moves from one extreme of its motion to the other once every second. What is the frequency of the system? A. 0.2 Hz B. 0.5 Hz C. 2 Hz D. 5 Hz Oscillators Many real physical systems oscillate when they are disturbed. Examples: • musical instruments • geological formations in an earthquake • wind-driven skyscrapers • atoms in a solid Equilibrium and amplitude Most oscillators have a resting state or equilibrium position. The amplitude is the maximum distance x that the mass is displaced from equilibrium during the oscillation. NOTES Amplitude For a vertical spring, equilibrium is where the mass hangs at rest (with spring stretch x0 ). The amplitude is the maximum distance x that the mass is displaced from equilibrium during the oscillation. NOTES Units of amplitude NOTES For a spring and mass, or a pendulum, the amplitude is measured in meters (or centimeters). With other types of oscillators, the units for amplitude might be a voltage or a pressure. A Equilibrium A Amplitude Force and position NOTES A mass attached to a spring oscillates on a frictionless surface. • At equilibrium, the restoring force is zero. Fs • When the mass is displaced to the right, the restoring force points left. • When the mass is displaced to the left, the restoring force points right. Fs The restoring force For a vertical spring and mass system . . . • When the mass is above equilibrium, the restoring force points down. • At equilibrium, the net force is zero. • When the mass is below equilibrium, the restoring force points up. NOTES Energy in an oscillator Any force that disturbs the system adds energy. This added energy is what causes oscillations. The energy oscillates between different forms. • For pendulums, the energy oscillates between gravitational potential energy and kinetic energy. • In spring and mass systems, the energy oscillates between elastic potential energy and kinetic energy. NOTES The spring and mass oscillator At which position(s) is this block moving the fastest? A At which position(s) is it at rest? Where does it have maximum kinetic energy? Maximum elastic potential energy? B C The role of inertia As the oscillator passes through When the equilibrium, the restoring force is zero but the mass keeps moving. WHY? Factors that affect frequency Which of these changes will affect the frequency of a mass on a spring? • changing the mass? • changing the spring constant? • changing the amplitude? Observations • When mass increases, frequency decreases. • When the spring constant increases (stiffer spring with higher k), frequency increases. • The amplitude does not affect the frequency. NOTES Friction and damping Ideal oscillators continue oscillating forever. In real oscillators, friction gradually converts some mechanical energy to heat. This is called damping. NOTES Damping Sometimes engineers don’t want springs to keep oscillating. They employ damping to remove the oscillatory energy from the spring. Shock absorbers in cars are a good example: they are designed to damp out oscillations and produce a smooth ride. The pendulum NOTES For a pendulum, the equilibrium point is the center of its swing, where it hangs at rest. The amplitude A is the maximum distance it is displaced sideways from equilibrium. The period T is the time it takes to complete one full cycle back and forth. A The pendulum At which position is the bob moving the fastest? Where is it at rest? Where does it have maximum kinetic energy? Where does it have maximum potential energy? A B C Restoring force NOTES When a pendulum is displaced, a restoring force brings it back to equilibrium. • When displaced to the right, the restoring force (a component of the weight) points left. • At equilibrium the restoring force shrinks to zero. • When displaced to the left, the restoring force points right. T T T mgy mgx mgx mg mgy The period of a pendulum Review: Which of these variables affects the period of a pendulum? • mass of the bob? • amplitude of the oscillation? • length of the string? Observations • The period does not depend on mass. • The period does not depend on amplitude. • The period does depend on the string length. A longer string gives a longer period. NOTES Assessment 1. This is the position vs. time graph for a harmonic oscillator. a. How many cycles occur in 10 seconds? b. What is the amplitude of the motion? c. What are the period and frequency of the motion? Assessment 2. Draw free-body diagrams for this oscillator at each position shown. A B C Assessment 3. Describe the forms of energy present in this oscillator at each of the three positions shown. A B C Assessment 4. A mass and spring oscillator is pulled back and released. As it passes the equilibrium position, the net force on it is zero. Why does it continue to move past this point, if the net force on it is zero? Assessment 5. The position vs. time graph for a simple pendulum is shown. a. At what times is its velocity a maximum? a. At what times is its velocity zero? a. When its velocity is zero, is its displacement zero, or a maximum?