Oscillations and Waves

Transcription

1 Oscillations and Waves Oscillation: Wave: Examples of oscillations: 1. mass on spring (eg. bungee jumping) 2. pendulum (eg. swing) 3. object bobbing in water (eg. buoy, boat) 4. vibrating cantilever (eg. diving board) 5. earthquake 6. bouncing ball 7. musical instruments (eg. strings, percussion, brass, woodwinds, vocal chords) 8. heartbeat Mean Position (Equilibrium Position) position of object at rest Displacement (x, meters) distance in a particular direction of a particle from its mean position Amplitude (A or x 0, meters) maximum displacement from the mean position Period (T, seconds) time taken for one complete oscillation Frequency (f, Hertz) number of oscillations that take place per unit time Phase Difference difference in phase between the particles of two oscillating systems, measured in radians Relationship between period and frequency: Angular Frequency - 1. A pendulum completes 10 swings in 8.0 seconds. a) Calculate its period. Formula: b) Calculate its frequency. Symbol: Units: c) Calculate its angular frequency. 1

2 Example of an Oscillating System A mass oscillates on a horizontal spring without friction as shown below. At each position, analyze its displacement, velocity and acceleration. Force from the Spring: 1. When is the velocity of the mass at its maximum value? 2. When is the acceleration of the mass at its maximum value? 2

3 The Displacement Function A mass on a spring is allowed to oscillate up and down about its mean position without friction. Two traces of the displacement (x) of the mass versus time (t) are shown. Initial condition: Function: Initial condition: Function: Analyzing the Displacement Function 1. Analyze the displacement function shown at right. a) What is the amplitude? x 0 = b) What is the period? T = c) What is the frequency? f = d) What is the angular frequency? ω = 2. What is the displacement of the mass when: a) t = 1.0 s? b) t = 2.0 s? c) t = 2.5 s? e) Write the displacement function. x =. 3

4 Velocity and Acceleration for Simple Harmonic Motion a) Displacement Function b) Velocity Function c) Acceleration Function Defining Equation for SHM: Negative Sign: Simple Harmonic Motion (SHM) 4

5 1. The graph shown at right shows the displacement of an object in SHM. Use the graph to find the: a) period of oscillation b) amplitude of oscillation c) displacement function Alternate Velocity Function d) maximum velocity e) velocity at 1.3 seconds f) maximum acceleration 2. Use the alternate form of the velocity function to find the velocity of the object at 1.3 s. g) acceleration at 1.3 seconds 5

6 Example of SHM Mass on a Horizontal Spring A mass m oscillates horizontally on a spring without friction, as shown. Is this SHM? Angular frequency, period, and frequency for a mass on a spring 1. A 2.00 kg mass oscillates back and forth 0.500m from its rest position on a horizontal spring whose constant is 40.0 N/m. a) Calculate the angular frequency, period and frequency of this system. b) Write the displacement, velocity and acceleration functions for this system. 6

7 Alternate Forms of the Equations of Motion for SHM 1. Write the equations of motion for the graphs shown below. 2. Write the equations of motion for the graphs shown below. 3. What is the difference between the motions described by the two sets of equations? 4. a) Write the equations of motion for the system whose displacement is shown on the graph at right. b) State two times when the: i) speed is maximum ii) magnitude of the acceleration is maximum. 7

8 Example of SHM Simple Pendulum 1. A mass is allowed to swing freely from the end of a light-weight string. Show that the motion of this simple pendulum is approximately simple harmonic motion. 2. Determine the angular frequency, period and frequency for the pendulum. 3. A 20.0 g pendulum on an 80.0 cm string is pulled back 5.0 cm and then swings. Determine its: a) angular frequency d) maximum velocity b) displacement function e) maximum acceleration c) velocity function 8

9 Energy and Simple Harmonic Motion A mass oscillates back and forth on a spring. Analyze the energy in the system at each location. When the mass is at its mean position... When the mass is at any position... 9

10 1. A 2.00 kg mass is oscillating on a spring and its displacement function is shown. a) At what time(s) does the mass have the most kinetic energy? b) Determine the maximum kinetic energy of the mass. c) At what time(s) does the mass have maximum potential energy? Determine this value. d) What is the total energy of the system at 1.5 seconds? e) Determine the kinetic and potential energy of the system at 1.5 seconds. 10

11 Energy Graphs and SHM Energy-Displacement Functions Energy-Time Functions Note that in simple harmonic motion, the energy of a system is proportional to:

12 Damping Damping in Oscillations Effect of damping: Sketch the displacement function for a system without and with damping. Without Damping With Damping Light damping (under-damping): small resistive force so only a small percentage of energy is removed each cycle period is not affected can take many cycles for oscillations to die out eg. car shock absorbers Degrees of Damping Heavy damping (over-damping): large resistive force can completely prevent any oscillations from taking place takes a long time for object to return to mean position eg.- oscillations in viscous fluid Critical damping: intermediate resistive force so time taken for object to return to mean position is minimum minimal or no overshoot eg. electric meters with pointers, automatic door closers 12

13 Resonance Natural Frequency of Vibration: when a system is displaced from equilibrium and allowed to oscillate freely, it will do so at its natural frequency of vibration Forced Oscillations a system may be forced to oscillate at any given frequency by an outside driving force that is applied to it Resonance Amplitude vs. frequency graph for forced oscillations Factors that affect the frequency response and sharpness of curve: 1) 2) 3) 4) 1. Sketch the frequency response for a lightly damped system whose natural frequency is 20 Hz that experiences forced oscillations. 13

14 Waves Pulse single oscillation or disturbance Continuous traveling wave succession of oscillations (series of periodic pulses) Both pulses and traveling waves: transfer energy though there is no net motion of the medium through which the wave passes. Mechanical Waves: require a medium to transfer energy eg. sound waves, water waves, waves on strings, earthquake waves Electromagnetic Waves: do not require a medium to transfer energy eg. light waves, all EM waves A transverse wave is one in which the direction of the oscillation of the particles of the medium is perpendicular to the direction of travel of the wave (the energy). Examples: light, violin and guitar strings, ropes, earthquake S waves A longitudinal wave is one in which the direction of the oscillation of the particles of the medium is parallel to the direction of travel of the wave (the energy). Example: sound, earthquake P waves Compression: region where particles of medium are close together Rarefaction: region where particles of medium are far apart Note that transverse mechanical waves cannot propagate (travel) through a gas only longitudinal waves can. Displacement (x, meters) distance in a particular direction of a particle from its mean position Amplitude (A or x 0, meters) maximum displacement from the mean position Period (T, seconds) time taken for one complete oscillation - time for one complete wave (cycle) to pass a given point Frequency (f, Hertz) number of oscillations that take place per unit time Wavelength (λ, meters) shortest distance along the wave between two points that are in phase -the distance a complete wave (cycle) travels in one period. Compare the motion of a single particle to the motion of the wave as a whole (the motion of the energy transfer). Particle Speed: Wave Speed: 14

15 1. Motion of the Wave 2. Motion of a Particle Control variable: in one medium - wave speed Wave speed depends on the properties of the medium, not how fast the medium vibrates. To change wave speed, you must change the medium or its properties. Control variable: across a boundary - frequency As a wave crosses a boundary between two different media, the frequency of a wave remains constant not the speed or wavelength. Light: Sound: Waves in Two Dimensions Wavefront line (or arc) joining neighboring points that have the same phase or displacement Ray line indicating direction of wave motion (direction of energy transfer). At great distances, the wavefronts are approximately parallel and are known as plane waves. Rays are perpendicular to wavefronts. Intensity - Formula: NOTE: Symbol: Units: x 10-5 W of sound power pass through each surface as shown. Surface 1 has area 4.0 m 2 and surface 2 is twice as far away from the source. Calculate the sound intensity at each location. 15

16 Reflection and Refraction Sketch the incident and reflected rays as well as the reflected wavefront. Law of Reflection The angle of incidence is equal to the angle of reflection when both angles are measured with respect to the normal line (and the incident ray, reflected ray and normal all lie in the same plane). Mirror Refraction: the change in direction of a wave (due to a change in speed) when it crosses a boundary between two different media at an angle Air to glass: Glass to air: Refractive Index (Index of refraction)(n): ratio of sine of angle of incidence to sine of angle of refraction, for a wave incident from air Snell s Law: the ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant, for a given frequency 16