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Oscillations

LECTURE 2. Oscillations. Time variations that repeat themselves at regular intervals - periodic or cyclic behaviour Examples: Pendulum (simple); heart (more complicated) Terminology: Period : time for one cycle of motion [s] Frequency : number of cycles per second [s -1 = hertz (Hz)].

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Oscillations

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  1. LECTURE 2 Oscillations Time variations that repeat themselvesat regular intervals - periodic or cyclic behaviour Examples:Pendulum (simple); heart (more complicated) Terminology: Period: time for one cycle of motion [s] Frequency: number of cycles per second [s-1 = hertz (Hz)] How can you determine the mass of a single E-coli bacterium or a DNA molecule ? CP458 CP Ch 14

  2. Signal from ECG period T Period: time for one cycle of motion [s] Frequency: number of cycles per second [s-1 = Hz hertz] 1 kHz = 103 Hz 106 Hz = 1 MHz 1GHz = 109 Hz CP445

  3. Example: oscillating stars Brightness Time CP445

  4. Simple harmonic motion SHM x = 0 spring restoring force +X • object displaced, then released • objects oscillates about equilibrium position • motion is periodic • displacement is a sinusoidal function of time (harmonic) • T = period = duration of one cycle of motion • f = frequency = # cycles per second • restoring force always acts towards equilibrium position CP447

  5. origin 0 equilibrium position displacement x [m] velocity v [m.s ] - 1 - + x x max max acceleration a [m.s ] - 2 Force F [N] e Motion problems – need a frame of reference Vertical hung spring: gravity determines the equilibrium position – does not affect restoring force for displacements from equilibrium position – mass oscillates vertically with SHM Fe = - k y CP447

  6. w q = d / d t w p = 2 / T amplitude A A q T = 1 / f f = 1 / T w p = 2 f w p = 2 / T One cycle: period T [s] w - Cycles in 1 s: frequency f [Hz] angular frequency Connection SHM – uniform circular motion [rad.s-1] CP453

  7. SHM & circular motion Displacement is sinusoidal function of time uniform circular motion radius A, angular frequency  x A  xmax x component is SHM: CP453

  8. Simple harmonic motion T amplitude displacement time T T Displacement is a sinusoidal function of time By how much does phase change over one period? CP451

  9. Simple harmonic motion xmax A equation of motion (restoring force) substitute oscillation frequency and period CP457

  10. Simple harmonic motion acceleration is  rad (180) out of phase with displacement At extremes of oscillations, v = 0 When passing through equilibrium, v is a maximum CP457

  11. How do you describe the phase relationships between displacement, velocity and acceleration? CP459

  12. Problem 2.1 • If a body oscillates in SHM according to the equation • where each term is in SI units. What are • the amplitude • the frequency and period • the initial phase at t = 0? • the displacement at t = 2.0 s?

  13. Problem 2.2 An object is hung from a light vertical helical spring that subsequently stretches 20 mm. The body is then displaced and set into SHM. Determine the frequency at which it oscillates.

  14. Answers to problems 2.1 5.0 m 0.064 Hz 16 s 0.10 rad 3.1 m 2.2  = 22 rad.s-1f = 3.5 Hz

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