Simple Harmonic Motion (SHM) SIMPLE HARMONIC MOTION AND WAVES - Periodic motion any type of motion that repeats itself in a regular cycle. Ex: a pendulum swinging, a mass bobbing up and down on a spring. - Simple Harmonic Motion any periodic motion that is the result of a force trying to bring an object back to its equilibrium point. A back and forth motion over the same path with equilibrium at its center. - Period (T) the amount of time it takes for one cycle to occur. Units: seconds (s) - Frequency (f) The amount of cycles that occur in one second. Units: Hertz (Hz) - Period and frequency are inverses of each other. - Amplitude the maximum distance from rest position the object reaches. Example Problem 1 A pendulum is observed to complete 23 full cycles in 58 seconds. Determine the period and the frequency of the pendulum. Pendulum - Two forces act on a pendulum: gravity and tension. - Gravity is the restoring force that brings the mass back to its rest position.
T Period Seconds (s) l Length of pendulum Meters (m) g Acceleration due to gravity Meters per second squared (m/s 2 ) Hooke s Law - When a mass bobs on a spring, it is another good example of SHM. The amplitude of oscillation will depend on the spring constant. The spring constant describes how stiff the spring is. - Hooke s Law allows us to determine the amount of force necessary to stretch a spring if we know the distance stretched and the spring constant. F Force Newtons (N) k Spring constant Newtons per meter (N/m) x Distance stretched/compressed Meters (m) Waves - Waves disturbances that carry energy through matter or space. Only energy is transferred, not the matter through which it travels. - Electromagnetic waves can transmit energy through a vacuum Ex: light, radio, infrared - Mechanical Waves Require a medium through which to transmit energy. Cannot travel through a vacuum. Ex: sound Are produced when matter is oscillated/disturbed. Only the energy is transferred, not the matter Conceptual Example 1 In order for John to hear Jill, air molecules must move from the lips of Jill to the ears of John. True or False?
- Wavelength The length of one complete wave. From like position to like position (i.e. crest to crest, etc.) - Crest The high point of a wave. - Trough The low point of a wave. - Amplitude The displacement from the undisturbed medium. Half the vertical distance between crest and trough. Bigger amplitudes = more energy - Frequency The number of complete waves per second. - Period The amount of time for one complete wave to pass. - Velocity How quickly the energy moves - Damping The tendency of a wave to die out. Energy will dissipate over time. - Transverse wave when particles of the medium move perpendicular to the flow of energy. - Longitudinal wave when particles of the medium move parallel to the flow of energy. - Waves travelling through a solid can be either transverse or longitudinal. Waves travelling through a fluid (liquid or gas) are typically longitudinal. - Surface waves waves that form along the surface of the ocean. Waves deep below the surface are longitudinal, but near the surface they are neither longitudinal nor transverse. The H 2O particles undergo circular motion. - Because waves carry energy, the Law of Conservation of Energy tells us that every wave must have a source. Somewhere something had to cause that first particle to oscillate. - Wave Equation relates a wave s velocity to its frequency and wavelength. v Velocity Meters per second (m/s) f Frequency Hertz (Hz) λ Wavelength Meters (m)
Example Problem 2 Ocean waves are observed to travel along the water surface during a developing storm. A Coast Guard weather station observes that there is a vertical distance from high point to low point of 4.6 meters and a horizontal distance of 8.6 meters between adjacent crests. The waves splash into the station once every 6.2 seconds. Determine the frequency and the speed of these waves. Example Problem 3 An automatic focus camera is able to focus on objects by use of an ultrasonic sound wave. The camera sends out sound waves that reflect off distant objects and return to the camera. A sensor detects the time it takes for the waves to return and then determines the distance an object is from the camera. The camera lens then focuses at that distance. If a sound wave (speed = 340 m/s) returns to the camera 0.150 seconds after leaving the camera, then how far away is the object? Interference - Interference when two waves collide, they will affect each other s amplitudes as they cross paths. Can be constructive or destructive. - Constructive Interference when two waves collide and they result in an increase in amplitude. Happens when a crest meets a crest or a trough meets a trough. - Destructive Interference when two waves collide and they result in a decrease in amplitude. Happens when a crest meets a trough. Total destructive interference a crest meets a trough of the same amplitude, resulting in the amplitudes cancelling each other out.
- The Principle of Superposition When 2 waves interfere, the displacement that results in the medium at any location is the sum of the displacements of the 2 waves. In other words: Add the 2 displacements to find your new displacement. Standing Waves - Standing waves occur when a wave reflects back on itself and interferes in such a way that specific points along the medium seem to be standing still. Despite this illusion, energy is still being transmitted. - Nodes points on a standing wave that appear to be standing still. Areas of destructive interference - Antinodes points on a standing wave that undergo maximum displacement. They fluctuate between being crests and troughs. Areas of constructive interference Sound - Sound a pressure wave It is a mechanical and longitudinal wave. These pressure waves are made up of compressions and rarefactions. Compressions areas where the medium is denser. (High pressure) Rarefactions areas where the medium is less dense. (Low pressure)
- As a sound moves through a medium, each particle of that medium vibrates at the same frequency as the energy is passed along. - The eardrum is basically a highly sensitive pressure detector. It detects fluctuations in pressure. Range of human hearing: 20 Hz 20,0000 Hz (20 khz) Infrasound sounds below 20 Hz Ultrasound sounds above 20 khz Animals range of hearing varies. Speed of Sound - Do not confuse speed with frequency. The frequency of a sound does not affect its speed. - The speed of sound will depend on the properties of the medium. Elastic properties properties related to the tendency of a material to maintain its shape and not deform whenever a force/stress is applied to it. Ex: steel has a high elasticity On the molecular level it means that there is a strong attraction between the atoms/molecules within the medium. - Phase of matter has a huge effect on elasticity. For this reason, sound will travel faster depending on the state of matter. Conceptual Example 2 Why can a tremor of the ground from a distant explosion be felt before the sound of the explosion can be heard? - Inertial properties properties related to a material s tendency to resist changes in its state of motion. Ex: mass The greater the inertia of individual particles, the less responsive they are to interact with neighboring particles. Results in a slower wave. - Temperature will also affect the speed of sound within a medium. As temperature rises, so does kinetic energy of particles. This makes it easier for the energy to be passed from particle to particle. v Velocity Meters per second (m/s) T Temperature Degrees Celsius ( o C)
- At normal atm and 20 o C (68 o F), sound travels at 343 m/s (app. 750 mi/h). Example Problem 4 What is the speed of sound when it is 78 o F outside? What is the speed of sound when it is 32 o F outside? - The speed of sound can also be found using the wave equation. CAUTION: this doesn t mean that the speed of sound is dependent on frequency or wavelength. It depends on the medium. An increase in frequency will result in a decrease in wavelength. Applications - Lightning/explosions. Sound travels much slower than light. The speed of light is 3x10 8 m/s You can calculate the distance from a lightning strike or explosion using the time between seeing the flash and hearing the sound. Example Problem 5 You hear the thunder 3s after seeing the lightning strike. Assuming perfect conditions (20 o C), how far away was the lightning strike? - Echoes a sound reflecting back to its source. Ex: to determine the depth of a cave, you can yell and use the time it takes for you to hear your echo. It will take the same amount of time to reach the cave s back wall as it will to reflect back to you. Example Problem 6 You yell into a canyon and the echo comes back 1.4 s later. Assuming perfect conditions, how far away are you from the canyon wall?
Pitch and Intensity - Pitch the sensation of frequency. High pitch = high frequency Low pitch = low frequency - Intensity the amount of energy transported across a portion of the medium in a given amount of time. Intensity is an objective, scientific measurement. Loudness is its subjective counterpart. Generally, louder sounds are more intense. - But since power is energy over time: I Intensity Watts per meter squared (W/m 2 ) P Power Watts (W) A Area Meters squared (m 2 ) - As sound travels, it loses intensity, because as it travels it acts over a larger and larger area. - Threshold of human hearing the faintest sound the average human ear can hear. 1x10-12 W/m 2 The most intense sound we can hear without serious damage is more than 1 billion times that. Because of the large range, we measure intensity on multiples of ten. Logarithmic or decibel scale. 0 db = 1x10-12 W/m 2 Source Intensity (W/m 2 ) Intensity Level (db) # of Times Greater than ToH ToH 1x10-12 0 10 0 Rustling leaves 1x10-11 10 10 1 Whisper 1x10-10 20 10 2 Normal conversation 1x10-6 60 10 6 Busy street corner 1x10-5 70 10 7 Vacuum cleaner 1x10-4 80 10 8 Walkman at maximum 1x10-2 100 10 10 level Front row of rock concert 1x10-1 110 10 11 Threshold of pain 1x10 1 130 10 13 Military jet takeoff 1x10 2 140 10 14 Perforation of eardrum 1x10 4 160 10 16
Resonance - All objects tend to vibrate at a particular frequency or set of frequencies. Natural frequency the frequency/set of frequencies at which objects tend to vibrate at when struck. - Timbre The quality of a sound produced by a vibrating object. Dependent on the natural frequencies of the sound produced by the object. Pure tone tends to vibrate at a single frequency. Rich tone vibrate at a set of natural frequencies that have a whole number mathematical relationship between them. Noise vibrate at a set of multiple frequencies that have no simple mathematical relationship between them. - Changing the speed or wavelength will result in a change in the object s natural frequency. This is the job of a musician - Example: Guitar Can change speed of the wave by changing tension and density of the string. Can change wavelength by pressing on the frets on the neck of the guitar. Both will change frequency. - Example: Wind instruments By changing the length of the air column, you can alter the wavelength and therefore the frequency. - Forced vibration when interconnected objects are forced to vibrate. Ex: guitar box is forced to vibrate by the strings. - Resonance When one object vibrates at the same natural frequency as another, it can cause the second object to vibrate. Usually results in large amplitude vibrations. Doppler Effect - Doppler Effect Perception of wave s frequency and wavelength can change if there is relative motion between the source of the sound and the observer. Ex: an ambulance passing by a parked car - As the source moves towards you, the wavelength is smaller. I.e., higher frequency - As the source moves away from you, the wavelength is longer. I.e., lower frequency - The moving source will always hear the same frequency since it is stationary relative to the source.