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Simple Harmonic Motion (SHM). (and waves). What do you think Simple Harmonic Motion (SHM) is???. Defining SHM. Equilibrium position Restoring force Proportional to displacement Period of Motion Motion is back & forth over same path. Θ. F g. Describing SHM. Amplitude. Θ. F g.

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defining shm
Defining SHM
  • Equilibrium position
  • Restoring force
    • Proportional to displacement
  • Period of Motion
  • Motion is back & forth over same path
describing shm

Θ

Fg

Describing SHM
  • Amplitude
describing shm1

Θ

Fg

Describing SHM
  • Period (T)
  • Full swing
    • Return to original position
frequency
Frequency
  • Frequency- Number of times a SHM cycles in one second (Hertz = cycles/sec)
  • f = 1/T
shm descriptors
SHM Descriptors
  • Amplitude (A)
    • Distance from start (0)
  • Period (T)
    • Time for complete swing or oscillation
  • Frequency (f)
    • # of oscillations per second
oscillations
Oscillations
  • SHM is exhibited by simple harmonic oscillators (SHO)
  • Examples?
examples of shos
Examples of SHOs
  • Mass hanging from spring, mass driven by spring, pendulum
shm for a pendulum
SHM for a Pendulum
  • T = period of motion (seconds)
  • L = length of pendulum
  • g = 9.8 m/s2
slide14
EPE = ½ k x2
  • KE = ½ m v2
  • E = ½ m v2 + ½ k x2
  • E = ½ m (0)2 + ½ k A2E = ½ k A2
  • E = ½ m vo2 + ½ k (0)2E = ½ m vo2
velocity

Velocity
  • E = ½ m v2 + ½ k x2
  • ½ m v2 + ½ k x2 = ½ k A2
  • v2 = (k / m)(A2 - x2) = (k / m) A2 (1 - x2 / A2)
        • ½ m vo2 = ½ k A2
        • vo2 = (k / m) A2
  • v2 = vo2 (1 - x2 / A2)
  • v = vo 1 - x2 / A2
damped harmonic motion
Damped Harmonic Motion
  • due to air resistance and internal friction
  • energy is not lost but converted into thermal energy
damping
Damping
  • A: overdamped
  • B: critically damped
  • C: underdamped
resonance
Resonance
  • occurs when the frequency of an applied force approaches the natural frequency of an object and the damping is small (A)
  • results in a dramatic increase in amplitude