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Fundamentals of Physics

Fundamentals of Physics. Chapter 13 Oscillations Oscillations Simple Harmonic Motion Velocity of SHM Acceleration of SHM The Force Law for SHM Energy in SHM An Angular Simple Harmonic Oscillator Pendulums The Simple Pendulum The Physical Pendulum Measuring “g”

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Fundamentals of Physics

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  1. Fundamentals of Physics • Chapter 13 Oscillations • Oscillations • Simple Harmonic Motion • Velocity of SHM • Acceleration of SHM • The Force Law for SHM • Energy in SHM • An Angular Simple Harmonic Oscillator • Pendulums • The Simple Pendulum • The Physical Pendulum • Measuring “g” • SHM & Uniform Circular Motion • Damped SHM • Forced Oscillations & Resonance Fundamentals of Physics

  2. Oscillations • Oscillations - motions that repeat themselves. Oscillation occurs when a system is disturbed from a position of stable equilibrium. • Clock pendulums swing • Boats bob up and down • Guitar strings vibrate • Diaphragms in speakers • Quartz crystals in watches • Air molecules • Electrons • Etc. Fundamentals of Physics

  3. Oscillations • Oscillations - motions that repeat themselves. Fundamentals of Physics

  4. Simple Harmonic Motion • Harmonic Motion - repeats itself at regular intervals (periodic). • Frequency - # of oscillations per second • 1 oscillation / s = 1 hertz (Hz) • Period - time for one complete oscillation (one cycle) T T Fundamentals of Physics

  5. Simple Harmonic Motion Position Time Fundamentals of Physics

  6. Simple Harmonic Motion Angles are in radians. Fundamentals of Physics

  7. Amplitude, Frequency & Phase The frequency of SHM is independent of the amplitude. Fundamentals of Physics

  8. Velocity & Acceleration of SHM The phase of v(t) is shifted ¼ period relative to x(t), In SHM, a(t) is proportional to x(t) but opposite in sign. Fundamentals of Physics

  9. The Force Law for SHM • Simple Harmonic Motion is the motion executed by a particle of mass m subject to a force proportional to the displacement of the particle but opposite in sign. Hooke’s Law: “Linear Oscillator”: F ~ - x SimpleHarmonicMotion/HorizSpring.html Fundamentals of Physics

  10. The Differential Equation that Describes SHM • Simple Harmonic Motion is the motion executed by a particle of mass m subject to a force proportional to the displacement of the particle but opposite in sign. Hooke’s Law! Newton’s 2nd Law: The general solution of this differential equation is: Fundamentals of Physics

  11. What is the frequency? k = 7580 N/m m = 0.245 kg f = ? Fundamentals of Physics

  12. xm without m falling off? m = 1.0 kg M = 10 kg k = 200 N/m ms = 0.40 Maximum xm without slipping Fundamentals of Physics

  13. Simple Harmonic Motion SimpleHarmonicMotion/HorizSpring.html Fundamentals of Physics

  14. Vertical Spring Oscillations Fundamentals of Physics

  15. Energy in Simple Harmonic Motion Fundamentals of Physics

  16. range of motion Energy in Simple Harmonic Motion turning point turning point Fundamentals of Physics

  17. Gravitational Pendulum Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L. SimpleHarmonicMotion/pendulum Fundamentals of Physics

  18. SHM for small q Simple Pendulum Simple Pendulum: a bob of mass m hung on an unstretchable massless string of length L. acceleration ~ - displacement SHM Fundamentals of Physics

  19. A pendulum leaving a trail of ink: Fundamentals of Physics

  20. Physical Pendulum A rigid body pivoted about a point other than its center of mass (com). SHM for small q Pivot Point acceleration ~ - displacement SHM Center of Mass quick method to measure g Fundamentals of Physics

  21. Angular Simple Harmonic Oscillator Torsion Pendulum: t ~ q Hooke’s Law Spring: Fundamentals of Physics

  22. Simple Harmonic Motion Any Oscillating System: “inertia” versus “springiness” Fundamentals of Physics

  23. SHM & Uniform Circular Motion The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM. The execution of uniform circular motion describes SHM. http://positron.ps.uci.edu/~dkirkby/music/html/demos/SimpleHarmonicMotion/Circular.html Fundamentals of Physics

  24. SHM & Uniform Circular Motion The reference point P’ moves on a circle of radius xm. The projection of xm on a diameter of the circle executes SHM. radius = xm x(t) UC Irvine Physics of Music Simple Harmonic Motion Applet Demonstrations Fundamentals of Physics

  25. x(t) v(t) a(t) SHM & Uniform Circular Motion The reference point P’ moves on a circle of radius xm. The projection of xm on a diameter of the circle executes SHM. radius = xm Fundamentals of Physics

  26. planet earth SHM & Uniform Circular Motion The projection of a point moving in uniform circular motion on a diameter of the circle in which the motion occurs executes SHM. Measurements of the angle between Callisto and Jupiter: Galileo (1610) Fundamentals of Physics

  27. Damped SHM SHM in which each oscillation is reduced by an external force. Restoring Force SHM Damping Force In opposite direction to velocity Does negative work Reduces the mechanical energy Fundamentals of Physics

  28. Damped SHM differential equation Fundamentals of Physics

  29. Damped Oscillations 2nd Order Homogeneous Linear Differential Equation: Eq. 15-41 Solution of Differential Equation: where: b = 0  SHM Fundamentals of Physics

  30. Damped Oscillations “the natural frequency” Exponential solution to the DE Fundamentals of Physics

  31. Auto Shock Absorbers Typical automobile shock absorbers are designed to produce slightly under-damped motion Fundamentals of Physics

  32. Forced Oscillations Each oscillation is driven by an external force to maintain motion in the presence of damping: wd = driving frequency Fundamentals of Physics

  33. Forced Oscillations Each oscillation is driven by an external force to maintain motion in the presence of damping. 2nd Order Inhomogeneous Linear Differential Equation: “the natural frequency” Fundamentals of Physics

  34. Forced Oscillations & Resonance 2nd Order Homogeneous Linear Differential Equation: Steady-State Solution of Differential Equation: where: w = natural frequency wd = driving frequency Fundamentals of Physics

  35. less damping more damping Forced Oscillations & Resonance The natural frequency, w, is the frequency of oscillation when there is no external driving force or damping. w = natural frequency wd = driving frequency When w = wd resonance occurs! Fundamentals of Physics

  36. Oscillations Fundamentals of Physics

  37. Resonance Fundamentals of Physics

  38. Stop the SHM caused by winds on a high-rise building 400 ton weight mounted on a spring on a high floor of the Citicorp building in New York. The weight is forced to oscillate at the same frequency as the building but 180 degrees out of phase. Fundamentals of Physics

  39. Forced Oscillations & Resonance Mechanical Systems e.g. the forced motion of a mass on a spring Electrical Systems e.g. the charge on a capacitor in an LRC circuit Fundamentals of Physics

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