Lecture Outline
Chapter 13
Physics, 4th Edition
James S. Walker
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Chapter 13
Oscillations about
Equilibrium
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Units of Chapter 13
• Periodic Motion
• Simple Harmonic Motion
• Connections between Uniform Circular
Motion and Simple Harmonic Motion
• The Period of a Mass on a Spring
• Energy Conservation in Oscillatory
Motion
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Units of Chapter 13
• The Pendulum
• Damped Oscillations
• Driven Oscillations and Resonance
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13-1 Periodic Motion
Period: time required for one cycle of periodic
motion
Frequency: number of oscillations per unit
time
This unit is
called the Hertz:
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13-2 Simple Harmonic Motion
A spring exerts a restoring force that is
proportional to the displacement from
equilibrium:
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13-2 Simple Harmonic Motion
A mass on a spring has a displacement as a
function of time that is a sine or cosine curve:
Here, A is called
the amplitude of
the motion.
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13-2 Simple Harmonic Motion
If we call the period of the motion T – this is the
time to complete one full cycle – we can write
the position as a function of time:
It is then straightforward to show that the
position at time t + T is the same as the
position at time t, as we would expect.
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
An object in simple
harmonic motion has the
same motion as one
component of an object
in uniform circular
motion:
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
Here, the object in circular motion has an
angular speed of
where T is the period of motion of the
object in simple harmonic motion.
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
The position as a function of time:
The angular frequency:
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13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
The velocity as a function of time:
And the acceleration:
Both of these are found by taking
components of the circular motion quantities.
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13-4 The Period of a Mass on a Spring
Since the force on a mass on a spring is
proportional to the displacement, and also to
the acceleration, we find that
.
Substituting the time dependencies of a and x
gives
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13-4 The Period of a Mass on a Spring
Therefore, the period is
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13-5 Energy Conservation in Oscillatory
Motion
In an ideal system with no nonconservative
forces, the total mechanical energy is
conserved. For a mass on a spring:
Since we know the position and velocity as
functions of time, we can find the maximum
kinetic and potential energies:
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13-5 Energy Conservation in Oscillatory
Motion
As a function of time,
So the total energy is constant; as the
kinetic energy increases, the potential
energy decreases, and vice versa.
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13-5 Energy Conservation in Oscillatory
Motion
This diagram shows how the energy
transforms from potential to kinetic and
back, while the total energy remains the
same.
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13-6 The Pendulum
A simple pendulum consists of a mass m (of
negligible size) suspended by a string or rod of
length L (and negligible mass).
The angle it makes with the vertical varies with
time as a sine or cosine.
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13-6 The Pendulum
Looking at the forces
on the pendulum bob,
we see that the
restoring force is
proportional to sin θ,
whereas the restoring
force for a spring is
proportional to the
displacement (which
is θ in this case).
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13-6 The Pendulum
However, for small angles, sin θ and θ are
approximately equal.
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13-6 The Pendulum
Substituting θ for sin θ allows us to treat the
pendulum in a mathematically identical way to
the mass on a spring. Therefore, we find that
the period of a pendulum depends only on the
length of the string:
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13-6 The Pendulum
A physical pendulum is a
solid mass that oscillates
around its center of mass,
but cannot be modeled as a
point mass suspended by a
massless string. Examples:
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13-6 The Pendulum
In this case, it can be shown that the period
depends on the moment of inertia:
Substituting the moment of inertia of a point
mass a distance l from the axis of rotation
gives, as expected,
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13-7 Damped Oscillations
In most physical situations, there is a
nonconservative force of some sort, which will
tend to decrease the amplitude of the
oscillation, and which is typically proportional
to the speed:
This causes the amplitude to decrease
exponentially with time:
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13-7 Damped Oscillations
This exponential decrease is shown in the
figure:
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13-7 Damped Oscillations
The previous image shows a system that is
underdamped – it goes through multiple
oscillations before coming to rest. A critically
damped system is one that relaxes back to the
equilibrium position without oscillating and in
minimum time; an overdamped system will
also not oscillate but is damped so heavily
that it takes longer to reach equilibrium.
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13-8 Driven Oscillations and Resonance
An oscillation can be driven by an oscillating
driving force; the frequency of the driving force
may or may not be the same as the natural
frequency of the system.
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13-8 Driven Oscillations and Resonance
If the driving frequency
is close to the natural
frequency, the
amplitude can become
quite large, especially
if the damping is small.
This is called
resonance.
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Summary of Chapter 13
• Period: time required for a motion to go
through a complete cycle
• Frequency: number of oscillations per unit time
• Angular frequency:
• Simple harmonic motion occurs when the
restoring force is proportional to the
displacement from equilibrium.
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Summary of Chapter 13
• The amplitude is the maximum displacement
from equilibrium.
• Position as a function of time:
• Velocity as a function of time:
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Summary of Chapter 13
• Acceleration as a function of time:
• Period of a mass on a spring:
• Total energy in simple harmonic motion:
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Summary of Chapter 13
• Potential energy as a function of time:
• Kinetic energy as a function of time:
• A simple pendulum with small amplitude
exhibits simple harmonic motion
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Summary of Chapter 13
• Period of a simple pendulum:
• Period of a physical pendulum:
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Summary of Chapter 13
• Oscillations where there is a nonconservative
force are called damped.
• Underdamped: the amplitude decreases
exponentially with time:
• Critically damped: no oscillations; system
relaxes back to equilibrium in minimum time
• Overdamped: also no oscillations, but
slower than critical damping
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Summary of Chapter 13
• An oscillating system may be driven by an
external force
• This force may replace energy lost to friction,
or may cause the amplitude to increase greatly
at resonance
• Resonance occurs when the driving frequency
is equal to the natural frequency of the system
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