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 massspring 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 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
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 f0:
– Spring - mass system: f0 = (1/2π)(k/m)½
• Pendulum: f0 = (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  f0.
• Detailed math shows that amplitude of forced
vibration depends on f - f0 as:
Aforced  1/[(f - f0 )2 + constant]
• Amplitude of forced vibration:
Aforced  1/[(f - f0 )2 + const]
• The closer the external
frequency f gets to
natural frequency f0 ,
the larger Aforced gets!
Aforced is a maximum
when f  f0. This is
called Resonance!
• Careful!! If Aforced
gets too big,
whatever is vibrating will break apart!