Transcript Document

Lecture Outlines
Chapter 13
Physics, 3rd Edition
James S. Walker
© 2007 Pearson Prentice Hall
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Chapter 13
Oscillations about
Equilibrium
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
Units of Chapter 13
• The Pendulum
• Damped Oscillations
• Driven Oscillations and Resonance
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:
13-2 Simple Harmonic Motion
A spring exerts a restoring force that is
proportional to the displacement from
equilibrium:
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.
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.
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:
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.
13-3 Connections between Uniform Circular
Motion and Simple Harmonic Motion
The position as a function of time:
The angular frequency:
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.
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
13-4 The Period of a Mass on a Spring
Therefore, the period is
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:
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.
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.
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.
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).
13-6 The Pendulum
However, for small angles, sin θ and θ are
approximately equal.
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:
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:
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,
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:
13-7 Damped Oscillations
This exponential decrease is shown in the
figure:
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.
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.
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.
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.
Summary of Chapter 13
• The amplitude is the maximum displacement
from equilibrium.
• Position as a function of time:
• Velocity as a function of time:
Summary of Chapter 13
• Acceleration as a function of time:
• Period of a mass on a spring:
• Total energy in simple harmonic motion:
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
Summary of Chapter 13
• Period of a simple pendulum:
• Period of a physical pendulum:
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
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