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Vibrations and Waves: Hooke's Law Reviewed

This chapter provides a review of Hooke's Law and its application to vibrations and waves. It covers topics such as sinusoidal oscillation, graphing displacement vs. time, velocity and acceleration vs. time, springs and masses, pendulum, damped oscillations, longitudinal and transverse waves, superposition principle, standing waves, resonances, and loose ends. Examples are included to enhance understanding.

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Vibrations and Waves: Hooke's Law Reviewed

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  1. Chapter 13 Vibrations and Waves

  2. Hooke’s Law Reviewed • When x is positive ,F is negative ; • When at equilibrium (x=0), F = 0 ; • When x is negative ,F is positive ;

  3. Sinusoidal Oscillation Pen traces a sine wave

  4. A T A : amplitude (length, m) T : period (time, s) Graphing x vs. t

  5. Some Vocabulary f = Frequency w = Angular Frequency T = Period A = Amplitude f = phase

  6. 90-degrees changes cosine to sine Phases Phase is related to starting point

  7. x v a Velocity and Acceleration vs. time • Velocity is 90°“out of phase” with x: When x is at max,v is at min .... • Acceleration is 180° “out of phase” with x a = F/m = - (k/m) x

  8. Find vmax with E conservation Find amax using F=ma v and a vs. t

  9. What is w? Requires calculus. Since

  10. Springs and masses

  11. Formula Summary

  12. Example13.1 An block-spring system oscillates with an amplitude of 3.5 cm. If the spring constant is 250 N/m and the block has a mass of 0.50 kg, determine (a) the mechanical energy of the system (b) the maximum speed of the block(c) the maximum acceleration. a) 0.153 J b) 0.783 m/s c) 17.5 m/s2

  13. Example 13.2 A 36-kg block is attached to a spring of constant k=600 N/m. If the block is pulled 3.5 cm away from its equilibrium positions and released from rest at t=0, then at t=0.75 seconds,a) what is the position of the block? b) what is the velocity of the block? a) -3.489 cm b) -1.138 cm/s

  14. Example 13.3 A 36-kg block is attached to a spring of constant k=600 N/m. If the block is pulled 3.5 cm away from its equilibrium position and is pushed so that is has an initial velocity of 5.0 cm/s at t=0, then at t=0.75 seconds,a) what is the position of the block? a) 0.00179 cm

  15. Pendulum Demo

  16. Simple Pendulum Looks like Hooke’s law (k  mg/L)

  17. Simple Pendulum

  18. Simple pendulum Frequency is independent of mass and amplitude! (for small amplitudes)

  19. Example 13.4 A man enters a tall tower, needing to know its height h. He notes that a long pendulum extends from the roof almost to the ground and that its period is 15.5 s. (a) How tall is the tower? (b) If this pendulum is taken to the Moon, where the free-fall acceleration is 1.67 m/s2, what is the period of the pendulum there? a) 59.7 m b) 37.6 s

  20. Damped Oscillations In real systems, friction slows motion

  21. Longitudinal (Compression) Waves Sound waves are longitudinal waves

  22. Compression and Transverse Waves Demo

  23. Transverse Waves Elements move perpendicular to wave motion Elements move parallel to wave motion

  24. wavelength x Snapshot of a Transverse Wave

  25. Snapshot of Longitudinal Wave l y could refer to pressure or density

  26. Fixing x=0, Moving Wave moves to right with velocity v

  27. Moving Wave: Formula Summary

  28. Example 13.5 A wave traveling in the positive x direction has a frequency of f = 25.0 Hz as shown in the figure. Find the a) amplitude b) wavelength c) period d) speed of the wave. a) 9.0 cm b) 20.0 cmc) 0.04 s d) 500 cm/s

  29. Example 13.6 Consider the following expression for a pressure wave,where it is assumed that x is in cm,t is in seconds and P will be given in N/m2. a) What is the amplitude? b) What is the wavelength? c) What is the period? d) What is the velocity of the wave? a) 43.5 N/m2 b) 3.59 cm c) 1.197 s d) -3.0 cm/s

  30. Example 13.7 Which of these waves move in the positive x direction?

  31. Speed of a Wave in a Vibrating String • For different kinds of waves: (e.g. sound) • Always a square root • Numerator related to restoring force • Denominator is some sort of mass density

  32. Example 13.9 A simple pendulum consists of a ball of mass 5.00 kg hanging from a uniform string of mass 0.060 0 kg and length L. If the period of oscillation for the pendulum is 2.00 s, determine the speed of a transverse wave in the string when the pendulum hangs vertically. 28.5 m/s

  33. Superposition Principle Traveling waves can pass through each other without being altered.

  34. Reflection – Fixed End Reflected wave is inverted

  35. Reflection – Free End Reflected pulse not inverted

  36. Standing Waves (Read 14.8) Consider a wave and its reflection:

  37. Standing Waves • Factorizes into x-piece and t-piece • Always ZERO at x=0 or x=ml/2

  38. Resonances Integral number of half wavelengths in length L

  39. Nodes and anti-nodes • A node is a minimum in the pattern • An antinode is a maximum

  40. 2nd harmonic 3rd harmonic Fundamental (n=1) Fundamental, 2nd, 3rd... Harmonics

  41. Example 13.10 A cello string vibrates in its fundamental mode with a frequency of 220 vibrations/s. The vibrating segment is 70.0 cm long and has a mass of 1.20 g. a) Find the tension in the string b) Determine the frequency of the string when it vibrates in three segments. a) 163 Nb) 660 Hz

  42. Loose Ends (Organ pipes open at one end)

  43. Example 13.11 An organ pipe of length 1.5 m is open at one end. What are the lowest two harmonic frequencies? DATA: Speed of sound = 343 m/s 57.2 Hz, 171.5 Hz

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