Chapter 16: Oscillations

Transcription

1 Chapter 16: Oscillations Brent Royuk Phys-111 Concordia University Periodic Motion Periodic Motion is any motion that repeats itself. The Period (T) is the time it takes for one complete cycle of motion. What is the period of rotation of the hour hand on a clock? The Frequency is the number of cycles per unit of time. The period of the reoccurrence of Monday is one week. What is Monday s frequency? f = 1 T 2 Simple Harmonic Motion One particular type of periodic motion is SHM. Hooke s Law F = - kx A restoring force A linear restoring force always produces SHM Vocabulary Equilibrium position Periodic Motion vs. SHM 3 Displacement (x), Amplitude (A), Period (T), Frequency (f) 1

2 SHM The Strip Chart sine (or cosine) curve Equation of motion: # x = A cos 2π T t & % ( $ ' What happens at time (t + T)? 4 SHM Generating the sine (or cosine) curve Animation courtesy of Dr. Dan Russell, Kettering University 5 SHM and Circular Motion Casting the Shadow Rewrite : # x = A cos 2π T t & % ( $ ' Use ω = 2πf,so ( ) x = A cos ωt Example: An oscillating mass on a spring has a period of 3.2 s and an amplitude of 2.4 cm. What is the equation of motion? When is the first time the mass is as x = -2.4 cm? 6 2

3 Position, Velocity & Acceleration Given a position graph for SHM, what would corresponding velocity and acceleration graphs look like? 7 Position, Velocity & Acceleration x = A cos ωt ( ) ( ) ( ) v = Aω sin ωt a = Aω 2 cos ωt What is v max? a max? Why? 8 Graph Animation 9 3

4 The Period of a Mass on a Spring [ ( )] m Aω 2 cos ωt F = kx ma = kx = k A cos ωt ω 2 = k m ω = [ ( )] Are vertical springs different than horizontal springs? k m 10 Energy of a Mass on a Spring K = 1 2 mv 2 = 1 m[ Aω sin(ωt) ] 2 = ma2 ω 2 sin 2 (ωt) U = 1 2 kx2 = 1 2 k [ A cos(ωt) ] 2 = 1 2 ka2 cos 2 (ωt) ω 2 = k / m K = 1 2 ka2 sin 2 (ωt) E = U + K E = 1 2 ka2 sin 2 (ωt) ka2 cos 2 (ωt) E = 1 2 ka2 11 Energy of a Mass on a Spring 12 4

5 Example A 240-g object is attached to a spring with k = 140 N/m and is compressed 12 cm and released. As it oscillates, what is its maximum speed? What is its speed and acceleration when it is at a point 6.0 cm to the left of its equilibrium position? What is its period of oscillation? 13 The Pendulum F x = T sin θ = Tx L F y = T cos θ mg cos θ 0 for small angles, so T mg F x mgx L a x = g L x = ω 2 x = ma x ω = g L T = 2π L g 14 The Pendulum = T = Tx L F y = T cosθ F = mg F = mg sinθ sinθ θ for small angles, and s = Lθ, or θ = s L & F = mg sinθ mgθ = mg ) ( + ' L s. * Comparing to a mass on a spring, T = 2π m mg L = 2π L g 15 5

6 The Pendulum So a pendulum exhibits SHM for small angle oscillations. What about the mass of the pendulum? What is the length of a simple pendulum with a period of exactly one second? The Physical Pendulum T = 2π I mgl = 2π L g I ml 2 16 Damped Oscillations 17 Driven Oscillations To increase the amplitude of an oscillation, you must add energy to the system. The timing is important. If you add energy (drive) the system at its natural (resonant) frequency, you can dramatically increase the amplitude of the oscillations. Example: Pushing a child on a swing 18 6

7 Driven Oscillations 19 Waves What is a wave? Examples Water, sound, slinky, ER Transverse vs. Longitudinal 22 Wave Properties The magic of waves. Great distances What are they made of? Wave Anatomy Crest, trough, speed, frequency, wavelength, amplitude, period. The Wave Equation: v = fλ What is the wavelength of a sound wave produced by a violin playing the note A above middle C when the speed of sound is 350 m/s? 23 7

8 The Four Wave Behaviors 1. Reflection Waves bounce off obstacles 2. Refraction Waves bend when entering a new medium at an angle. 3. Diffraction Waves bend around corners and spread out from small openings. 4. Superposition (Interference) Waves pass through each other, and their amplitudes add. 24 Superposition 25 Constructive vs. Destructive Interference 26 8

9 Interference 27 Standing Waves 28 Standing Waves Nodes, antinodes, fundamental, harmonics 29 9

10 Standing Waves f n = nf 1 λ 1 = 2L 30 Example A guitar string 60 cm long vibrates with a standing wave that has three antinodes. (a) Which harmonic is this? (b) What is the wavelength of this wave? (c) If this harmonic is excited with a frequency of 600 Hz, what is the frequency of the fundamental? 31 Drum Modes Animations courtesy of Dr. Dan Russell, Kettering University 33 10

11 Beats Turn signals analogy The beat frequency 35 11