The Frequency and Period of an Oscillator

1
The Frequency and Period of an Oscillator
2
Objectives

• Convert from frequency to period, or period to
  frequency.
• Create graphs of position vs. time for an
  oscillator.
• Determine amplitude, period, and frequency from a
  graph of oscillatory motion.
• Investigate the factors that determine the period
  of a pendulum and a spring/mass system.
• Describe the restoring forces of oscillatory
  motion in various types of media.
• Describe the energy transformations of
  oscillatory motion in various types of media.

3
Physics terms

• oscillation
• equilibrium
• amplitude
• damping
• period
• frequency
• phase
• oscillator

4
Equations
NOTES
The period of an oscillator is the time to
complete one cycle.
The frequency of an oscillator is the inverse of
its period.
5
Oscillators
NOTES
An oscillator is a system with motion that
repeats in cycles.
6
Oscillators
The graph shows repeated cycles.
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Period
NOTES
A full cycle is one complete back and forth
motion. The period is the time it takes to
complete one full cycle. Period T is measured in
seconds.
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Period

• What is the period of the following oscillators?
• Earth in its rotation
• 86,400 seconds, 24 hours, or 1 day
• your heartbeat
• the minute hand on a clock
• a classroom pendulum

9
Frequency
NOTES

• Frequency is how many cycles are completed each
  second.
• Frequency f is measured in hertz, or Hz.
• 100 800 Hz

10
Frequency

• Frequency is how many cycles are completed in one
  second. What is the frequency of these
  oscillators?
• your heartbeat
• a fan that rotates 360 times a minute
• 0.5 1 per second of 0.5 1 Hz
• the vibration of a guitar string
• 100 800 Hz

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Frequency and period
The period of an oscillator is one over its
frequency.
The frequency of an oscillator is one over its
period.
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Engaging with the concepts
Javier is on a swing. His feet brush the ground
every 3.0 seconds. What is Javiers frequency?
Frequency
13
Engaging with the concepts
Marie has a spring-mass system with a frequency
of 4 Hz. What is the systems period?
4
Period
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What causes oscillations?
NOTES
Oscillations occur in systems with stable
equilibrium. Stable systems have restoring forces
that act to return them to the equilibrium
position if they are displaced.
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What causes oscillations?
NOTES
What provides the restoring force for a mass on a
spring?
What provides the restoring force for a simple
pendulum?
The force of gravity
The spring force
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Amplitude
NOTES
Amplitude is the maximum displacement from the
average.
A 4 meters
17
Period
NOTES
Period is the time per cycle.
T 9 seconds
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Frequency
NOTES
Frequency is the number of cycles in 1 second.
f 0.11 Hz
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Assessment

• Determine the amplitude, period, and frequency
  from the graph.

20
Assessment

• An object has a frequency of 50 Hz. What is the
  period?
• A spring mass system moves from one extreme of
  its motion to the other once every second. What
  is the frequency of the system?
• A. 0.2 Hz B. 0.5 Hz C. 2 Hz
  D. 5 Hz

21
Oscillators

• Many real physical systems oscillate when they
  are disturbed.
• Examples
• musical instruments
• geological formations in an earthquake
• wind-driven skyscrapers
• atoms in a solid

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Equilibrium and amplitude
NOTES
Most oscillators have a resting state or
equilibrium position. The amplitude is the
maximum distance x that the mass is displaced
from equilibrium during the oscillation.
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Amplitude
NOTES
For a vertical spring, equilibrium is where the
mass hangs at rest (with spring stretch x0
). The amplitude is the maximum distance x that
the mass is displaced from equilibrium during the
oscillation.
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Units of amplitude
NOTES
For a spring and mass, or a pendulum, the
amplitude is measured in meters (or
centimeters). With other types of oscillators,
the units for amplitude might be a voltage or a
pressure.
A A
Equilibrium
25
Amplitude
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Force and position
NOTES
A mass attached to a spring oscillates on a
frictionless surface.

• At equilibrium, the restoring force is zero.
• When the mass is displaced to the right, the
  restoring force points left.
• When the mass is displaced to the left, the
  restoring force points right.

Fs
Fs
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The restoring force
NOTES
For a vertical spring and mass system . . .

• When the mass is above equilibrium, the restoring
  force points down.
• At equilibrium, the net force is zero.
• When the mass is below equilibrium, the restoring
  force points up.

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Energy in an oscillator
NOTES

• Any force that disturbs the system adds energy.
  This added energy is what causes oscillations.
• The energy oscillates between different forms.
• For pendulums, the energy oscillates between
  gravitational potential energy and kinetic
  energy.
• In spring and mass systems, the energy oscillates
  between elastic potential energy and kinetic
  energy.

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The spring and mass oscillator
At which position(s) is this block moving the
fastest? At which position(s) is it at rest?
Where does it have maximum kinetic
energy? Maximum elastic potential energy?

A B C
30
The role of inertia
As the oscillator passes through equilibrium, the
restoring force is zero but the mass keeps
moving. WHY?
When the
31
Factors that affect frequency
Which of these changes will affect the frequency
of a mass on a spring?

• changing the mass?
• changing the spring constant?
• changing the amplitude?

32
Observations
NOTES

• When mass increases, frequency decreases.
• When the spring constant increases (stiffer
  spring with higher k), frequency increases.
• The amplitude does not affect the frequency.

33
Friction and damping
NOTES
Ideal oscillators continue oscillating
forever. In real oscillators, friction gradually
converts some mechanical energy to heat. This
is called damping.
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Damping
Sometimes engineers dont want springs to keep
oscillating. They employ damping to remove the
oscillatory energy from the spring. Shock
absorbers in cars are a good example they are
designed to damp out oscillations and produce a
smooth ride.
35
The pendulum
NOTES
For a pendulum, the equilibrium point is the
center of its swing, where it hangs at
rest. The amplitude A is the maximum distance it
is displaced sideways from equilibrium. The
period T is the time it takes to complete one
full cycle back and forth.
A
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The pendulum
At which position is the bob moving the fastest?
Where is it at rest? Where does it have
maximum kinetic energy? Where does it have
maximum potential energy?
A B C
37
Restoring force
NOTES
When a pendulum is displaced, a restoring force
brings it back to equilibrium.

• When displaced to the right, the restoring force
  (a component of the weight) points left.
• At equilibrium the restoring force shrinks to
  zero.
• When displaced to the left, the restoring force
  points right.

T
T
T
mgx
mgx
mgy
mgy
mg
38
The period of a pendulum
Review Which of these variables affects the
period of a pendulum?

• mass of the bob?
• amplitude of the oscillation?
• length of the string?

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Observations
NOTES

• The period does not depend on mass.
• The period does not depend on amplitude.
• The period does depend on the string length.

A longer string gives a longer period.
40
Assessment

• This is the position vs. time graph for a
  harmonic oscillator.

1. How many cycles occur in 10 seconds?
2. What is the amplitude of the motion?
3. What are the period and frequency of the motion?
   

41
Assessment

• Draw free-body diagrams for this oscillator at
  each position shown.

A B C
42
Assessment

• Describe the forms of energy present in this
  oscillator at each of the three positions shown.

A B C
43
Assessment

• A mass and spring oscillator is pulled back and
  released. As it passes the equilibrium position,
  the net force on it is zero.

Why does it continue to move past this point, if
the net force on it is zero?
44
Assessment

1. The position vs. time graph for a simple pendulum
   is shown.

• At what times is its velocity a maximum?
• At what times is its velocity zero?
• When its velocity is zero, is its displacement
  zero, or a maximum?