Agenda Today: HW quiz, More simple harmonic motion and waves Thursday: More waves Midterm scores will be posted by Thursday. Chapter 13 Problem 28 Calculate the buoyant force due to the surrounding air on a man weighing 800N. Assume his average density is 1,000 kg/m 3, about the same as water. Density of air: 1.2 kg/m 3 ρ = m/v F B = m liquid g Frequency and Period The frequency of oscillation depends on physical properties of the oscillator; it does not depend on the amplitude of the oscillation. Equations of Motion for SHM: Acceleration is NOT constant! We can t use our equations from kinematics. f = 1 2π k m f = 1 2π g L x(t) = A cos (2π f t) v(t) = -v max sin (2π f t) a(t) = -a max cos (2π f t) v max = 2π f A a max = (2π f) 2 A What if we have friction or drag? This is called a damped oscillation: the amplitude will steadily decrease over time Spring: eventually will resettle back at equilibrium (friction with sliding surface, air resistance, friction at attachments) Pendulum: eventually will stop swinging (air resistance, friction at attachments) Driven Oscillations; Resonance Imagine pushing a child on a swingset, applying force on every swing. What if you push at exactly the right time, with exactly the right force one every swing? This exact right time is the resonant frequency Example: Conservation of energy problem from the third midterm. 1
Energy Changes Mass-on-a-spring: Exchange between kinetic energy (½ mv 2 ) and potential spring energy (½ kx 2 ) Pendulum: Exchange between kinetic energy (½ mv 2 ) and gravitational potential energy (mgh) This provides a faster way to find the position or velocity of the oscillator at a particular time! Example: A 0.5-kg, 0.75-m pendulum is pulled back at an angle of 10º. What are the frequency and period of the pendulum? If it just skims the ground at the bottom of the swing, how much energy does the pendulum have? How fast is it going at the bottom of its swing? Properties of Waves More Wave Properties Amplitude: Distance from midpoint to crest of wave Wavelength is the distance between two wave peaks Frequency is the number of times per second that a wave vibrates up and down Period: Time to complete one vibration wave speed = wavelength x frequency Speed of a Wave v = λƒ = λ/t Is derived from the basic speed equation of distance/time This is a general equation that can be applied to many types of waves For sound in air, v = 343 m/s For light in a vacuum, v = c = 3.0 x 10 8 m/s Types of Waves Transverse Waves: like ocean waves or waves created by shaking a string Longitudinal Waves: compression waves, such as sound 2
Types of Waves Transverse In a transverse wave, each element that is disturbed moves in a direction perpendicular to the wave motion Types of Waves Longitudinal In a longitudinal wave, the elements of the medium undergo displacements parallel to the motion of the wave A longitudinal wave is also called a compression wave What exactly is doing the waving? Many waves require a medium: substance or object to move around Sound waves Water waves Waves in a string Waves in a spring/slinky Waves transmit energy without moving the entire medium through space Agenda Today: Wave speed, sound vs. light, interference and standing waves Tuesday: HW #12 quiz, more superposition and interference, doppler effect Waveform A Picture of a Wave The brown curve is a snapshot of the wave at some instant in time The blue curve is later in time The high points are crests of the wave The low points are troughs of the wave Sinusoidal Waves A wave can take on a variety of shapes, we ll mostly look at sinusoidal waves These waveforms can be described by a sine function that changes in both space and time:!! y(x,t) = Acos 2π x λ + t $ $ # # && " " T %% 3
Longitudinal Wave Represented as a Sine Curve A longitudinal wave can also be represented as a sine curve Compressions correspond to crests and stretches correspond to troughs Also called density waves or pressure waves Waves Lecture-Tutorial Work with a partner or two Read directions and answer all questions carefully. Take time to understand it now! Come to a consensus answer you all agree on before moving on to the next question. If you get stuck, ask another group for help. If you get really stuck, raise your hand and I will come around. Speed of a Wave on a String The speed on a wave stretched under some tension, F F m v = where µ = µ L µ is called the linear density F is the tension force in the string The speed depends only upon the properties of the medium through which the disturbance travels Speed of sound waves Room temperature air allows sound waves to travel at a speed of about 343 m/s. Temperature of air can change the speed of sound! General rule of thumb: sound travels more slowly in cold air, faster in hot air Determine distance to a lightning strike by counting seconds until you hear the thunder (every 5 s ~ 1 mile) Unless otherwise specified, assume room temperature air. Sound waves: Human range of hearing is from about 20-20,000 Hz Frequency pitch Example: A violin string playing an A note vibrates at a frequency of 440 Hz. It has a linear density of µ = 7 x 10-4 kg/m. What is the wave speed if the string is under a tension of 10 N? What is the wavelength in the string? What is the wavelength of the 440 Hz sound wave travelling through the air? 4
Sound vs. Light Sound waves and light waves are not the same thing! You don t actually hear radio waves, you hear sound waves that were picked up by the radio Ears are sensitive to vibrations/ compressions in the air Eyes are sensitive to light waves (but only a very small portion of them ) Wave Model of Light Can characterize light as either a wave or a particle (modern physics says both!) Light is electromagnetic radiation: waves that arise from the Light needs no medium to travel! (Michelson & Morley showed this in 1887, effectively disproving the existence of the luminiferous ether ) The Electromagnetic Spectrum Gamma Rays X-rays Ultraviolet Light Visible Light (ROY G BIV) λ = 400-700nm Infrared Light Microwaves & Radio Waves Interference & Superposition Two traveling waves can meet and pass through each other without being destroyed or even altered Waves obey the Superposition Principle If two or more traveling waves are moving through a medium, the resulting wave is found by adding together the displacements of the individual waves point by point Actually only true for waves with small amplitudes Constructive Interference Two waves, a and b, have the same frequency and amplitude Are in phase The combined wave, c, has the same frequency and a greater amplitude 5
Constructive Interference in a String Two pulses are traveling in opposite directions The net displacement when they overlap is the sum of the displacements of the pulses Note that the pulses are unchanged after the interference Destructive Interference Two waves, a and b, have the same amplitude and frequency They are 180 out of phase When they combine, the waveforms cancel Destructive Interference in a String Two pulses are traveling in opposite directions The net displacement when they overlap is decreased since the displacements of the pulses subtract Note that the pulses are unchanged after the interference Reflection of Waves Fixed End Whenever a traveling wave reaches a boundary, some or all of the wave is reflected When it is reflected from a fixed end, the wave is inverted The shape remains the same Reflected Wave Free End Standing Waves on a String When a traveling wave reaches a boundary, all or part of it is reflected When reflected from a free end, the pulse is not inverted Nodes must occur at the ends of the string because these points are fixed 6
Standing Waves on a String The lowest frequency of vibration (b) is called the fundamental frequency ƒ 1, ƒ 2, ƒ 3 form a harmonic series n F ƒn = nƒ 1 = 2L µ 7