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Waves and Harmonic Motion

Waves and Harmonic Motion. AP Physics M. Blachly. Review: SHO Equation. Consider a SHO with a mass of 14 grams: Positions are given in mm. SHO Equation. Find the following: T k At t = 3, find y E total a. Period of a Harmonic Oscillator.

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Waves and Harmonic Motion

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  1. Waves and Harmonic Motion AP Physics M. Blachly

  2. Review: SHO Equation • Consider a SHO with a mass of 14 grams: • Positions are given in mm

  3. SHO Equation • Find the following: • T • k At t = 3, find • y • Etotal • a

  4. Period of a Harmonic Oscillator • For a system with a mass m, and a linear restoring force characterized by k, the period is:

  5. Pendulum • For small angles, gravity acts as a linear restoring force.

  6. Pendulum

  7. Pendulum • A 500 g. mass is suspended from a 2 m long string. It is then pulled to a position of x=4 cm and released. Write the equation for the position of this pendulum at any time.

  8. Damping • The damped Harmonic Oscillator is a system where energy is not conserved. • Applet: http://www.lon-capa.org/~mmp/applist/damped/d.htm • In the real world, there is always some damping, from friction, air resistance, internal deformations of the medium, etc.

  9. Damped SHO Examples • Car Spring and shocks • Lots of damping is good • Radio receivers • Very little damping is good

  10. Wave Motion • Types of Waves: • Transverse • Longitudinal • Torsional • Characteristics of Waves • Frequency • Period • Amplitude • Wavelength • velocity

  11. Wave Speed

  12. Wave Phenomena • Superposition • Interference • Beats

  13. Transverse: The medium oscillates perpendicular to the direction the wave is moving. • Water (more or less) • Slinky demo • Longitudinal: The medium oscillates in the same direction as the wave is moving • Sound • Slinky demo Types of Waves 8

  14. 5m Slinky TnR Suppose that a longitudinal wave moves along a Slinky at a speed of 5 m/s. Does one coil of the slinky move through a distance of five meters in one second? 1. Yes 2. No 12

  15. Wavelength  Amplitude A A Harmonic Waves y(x,t) = A cos(wt –kx) Wavelength: The distance  between identical points on the wave. Amplitude: The maximum displacement A of a point on the wave. Angular Frequency w: w = 2 p f Wave Number k: k = 2 p / l Recall: f = v / l y x 20

  16. Period: The time T for a point on the wave to undergo one complete oscillation. • Speed: The wave moves one wavelength  in one period T so its speed is v = / T. Period and Velocity 22

  17. correct TnR Suppose a periodic wave moves through some medium. If the period of the wave is increased, what happens to the wavelength of the wave assuming the speed of the wave remains the same? 1. The wavelength increases 2. The wavelength remains the same 3. The wavelength decreases  = v T 26

  18. ACT • The wavelength of microwaves generated by a microwave oven is about 3 cm. At what frequency do these waves cause the water molecules in my burrito to vibrate ? (a) 1 GHz (b) 10 GHz (c) 100 GHz 1 GHz = 109 cycles/sec The speed of light is c = 3x108 m/s 29

  19. H H Makes water molecules rotate O Solution • Recall that v = lf. 1 GHz = 109 cycles/sec The speed of light is c = 3x108 m/s 30

  20. Visible f = 10 GHz “water hole” Water and Waves Absorption coefficientof water as a functionof frequency. 31

  21. Interference and Superposition • When too waves overlap, the amplitudes add. • Constructive: increases amplitude • Destructive: decreases amplitude 34

  22. Reflection • A slinky is connected to a wall at one end. A pulse travels to the right, hits the wall and is reflected back to the left. The reflected wave is A) Inverted B) Upright • Fixed boundary reflected wave is inverted • Free boundary reflected wave upright 37

  23. Standing Waves Fixed Endpoints • Fundamental n=1 • ln = 2L/n • fn = n v / (2L)

  24. Wave Speed

  25. L = l / 2 f1= fundamental frequency (lowest possible) Standing Waves: A guitar’s E-string has a length of 65 cm and is stretched to a tension of 82N. If it vibrates with a fundamental frequency of 329.63 Hz, what is the mass of the string? v2 = F / m m = F / v2 m= F L / v2 = 82 (0.65) / (428.5)2 = 2.9 x 10-4 kg v = l f = 2 (0.65 m) (329.63 s-1) = 428.5 m/s

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