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# 4.1. Oscillations

Back in Chapter 2 we started looking at motion, but mostly concerned with stuff moving in a straight line. Now we start to look at stuff oscillating; i.e. moving backwards and forwards. In Physics there are loads of examples of oscillatory motion, but one of the most commonly studied examples is that of a pendulum. A pendulum has a very well defined time period (the time taken to swing back and forth) - the reason behind their historical use in clocks. This regular time period is determined by the length of the pendulum (and perhaps counter intuitively, is NOT affected by the mass of the bob). Some YouTubers have also used some of these interesting properties to make some cool videos of ‘pendulum waves’ – see below.

Now, this subchapter lays a lot of the groundwork for a lot of the upcoming waves material. It has been broken up as follows:

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Periodic Motion

#### Periodic Motion

Periodic motion looks at movement backwards and forwards at a certain frequency. Let's start by looking at a nice simple example - a rubber duck bobbing up and down on water.

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Here we can introduce a few key bits of terminology:

• The Time Period : the time taken for one complete oscillation, in seconds.

• Frequency : The number of oscillations per second, measured in Hertz, Hz.

These are linked by the equation

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f = 1/T  (...or T = 1/f)

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If we were to plot the position of the rubber duck over time, we obtain the following relationship as shown below. The duck oscillates about the 'equilibrium position'. The vertical position of the duck is also known as the displacement, measured as the vertical distance from the equilibrium position.

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This representation is a displacement-time representation - showing how the rubber duck oscillates up and down over time. The amplitude is the position of maximum displacement (measured from the equilibrium position). The time period is the time taken for the duck to move through one complete oscillation back to the same point.

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Clearly, this graphical representation is very similar to what you will have looked at in maths with your trigonometric relationships. The graphs of both sine and cosine oscillate in a similar way. In fact, we can represent waves and oscillating motion with either sine or cos graphs. If you are rusty on these, it's worth brushign up on them here.

## Video Lessons

 Chris Doner Oscillations IB Specific Khan Academy Amplitude and Period Physics Online Wave Definitions

## Resources

 IB Physics Topic 4 Notes IB-Physics.net Chapter 4 Summary IB Revision Notes Mr. G 4.1 Teaching Notes 4.1 Student Notes Physics and Maths Tutor Waves Definitions Waves Key Points Waves Detailed Notes Waves Progressive & Standing Waves A Level Resources - content slightly different

## Questions

 Cambridge University Press Topic 4: Add Qs Topic 4: Add Qs MS Topic 4: MCQs CUP Website Link Freely available online Grade Gorilla Topic 4 (Waves A) End Quiz Quick IB Specific Mixed MCQs Mr. G 4.1 Formative Assessment Topic 4 Summary Qs IB Specific Questions
Simple Harmonic Motion

#### Simple Harmonic Motion

Simple harmonic motion (or SHM) is a type of oscillatory motion. Objects following simple harmonic motion have a displacement-time graph similar to that of the rubber duck - obeying our trigonometric mathematical relationships.

There are many examples of objects that follow SHM, such as:

• A pendulum swinging

• A mass bouncing on a spring

• A ball rolling in a curved dish

• A ruler vibrating off the side of a bench when 'pinged'.

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Let's start with looking at a mass on a spring - it's a really common example of SHM behaviour and introduces some of the main points quite nicely. PHET have a nice simulation on Mass-Spring Systems. Click the second tab marked Vectors and take a look in particular at how the Net Force and Acceleration vectors vary with time in relation to the equilibrium position.

We see that as the mass moves above the equilibrium position (i.e. has a positive displacement), the restoring force acts downwards towards the equilibrium position. The acceleration acts in the same direction as this restoring force (in a downwards, negative direction). The greater the displacement from the equilibrium, the greater the restoring force.

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This is one of the key characteristics of SHM.

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The acceleration (and restoring force) is proportional to, and in the opposite direction to the displacement.

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x ∝ -a

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This is the same for all objects following simple harmonic motion. For a pendulum, the restoring force always acts towards the equilibrium position. Graphically, this obeys the following relationship.

Plotting the position of the mass spring system against time and we observe an identical relationship to the rubber duck floating on a water wave. The key words we associate with wave motion - e.g. frequency and time period can equally be used to describe this oscillatory motion of a bouncing mass on a spring.

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We look in much more detail at Simple Harmonic Motion in Section 9.1 (HL Only), where we bring in some of the trigonometric maths to model this behaviour.

Geogebra has a couple of nice simulations that clearly demonstrate the mathematical relationships behind SHM.

1) SHM Tutorial : This tutorial nicely explains a few of the key mathematical ideas here.

2) SHM Graphs : This clearly shows the trigonometric relationships we can use to model simple harmonic motion.

## Video Lessons

 Chris Doner Oscillations (SHM) IB Specific Khan Academy Simple Harmonic Motion Intuition Science Shorts Simple Harmonic Motion Intro Equations go beyond IB Study Nova SHM Intro (Lecture)

## Resources

 IB Physics Topic 4 Notes IB-Physics.net Chapter 4 Summary IB Revision Notes Isaac Physics SHM Introduction Only first 2 sections SL Khan Academy SHM Introduction Mr. G 4.1 Teaching Notes 4.1 Student Notes Physics and Maths Tutor Further Mechanics Definitions Further Mechanics Key Points Further Mechanics Detailed Notes Further Mechanics Flashcards A Level Resources - content slightly different

## Questions

 Cambridge University Press Topic 4: Add Qs Topic 4: Add Qs MS Topic 4: MCQs CUP Website Link Freely available online Grade Gorilla 4.1 (SHM) MCQ Topic 4 (Waves A) End Quiz Quick IB Specific Mixed MCQs Isaac Physics Oscillators Some may be beyond SL Khan Academy SHM Introduction Q Mr. G 4.1 Formative Assessment Topic 4 Summary Qs IB Specific Questions
Phase and Phase Difference

#### Phase and Phase Difference

This last section introduces a new idea that is important to get our heads around before looking at the new ideas related to waves.

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A Level Physics Online introduces the concepts of Phase and Phase Difference below:

To begin with, we need to recap our graphs for sine and cosine. One graphs repeat after a period of 360° or  â€‹2π radians, as shown below.

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Now let's add a second wave of the same frequency and time period, shown in red in the graph below. We can see that these waves are not quite in-sync, and the red wave 'lags' slightly behind the orange wave.

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Here, we introduce a term called Phase Difference to measure the amount of 'lag' between one oscillation and another. Phase difference is typically measured in radians (rather than degrees). (e.g. Wave A and Wave B have a phase difference of π/4 radians between them).

A couple more examples are shown below.

• In the top of the two graphs, the oscillations are both oscillating in-time and in-sync. These two waves have no phase difference between them (Φ = 0) and are said to be IN PHASE.

• In the bottom graph, the oscillations are both exactly 'opposite' one another - i.e. one reaches its maximum positive displacement when the other reaches its maximum negative displacement. These two waves have a phase difference of π radians between them (Φ = π) and are said to be IN ANTIPHASE.

These ideas of phase difference will become important when we look at the next sections with some ideas to do with wave interference.

You can get a little glimpse at what happens when we change the phase difference between two waves by playing around with this Geogebra simulation.

## Video Lessons

 Chris Doner Oscillations (Phase) IB Specific Khan Academy Phase Constant Science Shorts Phase Difference

## Resources

 IB Physics Topic 4 Notes IB-Physics.net Chapter 4 Summary IB Revision Notes Isaac Physics Phase Phasors not required Mr. G 4.1 Teaching Notes 4.1 Student Notes Physics and Maths Tutor Waves Definitions Waves Key Points Waves Detailed Notes Waves Progressive & Standing Waves A Level Resources - content slightly different

## Questions

 Cambridge University Press Topic 4: Add Qs Topic 4: Add Qs MS Topic 4: MCQs CUP Website Link Freely available online Grade Gorilla Topic 4 (Waves A) End Quiz Quick IB Specific Mixed MCQs Isaac Physics Path Difference Mixed Questions Mr. G 4.1 Formative Assessment Topic 4 Summary Qs IB Specific Questions