Transcript 13-5 Energy Conservation in Oscillatory Motion As a function of time

Chapter 13
Oscillations about
Equilibrium
Copyright © 2010 Pearson Education, Inc.
Copyright © 2010 Pearson Education, Inc.
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
• The Pendulum
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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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