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Lecture 2 Harmonic Oscillator Ii LECTURE 2 HARMONIC OSCILLATOR II Instructor: Kazumi Tolich Lecture 2 2 ¨ Reading chapter 14-3 to 14-5 ¤ Energy in simple harmonic motion ¤ Oscillating systems n Vertical spring n Simple pendulum n Torsion pendulum n Physical pendulum ¤ Damped oscillations ¤ Driven oscillations ¤ Resonance Energy in SHM 3 ¨ The potential energy of a mass on a spring: Turning points ¨ The kinetic energy of a mass on a spring: ¨ The total energy in SHM is Quiz: 1 Vertical spring 5 ¨ A vertical spring undergoes simple harmonic motion about the new equilibrium position, �" with an # angular frequency of . $ Example: 1 6 ¨ An object of mass m = 2.0 kg is attached to the top of a vertical spring that is anchored to the floor. The unstressed length of the spring is L = 8.0 cm, and the length of the spring when the object is in equilibrium is L’ = 5.0 cm. When the object is resting at its equilibrium position, it is given a sharp downward blow with a hammer so that its initial speed is vi = 0.30 m/s. a) To what maximum height, h, above the floor does the object eventually rise? b) How long does it take for the object to reach its maximum height for the first time, t? c) Does the spring ever become unstressed? Simple pendulum 7 ¨ The weight of the bob provides the restoring torque. ¨ Simple pendula do not exhibit true simple harmonic motion for any angle. However, if the angle of oscillation is small, the motion is close to and can be modeled as simple harmonic motion. Demo 1 8 ¨ Simple Pendula with Different Lengths and Masses ¤ Demonstration of the relationship between L and T. Quiz: 2 & 3 9 Example 2 10 ¨ If the period of a 70.0-cm-long simple pendulum is 1.68 s, what is the value of g at the location of the pendulum? Torsion pendulum 11 ¨ A simple torsional pendulum consists of a disk with a moment of inertia I suspended from a wire with a torsional constant κ. ¨ The wire provides the restoring torque, and the disk undergoes simple harmonic motion. κ Demo 2 12 ¨ Torsion pendulum ¤ Demonstration of the relationship between I and T. For fun: torsion pendulum at UW 13 ¨ At University of Washington, they have used very precise torsion pendulum to measure gravity at separation of 0.1 mm. Physical pendulum 14 ¨ A physical pendulum is a rigid object free to rotate about a horizontal axis that is not through its center of mass that oscillates when displaced from equilibrium. ¨ Physical pendula do not exhibit true simple harmonic motion for any angle. However, if the angle of oscillation is small, the motion is close to and can be modeled as simple harmonic motion. Demo 3 15 ¨ Physical pendulum ¤ Demonstration of the relationship between T and I/D. Damped oscillations 16 ¨ An oscillation that eventually stops by dissipating its mechanical energy due to frictional forces is said to be “damped.” ¤ Overdamped ¤ Critically damped ¤ Underdamped Overdamped and critically damped motion 17 ¨ A system is overdamped if the damping is too large for the oscillator to complete one cycle of oscillation. The oscillator slowly moves toward the equilibrium position. ¨ Motion with the minimum damping for nonoscillatory motion is said to be critically damped. Underdamped oscillations 18 ¨ The motion of an underdamped oscillator can be described as an oscillation with the amplitude decaying exponentially. ¨ Energy of an oscillator is proportional to A2. So, the energy also decreases exponentially. ¨ Q factor of a weakly damping oscillator can be approximated by Reducing swaying in tall buildings 19 ¨ Dampers are placed near the top of tall buildings. ¨ Dampers include an object of large mass that oscillates under computer control at the same frequency as the building. ¨ The 730-ton suspended sphere below is for Taipei 101 (509.2 m). Driven oscillations 20 ¨ When mechanical energy is put into the system to keep a damped system going, the oscillator is said to be “driven” or “forced.” ¨ The natural frequency or resonant frequency of an oscillator, ω0, is its frequency when no driving or damping forces are present. ¨ When the driving frequency equals the natural frequency, ω = ω0 of the oscillator, the amplitude of the oscillator, A, is maximum, and the system resonates. ¨ Q-factor tells the sharpness of the resonance peak. For a weak damping, Noisy neighbors 21 ¨ Why can you hear bass notes from music played by your neighbors more distinctively than higher pitched notes? ¤ Sound is vibration of air molecules. ¤ The bass tones have lower frequencies than higher pitched notes. ¤ Large structures like walls and ceilings are more easily set into forced vibrations and resonance by bass notes since their natural frequencies are closer to the frequencies of bass notes. ¤ Another factor is that the higher frequency notes transfer sound energy into heat in the walls more rapidly and are thermally “eaten up.” Demo 4 22 ¨ Resonant driven pendula ¤ Demonstration of pendula with different resonance frequencies and various driving frequencies. Examples of resonance 23 ¨ Glass and speaker ¤ http://www.youtube.com/watch?v=17tqXgvCN0E&NR=1 ¨ London Millennium Bridge ¤ http://www.youtube.com/watch?v=eAXVa__XWZ8 ¤ The problem was fixed by the retrofitting of 37 fluid-viscous dampers (energy dissipating) to control horizontal movement and 52 tuned mass dampers (inertial) to control vertical movement. ¨ Tacoma Narrows Bridge ¤ http://www.youtube.com/watch?v=j-zczJXSxnw ¤ The bridge collapsed in 1940 only 4 months after its opening. Fluid (top) and mass dampers (bottom).
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