1. 1 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)] LECTURE 2 CP Ch 14 How can you determine the mass of a single E-coli bacterium or a DNA molecule ? CP458

2. 2 Signal from ECG period T Period: time for one cycle of motion [s] Frequency: number of cycles per second [s -1 = Hz hertz] CP445 1 kHz = 10 3 Hz 10 6 Hz = 1 MHz 1GHz = 10 9 Hz

3. 3 Example: oscillating stars Brightness Time CP445

4. 4 Simple harmonic motion SHM 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 spring restoring force CP447 x = 0 +X

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

6. 6   =d  /dt  = 2  /T One cycle: period T [s] Cycles in 1 s: frequency f [Hz] angular frequency  - T= 1 /f f T  = 2  f  =2  /T amplitudeA A CP453 Connection SHM – uniform circular motion [ rad.s -1 ]

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

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

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

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

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

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

13. 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. 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 -1 f = 3.5 Hz