PHYSICS 149: Lecture 22

Transcription

1 PHYSICS 149: Lecture 22 Chapter 11: Waves 11.1 Waves and Energy Transport 11.2 Transverse and Longitudinal Waves 11.3 Speed of Transverse Waves on a String 11.4 Periodic Waves Lecture 22 Purdue University, Physics 149 1

2 Midterm Exam 2 Wednesday, November 18, 6:30 PM 7:30 PM Place: PHY 223 Chapters 5-8 The exam is closed book. The exam is a multiple-choice test. There will be multiple-choice problems. Each problem is worth 10 points. Note that total possible score for the course is 1,000 points (see the course syllabus) The difficulty level is about the same as the level of CHIP HW. You may make a single crib sheet you may write on both sides of an sheet Lecture 21 Purdue University, Physics 149 2

3 ILQ 1 A bolt requires 110 N m of torque to be unscrewed. If the maximum force you can apply is 198 N, what is the shortest wrench you can use to unscrew the bolt? A) 28 cm B) 56 cm C) 1.8 m D) cm Lecture 22 Purdue University, Physics 149 3

4 ILQ 2 A uniform bar of mass m is supported by a pivot at its top, about which h the bar can swing like a pendulum. If a force F is applied perpendicularly to the lower end of the bar as in the diagram, how big must F be in order to hold the bar in equilibrium at an angle from the vertical? A) 2mg B) mg sinθ C) 2mg sinθ D) (mg/2) sinθ E) (mg/2) cosθθ Lecture 22 Purdue University, Physics 149 4

5 Linear and Angular Linear Angular Displacement x θ Velocity v ω Acceleration a α Inertia m I Kinetic Energy ½ m v 2 ½ I ω 2 Newton s 2 nd Law F = ma τ = Iα Momentum p = mv L = Iω Lecture 22 Purdue University, Physics 149 5

6 What is a Wave A wave is a disturbance that travels away from its source and carries energy. A wave can transmit energy from one point to another without transporting any matter between the two points. Examples: Sound waves (air moves back & forth) Stadium waves (people move up & down) Water waves (water moves up & down) Seismic waves (earth moving up & down) Light waves (what moves??) Lecture 22 Purdue University, Physics 149 6

7 Waves A wave is a disturbance that travels away from its source. Mechanical waves Waves traveling through a material medium Examples include water waves, sound waves, and the seismic waves caused by earthquakes. Particles in the medium oscillate (or vibrate) around their equilibrium position but do not travel. Electromagnetic waves Waves in which the disturbance consists of oscillating electromagnetic ti fields Example: visible light, radio waves, infrared waves, and ultraviolet waves. Lecture 22 Purdue University, Physics 149 7

8 Why are Waves Important How do we transport energy from one place to another through matter or a medium? Electromagnetic waves transport energy (electromagnetic energy in the form of light) from the Sun to the Earth. In wave motion energy is transported, but matter is not. Lecture 22 Purdue University, Physics 149 8

9 Energy Transport A wave can transmit energy from one point to another without transporting any matter between the two points. Lecture 22 Purdue University, Physics 149 9

10 Intensity Average power per unit area carried by the wave past a surface which is perpendicular to the direction of propagation of the wave. P Intensity decreases with distance I = 2 Unit: W/m 2 (recall 1 W = 1J/s) 4π r I A 2 Note: surface area of a sphere = 4πr 2 Lecture 22 Purdue University, Physics

11 Example: Intensity The intensity of sunlight that reaches the Earth s atmosphere is 1400 W/m 2. What is the intensity i of the sunlight that reaches Saturn? Saturn is 9.5 times as far from the Sun as Earth. Average power of the Sun: P = I E (4πr E2 ) Intensity of the sunlight at the Saturn: I S = P / (4πr S2 ) = I 2 )/(4πr 2 E (4πr E2 S2 ) Sun r E r S = I E (r E /r S ) 2 = (1400 W/m 2 ) (1 / 9.5) 2 = 16 W/m 2 Lecture 22 Purdue University, Physics

12 ILQ If the distance to a point source of sound is tripled, by what factor does the intensity of the sound change? A) I far = 9 I near B) I far = 3 I near C) I far = (1/3) I near D) I far = (1/9) I near Lecture 22 Purdue University, Physics

13 Types of Waves Transverse: The medium oscillates perpendicular to the direction the wave is moving. Water (more or less) Slinky energy transport Longitudinal: The medium oscillates in the same direction as the wave is moving. Sound Slinky energy transport Lecture 22 Purdue University, Physics

14 ILQ In a transverse wave, the individual particles of the medium a) move in ellipses. b) move in circles. c) move perpendicularly to the direction of the wave's travel. d) move parallel to the direction of the wave's travel. Lecture 22 Purdue University, Physics

15 ILQ Consider a wave on a string moving to the right, as shown. What is the direction of the velocity of a particle at the point labeled A? a) Right b) Zero c) Left d) Down Lecture 22 Purdue University, Physics

16 Simple Harmonic Motion Vibrations Vocal cords when singing/speaking String/rubber band Simple Harmonic Motion Restoring force proportional to displacement Springs F = -kx Lecture 22 Purdue University, Physics

17 The force exerted by a spring is proportional to the distance the spring is stretched or compressed from its relaxed position. F X = -k x x is the displacement from the relaxed position and k is the constant of proportionality. Hooke s Law relaxed position x=0 F X =0 x > 0 x=0 F X = -kx > 0 x < 0 x=0 F X = - kx < 0 Lecture 22 Purdue University, Physics x x x

18 Energy in SHM A mass is attached to a spring and set to motion. The maximum displacement is x=a ΣW nc = ΔK + ΔU 0 = ΔK K + ΔU U or Energy U+K is constant! t! Energy = ½ k x 2 + ½ m v 2 At maximum displacement x=a, v = 0 Energy = ½ k A At zero displacement x = 0 Energy = 0 + ½ mv 2 m Since Total Energy is same ½ k A 2 = ½ m v 2 m v m = sqrt(k/m) A PE S 0 x=0 Lecture 22 Purdue University, Physics m x x

19 Springs and Simple Harmonic Motion X=0 X=A; v=0; a=-a max X=0; v=-v max ; a=0 X=-A; v=0; a=a max X=0; v=vv max ; a=0 X=A; v=0; a=-a max X=-A X=A Lecture 22 Purdue University, Physics

20 Simple Harmonic Motion What does moving in a circle have to do with moving back & forth in a straight line? x= R cosθ =R cos (ωt) since θ = ω t x R R θ 3 3 x y 0 θ 4 7 -R π 2 π 7 3π 2 Lecture 22 Purdue University, Physics

21 Simple Harmonic Motion x(t) () = [A]cos(ωt) v(t) = -[Aω]sin(ωt) a(t) = -[Aω 2 ]cos(ωt) OR x(t) () = [A]sin(ωt) v(t) = [Aω]cos(ωt) a(t) = -[Aω 2 ]sin(ωt) x max = A v max = Aω a max = Aω 2 Period = T (seconds per cycle) Frequency = f = 1/T (cycles per second) Angular frequency = ω = 2πf = 2π/T For spring: ω 2 = k/m since F = ma = -kx Lecture 22 Purdue University, Physics

22 Period of a Spring Simple Harmonic Oscillator ω = 2 π f = 2 π / T x(t) () = [A] cos(ωt) v(t) = -[Aω] sin(ωt) a(t) () = -[Aω 2 ] cos(ωt) For a Spring F = -kx a max = (k/m) A Aω 2 = (k/m) A ω = sqrt(k/m) ω k = T = 2 π m m k Lecture 22 Purdue University, Physics

23 For small angles Pendulum Motion T = mg T x = -mg (x/l) Note: F proportional to x! Σ F x = m a x -mg g( (x/l) = m a x a x = -(g/l) x T g ω = L 2π L = = 2 π ω g Recall for SHO a = -ω 2 x L ω = sqrt(g/l) T = 2 π sqrt(l/g) Period does not depend on A, or m! x T m mg Lecture 22 Purdue University, Physics

24 Simple Harmonic Motion Occurs when having linear restoring force F= -kx x(t) = [A] cos(ωt) v(t) = -[Aω] sin(ωt) a(t) = -[Aω 2 ]cos(ωt) Springs F = -kx U = ½ k x 2 ω = sqrt(k/m) Pendulum (small oscillations) ω = sqrt(l/g) Lecture 22 Purdue University, Physics

25 Waves on a String velocity = F μ μ = linear mass density = m L F = Force Lecture 22 Purdue University, Physics

26 Velocity of Waves v T = = m/l T μ A spring and slinky are attached and stretched. Compare the speed of the wave pulse in the slinky with the speed of the wave pulse in the spring. A) v slinky > v spring B) v slinky = v spring C) v slinky < v spring Slinky is stretches t more, so it has a smaller mass/length μ. Lecture 22 Purdue University, Physics

27 Harmonic Waves y(x,t) = A sin(ωt kx) A = amplitude ω = angular frequency k = wave number = 2π/λ Lecture 22 Purdue University, Physics

28 Amplitude and Wavelength y(x,t) = A cos(ωt kx) Wavelength: The distance λ between identical points on the wave. Amplitude: The maximum displacement A of a point on the wave. Angular Frequency ω: ω = 2 π f Wave Number k: k = 2 π / λ Recall: f = v/λ λ Amplitude A y Wavelength λ A x Lecture 22 Purdue University, Physics

29 Period and Velocity Period: The time T for a point on the wave to undergo one complete oscillation. Speed: The wave moves one wavelength λ in one period T so its speed is v = λ / T. λf = v wave length x frequency (Hz) = velocity Hz = cycles/sec v= λ = λ f = ( 2 π f ) λ = ω T 2π k Lecture 22 Purdue University, Physics