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Sect. 11-4: The Simple Pendulum

Sect. 11-4: The Simple Pendulum. Note: All oscillatory motion is not SHM!! SHM gives x(t) = sinusoidal function. Can show, any force of Hooke’s “Law” form F  Displacement (F = - kx) will give SHM. Does NOT have to be mass-spring system!

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Sect. 11-4: The Simple Pendulum

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  1. Sect. 11-4: The Simple Pendulum • Note: All oscillatory motion is not SHM!! • SHM gives x(t) = sinusoidal function. • Can show, any force of Hooke’s “Law” form F  Displacement (F = - kx) will give SHM. DoesNOThave to be mass-spring system! • Example is simple pendulum (for small angles of oscillation only!)

  2. Simple Pendulum

  3. For simple pendulum (small θ): F = -(mg/L) x Hooke’s “Law” form, F = -kx, with k = (mg/L) Can apply all SHO results to pendulum withreplacementk  (mg/L) For example, SHO period is TSHO = 2π(m/k)½ Get period for pendulum (small θ), by putting k = (mg/L) in SHO period. This gives: TPend = 2π(L/g)½ Independent of m & amplitude. Use pendulum to measure g. Pendulum frequency: f = (1/T) = (1/2π)(g/L)½ Example 11-9

  4. Sect. 11-5: Damped Harmonic Oscillator • Qualitative discussion! • Real harmonic oscillators (including mass-spring system) experience friction (“damping”) of the oscillations. Motion is still sinusoidal, with diminished amplitude as time progresses (exponential envelope to sinusoidal function).

  5. Other types of x vs. t curves, depending on how big the frictional damping is:

  6. Car shock absorber, a practical application of a damped oscillator!

  7. Sect. 11-5: Forced Vibrations & Resonance • Qualitative (plus maybe a movie!). • Set a vibrating system in motion by a “shove” (& then letting go). • System will vibrate at its: “Natural” Frequencyf0: • Spring - mass system: f0 = (1/2π)(k/m)½ • Pendulum: f0 =(1/2π)(g/L)½

  8. Suppose, instead of letting system go after shoving it, we continually apply an oscillatingEXTERNAL force. • Frequency at which force oscillates  f • Oscillator will oscillate, even if f  f0. • Detailed math shows that amplitude of forced vibration depends on f - f0 as: Aforced  1/[(f - f0 )2 + constant]

  9. Amplitude of forced vibration: Aforced  1/[(f - f0 )2 + const] • The closer the external frequency f gets to natural frequency f0 , the larger Aforcedgets! Aforcedis a maximum when f  f0. This is called Resonance! • Careful!! If Aforced gets too big, whatever is vibrating will break apart!

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