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## Part Two: Oscillations, Waves, & Fluids

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**Examples of oscillations & waves:**Earthquake – Tsunami Electric guitar – Sound wave Watch – quartz crystal Radar speed-trap Radio telescope Part Two: Oscillations, Waves, & Fluids Examples of fluidmechanics: Flow speed vs river width Plane flight High-speed photo: spreading circular waves on water.**13. Oscillatory Motion**Describing Oscillatory Motion Simple Harmonic Motion Applications of Simple Harmonic Motion Circular & Harmonic Motion Energy in Simple Harmonic Motion Damped Harmonic Motion Driven Oscillations & Resonance**Dancers from the Bandaloop Project perform on vertical**surfaces, executing graceful slow-motion jumps. What determines the duration of these jumps? Wilberforce Pendulum pendulum motion: rope length & g**Disturbing a system from equilibrium results in oscillatory**motion. Absent friction, oscillation continues forever. Examples of oscillatory motion: Microwave oven: Heats food by oscillating H2O molecules in it. CO2 molecules in atmosphere absorb heat by vibrating global warming. Watch keeps time thru oscillation ( pendulum, spring-wheel, quartz crystal, …) Earth quake induces vibrations collapse of buildings & bridges . Oscillation**13.1. Describing Oscillatory Motion**Characteristics of oscillatory motion: • Amplitude A = max displacement from equilibrium. • PeriodT = time for the motion to repeat itself. • Frequencyf = # of oscillations per unit time. same period T same amplitude A [ f ] = hertz (Hz) = 1 cycle / s A, T, f do not specify an oscillation completely. Oscillation**Example 13.1. Oscillating Ruler**An oscillating ruler completes 28 cycles in 10 s & moves a total distance of 8.0 cm. What are the amplitude, period, & frequency of this oscillatory motion? Amplitude = 8.0 cm / 2 = 4.0 cm.**13.2. Simple Harmonic Motion**Simple Harmonic Motion (SHM): 2nd order diff. eq 2 integration const. Ansatz: angular frequency **A, B determined by initial conditions** ( t ) 2 x 2A**Amplitude & Phase**C = amplitude = phase Note: is independent of amplitude only for SHM. Curve moves to the right for < 0. Oscillation**Velocity & Acceleration in SHM**|x| = max at v = 0 |v| = max at a = 0**GOT IT? 13.1.**Two identical mass-springs are displaced different amounts from equilibrium & then released at different times. Of the amplitudes, frequencies, periods, & phases of the subsequent motions, which are the same for both systems & which are different? Same: frequencies, periods Different: amplitudes ( different displacement ) phases ( different release time )**Application: Swaying skyscraper**Tuned mass damper : Damper highly damped , Overall oscillation overdamped. Taipei 101 TMD: 41 steel plates, 730 ton, d = 550 cm, 87th-92nd floor. Also used in: • Tall smokestacks • Airport control towers. • Power-plant cooling towers. • Bridges. • Ski lifts. Movie Tuned Mass Damper**Example 13.2. Tuned Mass Damper**The tuned mass damper in NY’s Citicorp Tower consists of a 373-Mg (vs 101’s 3500 Mg) concrete block that completes one cycle of oscillation in 6.80 s. The oscillation amplitude in a high wind is 110 cm. Determine the spring constant & the maximum speed & acceleration of the block. **13.3. Applications of Simple Harmonic Motion**• The Vertical Mass-Spring System • The Torsional Oscillator • The Pendulum • The Physical Pendulum**The Vertical Mass-Spring System**Spring stretched by x1 when loaded. mass m oscillates about the new equil. pos. with freq**The Torsional Oscillator**= torsional constant Used in timepieces**The Pendulum**Small angles oscillation: Simple pendulum (point mass m):**Example 13.3. Rescuing Tarzan**Tarzan stands on a branch as a leopard threatens. Jane is on a nearby branch of the same height, holding a 25-m-long vine attached to a point midway between her & Tarzan. She grasps the vine & steps off with negligible velocity. How soon can she reach Tarzan? Time needed:**GOT IT? 13.2.**• What happens to the period of a pendulum if • its mass is doubled, • it’s moved to a planet whose g is ¼ that of Earth, • its length is quadrupled? no change doubles doubles**Conceptual Example 13.1. Nonlinear Pendulum**• A pendulum becomes nonlinear if its amplitude becomes too large. • As the amplitude increases, how will its period changes? • If you start the pendulum by striking it when it’s hanging vertically, • will it undergo oscillatory motion no matter how hard it’s hit? • If it’s hit hard enough, • motion becomes rotational. (a) sin increases slower than smaller longer period**The Physical Pendulum**Physical Pendulum = any object that’s free to swing Small angular displacement SHM**Example 13.4. Walking**When walking, the leg not in contact of the ground swings forward, acting like a physical pendulum. Approximating the leg as a uniform rod, find the period for a leg 90 cm long. Table 10.2 Forward stride = T/2 = 0.8 s**13.4. Circular & Harmonic Motion**Circular motion: 2 SHO with same A & but = 90 x = R x = R x = 0 Lissajous Curves**GOT IT? 13.3.**The figure shows paths traced out by two pendulums swinging with different frequencies in the x- & y- directions. What are the ratios x : y ? 1 : 2 3: 2 Lissajous Curves**13.5. Energy in Simple Harmonic Motion**SHM: = constant Energy in SHM**Potential Energy Curves & SHM**Linear force: parabolic potential energy: Taylor expansion near local minimum: Small disturbances near equilibrium points SHM**GOT IT? 13.4.**• Two different mass-springs oscillate with the same amplitude & frequency. • If one has twice as much energy as the other, how do • their masses & (b) their spring constants compare? • (c) What about their maximum speeds? • The more energetic oscillator has • twice the mass • twice the spring constant • (c) Their maximum speeds are equal.**13.6. Damped Harmonic Motion**sinusoidal oscillation Damping (frictional) force: Damped mass-spring: Amplitude exponential decay Ansatz: ** At t = 2m / b, amplitude drops to 1/e of max value.**(a) For is real, motion is oscillatory ( underdamped ) (c) For is imaginary, motion is exponential ( overdamped ) (b) For = 0, motion is exponential ( critically damped ) Damped & Driven Harmonic Motion**Example 13.6. Bad Shocks**A car’s suspension has m = 1200 kg & k = 58 kN / m. Its worn-out shock absorbers provide a damping constant b = 230 kg / s. After the car hit a pothole, how many oscillations will it make before the amplitude drops to half its initial value? Time required is # of oscillations: bad shock !**13.7. Driven Oscillations & Resonance**External force Driven oscillator Let d= driving frequency ( long time ) Prob 75: = natural frequency Damped & Driven Harmonic Motion Resonance:**Buildings, bridges, etc have natural freq.**If Earth quake, wind, etc sets up resonance, disasters result. Collapse of Tacoma bridge is due to self-excitation described by the van der Pol equation. Tacoma Bridge Resonance in microscopic system: • electrons in magnetron microwave oven • Tokamak (toroidal magnetic field) fusion • CO2 vibration: resonance at IR freq Green house effect • Nuclear magnetic resonance (NMR) NMI for medical use.