The frequency and period of an oscillator

The frequency and period of an oscillator

ObjectivesObjectives

• 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.

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

AssessmentAssessment

AssessmentAssessment2. An object has a frequency of 50 Hz. What is the period?

3. 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

Physics termsPhysics terms

• frequency

• period

• amplitude

• oscillator

EquationsEquations

The period of an oscillator is the time to complete one cycle.

The frequency of an oscillator is the inverse of its period.

BrainstormingBrainstorming

How is the motion of a pendulum different from the motion of a falling body?

How could you describe this motion exactly to someone who couldn’t see it?

What would you need to measure to describe it exactly?

OscillatorsOscillators

A pendulum swings back and forth.

The motion repeats itself over and over again in cycles.

What does the graph of position versus time look like?

OscillatorsOscillators

The graph shows repeated cycles.

OscillatorsOscillators

An oscillator is a system with motion that repeats in cycles.

OscillatorsOscillators

Watch a pendulum swing back and forth.

What is its cycle?

How long does each cycle take?

PeriodPeriod

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.

PeriodPeriodWhat is the period of the following oscillators?

1.Earth in its rotation

86,400 seconds, 24 hours, or 1 day

2.your heartbeat

3.the minute hand on a clock

4.a classroom pendulum

PeriodPeriodWhat is the period of the following oscillators?

1.Earth in its rotation

1 day, or 24 hours, or 86,400 seconds

2.your heartbeat

3.the minute hand on a clock

4.a classroom pendulum

PeriodPeriodWhat is the period of the following oscillators?

1.Earth in its rotation

1 day, or 24 hours, or 86,400 seconds

2.your heartbeat

about 1 second

3.the minute hand on a clock

4.a classroom pendulum

PeriodPeriodWhat is the period of the following oscillators?

1.Earth in its rotation

1 day, or 24 hours, or 86,400 seconds

2.your heartbeat

about 1 second

3.the minute hand on a clock

1 hour

4.a classroom pendulum

PeriodPeriodWhat is the period of the following oscillators?

1.Earth in its rotation

1 day, or 24 hours, or 86,400 seconds

2.your heartbeat

about 1 second

3.the minute hand on a clock

1 hour

4.a classroom pendulum

typically 1-2 seconds

FrequencyFrequencyFrequency is how many cycles are completed each second.

Frequency f is measured in hertz, or Hz.

1. 100 – 800 Hz

FrequencyFrequencyFrequency is how many cycles are completed in one second. What is the frequency of these oscillators?

1.your heartbeat

2.a fan that rotates 360 times a minute

1.0.5 – 1 per second of 0.5 – 1 Hz

3.the vibration of a guitar string

1.100 – 800 Hz

FrequencyFrequencyFrequency is how many cycles are completed in one second. What is the frequency of these oscillators?

1.your heartbeat

1 – 2 beats per second, or 1 – 2 Hz

2.a fan that rotates 360 times a minute

1.0.5 – 1 per second of 0.5 – 1 Hz

3.the vibration of a guitar string

1.100 – 800 Hz

FrequencyFrequencyFrequency is how many cycles are completed in one second. What is the frequency of these oscillators?

1.your heartbeat

1 – 2 beats per second, or 1 – 2 Hz

2.a fan that rotates 360 times a minute

6 cycles per second, or 6 Hz

3.the vibration of a guitar string

1.100 – 800 Hz

FrequencyFrequencyFrequency is how many cycles are completed in one second. What is the frequency of these oscillators?

1.your heartbeat

1 – 2 beats per second, or 1 – 2 Hz

2.a fan that rotates 360 times a minute

6 cycles per second, or 6 Hz

3.the vibration of a guitar string

100 – 800 Hz

1.100 – 800 Hz

Frequency and periodFrequency and period

The period of an oscillator is one over its frequency.

The frequency of an oscillator is one over its period.

Exploring the ideasExploring the ideas

Click this interactive calculator(page 390)

Engaging with the conceptsEngaging with the concepts

Javier is on a swing. His feet brush the ground every 3.0 seconds.

What is Javier’s frequency?

Frequency

Engaging with the conceptsEngaging with the concepts

Javier is on a swing. His feet brush the ground every 3.0 seconds.

What is Javier’s frequency?

Javier has a period of 6.0 s. A period is how long it takes to complete a full cycle.

6.0

Frequency

0.17

Observe the motion of the pendulum.

Engaging with the conceptsEngaging with the concepts

Marie has a spring-mass system with a frequency of 4 Hz. What is the system’s period?

4

Period

Engaging with the conceptsEngaging with the concepts

Observe the motion of the pendulum.

Marie has a spring-mass system with a frequency of 4 Hz. What is the system’s period?

4

Period

0.25

InvestigationInvestigation

How do amplitude, mass, and string length affect the period of a pendulum?

Turn to Investigation 14A on page 391.

InvestigationInvestigationPart 1: Period of a pendulum

1.Attach the protractor and pendulum to the stand and clamp as shown.

The string sets in the slot just below the thumb nuts.

InvestigationInvestigation

5 washers

Part 1: Period of a pendulum

1.Attach the protractor and pendulum to the stand and clamp as shown.

2.Set 5 washers on the hanger for the mass.

The string sets in the slot just below the thumb nuts.

InvestigationInvestigationPart 1: Period of a pendulum

1.Attach the protractor and pendulum to the stand and clamp as shown.

2.Set 5 washers on the hanger for the mass.

3.Set the pendulum swinging and observe the motion.

The string sets in the slot just below the thumb nuts.

InvestigationInvestigationPart 1: Period of a pendulum

4.Use the protractor to observe the amplitude of the motion.

4.With a stopwatch, measure the time it takes to complete 10 full cycles.

5.Change the amplitude, mass, and string length and see how each variable affects the period of your pendulum.

Amplitude

InvestigationInvestigation

a. Describe how you determined one full cycle of the pendulum.

b. How does the period of the pendulum depend on length, mass, and amplitude? Your answers should be supported by the data.

a. Propose a design for a pendulum that has a period of 2.0 seconds.

Questions for Part 1

Amplitude

1.Set 12 washers on the mass hanger. Attach the mass and spring.

InvestigationInvestigationPart 2: Mass and spring oscillator

InvestigationInvestigation

Place the meter rule against the stand. Note the marking on the ruler that aligns with the top washer in its equilibrium position.

Part 2: Mass and spring oscillator

1.Set 12 washers on the mass hanger. Attach the mass and spring.

2.Displace the mass 5 cm and release it. Record the time to complete 10 oscillations.

3.Repeat the experiment and record data for different masses and amplitudes.

InvestigationInvestigationPart 2: Mass and spring oscillator

4.Replace the first spring with a second spring of a different length. Set 12 washers on the mass hanger.

5.With a stopwatch, measure the time to complete 10 oscillations.

6.With a spring scale, measure the force needed to extend each spring 10 cm. Calculate the spring constants:

InvestigationInvestigationPart 2: Mass and spring oscillator

InvestigationInvestigation

a. How did you determine one full cycle of the motion?

a. How does the period of the mass-spring oscillator depend on mass and amplitude?

b. Explain the answer to part (b) using Newton's second law.

c. How does the period of the mass-spring oscillator depend on the spring constant?

Questions for Part 2

What causes oscillations?What causes oscillations?

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.

What causes oscillations?What causes oscillations?What provides the restoring force for a simple pendulum?

What provides the restoring force for a mass on a spring?

What causes oscillations?What causes oscillations?What provides the restoring force for a simple pendulum?

What provides the restoring force for a mass on a spring?

The force of gravity

The spring force

Finding patternsFinding patternsGraph these data points on your assignment sheet.

Is there a pattern?

Finding patternsFinding patternsGraph these data points on your assignment sheet.

Is there a pattern?

Finding patternsFinding patternsGraph these data points on your assignment sheet.

Is there a pattern? Yes!

AmplitudeAmplitudeAmplitude is the maximum displacement from the average.

A = 4 meters

PeriodPeriodPeriod is the time per cycle.

T = 9 seconds

FrequencyFrequencyFrequency is the number of cycles in 1 second.

f = 0.11 Hz

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

AssessmentAssessment

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

AssessmentAssessment

A = 7.5 cm

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

AssessmentAssessment

T = 8 seconds

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

AssessmentAssessment

f = 0.12 Hz

f 1

T

1

8 s0.12 Hz

2. An object has a frequency of 50 Hz. What is the period?

AssessmentAssessment

2. An object has a frequency of 50 Hz. What is the period?

T = 1/f = 1/50 Hz = 0.02 s

3. 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

AssessmentAssessment

2. An object has a frequency of 50 Hz. What is the period?

T = 1/f = 1/50 Hz = 0.02 s

3. 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

AssessmentAssessment

The period is 2 seconds, so the frequency is 0.5 Hz.