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! Example is simple pendulum (for small angles of oscillation only!)
Simple Pendulum
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 with replacement k (mg/L) For example, SHO period is T SHO = 2π(m/k) ½ Get period for pendulum (small θ), by putting k = (mg/L) in SHO period. This gives: T Pend = 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
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).
Other types of x vs. t curves, depending on how big the frictional damping is:
Car shock absorber, a practical application of a damped oscillator!
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” Frequency f 0 : –Spring - mass system: f 0 = (1/2π)(k/m) ½ Pendulum: f 0 = (1/2π)(g/L) ½
Suppose, instead of letting system go after shoving it, we continually apply an oscillating EXTERNAL force. Frequency at which force oscillates f –Oscillator will oscillate, even if f f 0. Detailed math shows that amplitude of forced vibration depends on f - f 0 as: A forced 1/[(f - f 0 ) 2 + constant]
Amplitude of forced vibration: A forced 1/[(f - f 0 ) 2 + const] The closer the external frequency f gets to natural frequency f 0, the larger A forced gets! A forced is a maximum when f f 0. This is called Resonance! Careful!! If A forced gets too big, whatever is vibrating will break apart!