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13.2.1 - Period and frequency

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Introduction to Periodic Motion

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Teacher
Teacher

Today, we're going to explore periodic motion. Can anyone tell me what periodic motion is?

Student 1
Student 1

Isn't it motion that repeats over time?

Teacher
Teacher

Exactly! Periodic motion repeats after regular intervals. Can anyone give an example?

Student 2
Student 2

Like a pendulum swinging back and forth?

Teacher
Teacher

Yes, that's a great example! So, what do we call the time taken for one complete cycle of this motion?

Student 3
Student 3

That's the period, right?

Teacher
Teacher

Correct! We denote the period by T. Remember, the unit for period is seconds.

Understanding Frequency

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Teacher
Teacher

Now, let's talk about frequency. Who can tell me how frequency is related to period?

Student 1
Student 1

Is frequency the number of cycles per second?

Teacher
Teacher

Correct! The relationship is given by the formula ν = 1/T. So what happens if the period increases?

Student 2
Student 2

Then the frequency decreases?

Teacher
Teacher

Exactly! If T gets larger, ν goes down. What's the SI unit for frequency?

Student 4
Student 4

It's hertz, right? One hertz equals one cycle per second.

Teacher
Teacher

Spot on!

Examples and Applications

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Teacher
Teacher

Let's apply what we've learned. How would we calculate the frequency of a heart that beats 75 times per minute?

Student 1
Student 1

We convert it to seconds, so 75 divided by 60 gives us 1.25 Hz.

Teacher
Teacher

That's correct! And what about the period?

Student 2
Student 2

It would be the reciprocal, so T = 1/1.25, which is 0.8 seconds.

Teacher
Teacher

Perfect! These calculations help us understand biological rhythms and mechanical systems.

Introduction & Overview

Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.

Quick Overview

This section discusses the concepts of period and frequency in relation to periodic motion, defining key terms and relationships between them.

Standard

The section elaborates on periodic motion, defining period as the time for one complete oscillation and frequency as the number of oscillations per unit time. It includes examples of different types of periodic motions and highlights the relationship between period and frequency.

Detailed

Detailed Summary

This section introduces the concepts of period and frequency as essential elements of periodic motion. Periodic motion refers to any motion that repeats itself at regular intervals. The period (T) is the smallest interval of time after which this motion repeats, measured in seconds. Different motions might have varying periods; for instance, vibrations in a quartz crystal can be measured in microseconds (µs), while the orbital period of planets might span days or even years.

The frequency (ν) represents the number of oscillations that occur per unit time, calculated as the reciprocal of the period: ν = 1/T. Frequency has the SI unit of hertz (Hz), which reflects the number of cycles per second. Moreover, the text states that frequency is not always expressed as an integer.

A practical example illustrates how the average human heart rate of 75 beats per minute equates to a frequency of 1.25 Hz and a period of 0.8 seconds. Overall, the relationship between period and frequency provides a foundation for understanding oscillatory behavior in various physical systems.

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Audio Book

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Definition of Period

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We have seen that any motion that repeats itself at regular intervals of time is called periodic motion. The smallest interval of time after which the motion is repeated is called its period. Let us denote the period by the symbol T. Its SI unit is second.

Detailed Explanation

The concept of 'period' refers to the duration for one complete cycle of a periodic motion. For example, if you were to swing back and forth, the time from the start of the swing to when you return to that starting point is one complete cycle, which is your period T. It's important to note that the SI unit of this period is seconds, meaning we can measure how long it takes using a clock.

Examples & Analogies

Think of a Ferris wheel: the period is the time it takes for a cabin to go all the way around and return to its starting position. If the cabin takes two minutes to complete this cycle, then the period T is 2 minutes.

Units of Period

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For periodic motions which are either too fast or too slow on the scale of seconds, other convenient units of time are used. The period of vibrations of a quartz crystal is expressed in units of microseconds (10–6 s) abbreviated as µs. On the other hand, the orbital period of the planet Mercury is 88 earth days. The Halley’s comet appears after every 76 years.

Detailed Explanation

Depending on the speed of the periodic motion, the unit of measurement for the period can change. For extremely fast motions, such as vibrations in crystals, we might use microseconds, which helps us measure very short periods. Conversely, for astronomical events like the orbits of planets or comets, we need much larger units like days or years.

Examples & Analogies

Imagine you have a very noisy clock that 'ticks' at a high frequency. The period might be in milliseconds (ms) for a busy clock, but for measuring something like the return of a comet, you might use years, highlighting the vast difference in time scales.

Frequency Relation to Period

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The reciprocal of T gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion. It is represented by the symbol ν. The relation between ν and T is ν = 1/T.

Detailed Explanation

Frequency ν refers to how many cycles occur in a second, essentially measuring the rate of repetition of the periodic motion. If we know the period T, we can find frequency by taking the inverse. For instance, if a pendulum takes 2 seconds to complete one swing, its frequency is half a cycle per second, or 0.5 Hz.

Examples & Analogies

Think of a drummer. If a drummer beats a drum once every second, their frequency is 1 Hz, which means one full motion each second. If the drummer speeds up and hits the drum twice in that same second, the frequency is now 2 Hz.

Unit of Frequency - Hertz

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The unit of ν is thus s–1. After the discoverer of radio waves, Heinrich Rudolph Hertz (1857 –1894), a special name has been given to the unit of frequency. It is called hertz (abbreviated as Hz). Thus, 1 hertz = 1 Hz = 1 oscillation per second = 1 s–1.

Detailed Explanation

Frequency is measured in hertz (Hz), where 1 Hz indicates one complete cycle per second. The unit hertz honors Heinrich Hertz, who contributed significantly to the understanding of waves. When we say something vibrates at 60 Hz, we mean it completes 60 cycles every second.

Examples & Analogies

Imagine a strobe light at a concert flashing 60 times a second. If the strobe light is at 60 Hz, it means that light is turning on and off 60 times each second, creating a rhythm with the music that you can visually perceive.

Frequency Is Not Always an Integer

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Note, that the frequency, ν, is not necessarily an integer.

Detailed Explanation

Frequency can take on any positive value, allowing for both whole numbers and fractions. This means that a system can oscillate at a frequency that isn’t a whole number—like 1.5 Hz, which would mean it completes one and a half cycles in a second. This flexibility in measurement is important for understanding more complex motions.

Examples & Analogies

Think about a spinning record. If it spins fast, it might complete 33 and a third rotations every minute, indicating a frequency that isn’t a neat whole number—this is quite common in real-world applications.

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Periodic Motion: Motion that repeats after regular intervals.

  • Period (T): The time for one complete cycle of motion.

  • Frequency (ν): The number of cycles per second, related to the period.

  • Hertz (Hz): The unit for measuring frequency.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • The swinging of a pendulum, where the period is the time taken for a complete swing.

  • The heartbeat of humans, where the average heart rate represents a periodic motion with specific frequency and period.

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • To find the period, don't be a fool, it's T, just wait for the next cycle's rule.

📖 Fascinating Stories

  • Imagine a grandfather clock ticking every second. Each tick represents periodic motion—time keeps moving forward at a regular period, and if you divide that time into seconds, you can find how often it ticks!

🧠 Other Memory Gems

  • T=1/ν: T's the time, inverse to the frequency.

🧠 Other Memory Gems

  • A toy swings back and forth with a period of 1.5 seconds. Calculate its frequency and the time it takes for it to complete 10 cycles.

  • If a planet completes cyclic orbits at different rates, with Mercury having a period of 88 days, express its frequency in terms of days.

🧠 Other Memory Gems

  • Frequency = 1/T = 1/1.5 = 0.67 Hz. Time for 10 cycles = 10 * 1.5 = 15 seconds.

  • Frequency ν = 1/T in days. Thus, ν = 1/88 = 0.0114 cycles per day.

🧠 Other Memory Gems

  • Think about the total number of oscillations and how they relate to time.

  • Remember to convert the period measured in days to frequency.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Periodic Motion

    Definition:

    Motion that repeats itself at regular intervals.

  • Term: Period (T)

    Definition:

    The smallest interval of time after which motion repeats.

  • Term: Frequency (ν)

    Definition:

    The number of repetitions occurring per unit time, measured in hertz (Hz).

  • Term: Hertz (Hz)

    Definition:

    Unit of frequency indicating one oscillation per second.