```Chapter 13 Lecture
Essential University Physics
Richard Wolfson
2nd Edition
Oscillatory Motion
Slide 13-1
In this lecture you’ll learn
• To describe the conditions
under which oscillatory
motion occurs
• To describe oscillatory motion
quantitatively in terms of
frequency, period, and
amplitude
• To explain simple harmonic
motion and why it occurs
universally in both natural and
technological systems
• To explain damped harmonic
motion and resonance
Slide 13-2
Oscillatory Motion
• A system in oscillatory motion undergoes repeating,
periodic motion about a point of stable equilibrium.
• Oscillatory motion is
characterized by
– Its frequency f or period T=1/f.
– Its amplitude A, or maximum
excursion from equilibrium.
• Shown here are positionversus-time graphs for two
different oscillatory motions
with the same period and
amplitude.
Slide 13-3
Simple Harmonic Motion
• Simple harmonic motion (SHM) results when the force
or torque that tends to restore equilibrium is directly
proportional to the displacement from equilibrium.
– The paradigm example is a mass
m on a spring of spring constant k.
d 2x
m 2  kx
dt
• The angular frequency
= 2p for this system is
w=
k m
– In SHM, displacement is a
sinusoidal function of time:
x(t )  A cos t
– Any amplitude A is possible.
• In SHM, frequency doesn’t
depend on amplitude.
Slide 13-4
Quantities in Simple Harmonic Motion
• Angular frequency, frequency, period:
k
w=
m
• Phase
w
1
f =
=
2p 2p
k
m
1
m
T = = 2p
f
k
– Describes the starting time of the displacement-versustime curve in oscillatory motion:
x(t) = Acos(t + )
Slide 13-5
Velocity and Acceleration in SHM
• Velocity is the rate of
change of position:
dx
v(t ) 
  A sin t
dt
• Acceleration is the rate of
change of velocity:
a(t ) 
dv
  2 A cos t
dt
Slide 13-6
Other Simple Harmonic Oscillators
• The torsional oscillator
– A fiber with torsional constant
 provides a restoring torque.
– Frequency depends on  and
rotational inertia:
w= k I
• Simple pendulum
– Point mass on massless cord of
length L:
d 2
I 2  mgL sin   mgL
dt
I  mL2 for point mass
w= g L
Slide 13-7
SHM and Circular Motion
• Simple harmonic motion can be viewed as one
component of uniform circular motion.
– Angular frequency  in SHM is the same as angular
velocity  in circular motion.
As the position vector r traces out a
circle, its x– and y– components are
sinusoidal functions of time.
Slide 13-8
Energy in Simple Harmonic Motion
• In the absence of nonconservative
forces, the energy of a simple
harmonic oscillator does not
change.
– But energy is transfered back and
forth between kinetic and potential
forms.
Slide 13-9
Simple Harmonic Motion is Ubiquitous!
• That’s because most systems near stable
equilibrium have potential-energy curves that are
approximately parabolic.
– Ideal spring: U = 12 kx 2 = 12 mw 2 x 2
– Typical potential-energy curve of an arbitrary system:
Slide 13-10
Damped Harmonic Motion
• With nonconservative forces present, SHM gradually damps
d 2x
dx
out:
m
 kx  b
dt 2
dt
– Amplitude declines exponentially toward zero:
x (t )  Ae
bt 2 m
cos(t   )
– For weak damping b, oscillations still occur at approximately the
undamped frequency
– With stronger damping, oscillations cease.
• Critical damping brings the system to equilibrium most quickly.
w=
k m.
(a) Underdamped, (b) critically damped,
and (c) overdamped oscillations.
Slide 13-11
Driven Oscillations
• When an external force acts on an oscillatory system, we
say that the system is undergoing driven oscillation.
• Suppose the driving force is F0cosdt, where d is the
driving frequency, then Newton’s law is
d 2x
dx
m 2  kx  b  F0 cos d t
dt
dt
• The solution is
x (t )  A cos(d t   )
where
A( ) 
0 
k
m
F0
m (d2  02 ) 2  b 2d2 / m2
: natural frequency
Slide 13-12
Resonance
• When a system is driven by an external force at near its
natural frequency, it responds with large-amplitude
oscillations.
– This is the phenomenon of resonance.
– The size of the resonant response increases as damping
decreases.
– The width of the resonance curve (amplitude versus
driving frequency) also narrows with lower damping.
Resonance curves for several
damping strengths; 0 is the
undamped natural frequency k/m.
Slide 13-13
Summary
• Oscillatory motion is periodic motion that results from a
force or torque that tends to restore a system to
equilibrium.
• In simple harmonic motion (SHM), the restoring force or
torque is directly proportional to displacement.
– The mass-spring system is the paradigm simple harmonic
oscillator.
x(t )  A cos t
w=
k m
– Damped harmonic motion occurs when nonconservative
forces act on the oscillating system.
– Resonance is a high-amplitude oscillatory response of a
system driven at near its natural oscillation frequency.
Slide 13-14
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