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 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 Example: oscillating stars Brightness Time CP445

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 +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   =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 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 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 Simple harmonic motion equation of motion substitute (restoring force) oscillation frequency and period CP457 x max  A

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 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 (a)the amplitude (b)the frequency and period (c)the initial phase at t = 0? (d)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 m Hz 16 s 0.10 rad 3.1 m 2.2  = 22 rad.s -1 f = 3.5 Hz