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Exploring Hooke’s Law: Basics of Elasticity and Wave Motion

This guide delves into Hooke's Law, established by Robert Hooke in 1678, which defines the relationship between elastic force and displacement in springs. It explains the spring constant (k), how it measures spring stiffness, and its applications in simple harmonic motion. The text covers wave types, including mechanical and electromagnetic waves, along with definitions for equilibrium, amplitude, period, and frequency. Practical examples illustrate concepts such as pendulum motion and the behavior of springs under various forces, enhancing understanding of vibrations and wave behavior.

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Exploring Hooke’s Law: Basics of Elasticity and Wave Motion

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  1. Waves

  2. Hooke’s Law Robert Hooke 1678 Felastic = k x Felastic – elastic force (Newtons) k – spring constant (N/m) x – displacement (meters)

  3. Spring Constant • Measures the stiffness of the spring • The greater the value of the k means a stiffer spring because a greater force is needed to stretch or compress that spring

  4. Hooke’s Law ex: Fel = k x #1 #2 k = F / x #1 k = 1N / 1 cm K = 1 N /cm #2 k = 2 N/2 cm K = 1 N/cm

  5. Plot Hooke’s Law from the previous information.What is the spring constant?

  6. Spring Constants • Hooke’s Law Problems

  7. Simple Harmonic Motion A vibration about an equilibrium position (in which a restoring force is proportional to displacement from equilibrium) F and a max F and a max v max

  8. The motion of Earth orbiting the sun is periodic. Is this motion simple harmonic? Why or why not? No; because Earth does not oscillate about an equilibrium position

  9. What is equilibrium, amplitude, period and frequency?

  10. Amplitude The maximum displacement from equilibrium

  11. Period The time that it takes a complete cycle to occur Measured in seconds T = 1 / f

  12. frequency The number of cycles or vibrations per unit of time Measured in Hertz ( Hz)= 1 / s f = 1 / T

  13. What is equilibrium, amplitude, period and frequency?

  14. What is the equilibrium of the spring?

  15. Period of a Pendulum or spring Pendulum T = 2 pL / g L = length(m) g= gravity( 10 m/s2) Spring T = 2 p m / k M= mass (kg) k = spring constant(N/m)

  16. Pendulum Ex: You need to know the height of a tower, but darkness obscures the ceiling. You note that a pendulum extending from the ceiling almost touches the floor and that its period is 12 s. How tall is the tower?What is the frequency of the pendulum? T = 2 pL / g (T) 2 = (2 pL / g)2 T 2 = 4 p2L / g L = gT 2 = 10 m/s2(12s)2 4 p2 4 p2 L = 37 m f = 1 / T = 1/ 12 = 0.08 Hz

  17. Spring Ex: A 5 N weight oscillates on a spring that has a displacement of 45 cm. What is the period of the spring?What is the frequency? k = f / x = 5 N / 0.45 m = 11 N / m T = 2pm/k = 2p.5kg/ 11N/m = 1.3 s f = 1 / T = 1 / 1.3 s = 0.77 Hz Pendulum and Spring Problems

  18. Types of Waves • Mechanical – a wave that propagates through a deformable elastic medium (needs a medium to travel) 2. Electromagnetic – does not need a medium to travel

  19. Mechanical Waves Longitudinal waves - Waves move parallel to the wave direction EX: Sound Wave

  20. Longitudinal Waves Anatomy

  21. Longitudinal Waves • Compression- region of high density and pressure • Rarefaction- region of low density and pressure

  22. Electromagnetic Wave Transverse wave - Waves move perpendicular to the wave direction EX: Light Wave

  23. Transverse Waves Anatomy

  24. Pulse Wave Throwing a stone in a pond would be a pulse

  25. Transverse Waves • Crest- top of the wave • Trough- bottom of the wave • Amplitude- half the height of a wave • Wavelength-point on a wave to the samepoint on the consecutive wave

  26. Speed of a Wave Speed of a mechanical wave is constant for any given medium Temperature determines speed V = f l V = velocity (m/s) F = frequency ( Hz) l = wavlength ( m)

  27. Wave Speed EX: A piano string tuned to middle C vibrates with a frequency of 262 Hz. Assuming the speed of sound is 343 m/s, find the wavelength of the sound waves produced by the string. V = f l f = v / l l = 343 m/s = 1.31 262 Hz

  28. Standing Wave

  29. Standing Waves • Standing wave- A wave pattern that results when two waves of the same frequency, wavelength, and amplitude travel in opposite directions and interfere • Node- a point in a standing wave that mainstains zero displacement • Antinode- a point in a standing wave, halfway between two nodes, at which the largest displacement occurs

  30. Harmonics Draw a 1st, 2nd and 3rd Harmonic

  31. Harmonic Pattern

  32. Free and Fixed Reflections

  33. Constructive Interference

  34. Constructive and Destructive Interference

  35. Can you tell where the constructive and destructive interference is in the water?

  36. Diffraction

  37. More Diffraction

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