# OCR AS/A Level Physics A

# Circular and simple harmonic motion

Navigate to resources by choosing units within one of the unit groups shown below.

## Introduction

### Overview

Delivery guides are designed to represent a body of knowledge about teaching a particular topic and contain:

- Curriculum Content: A clear outline of the content covered by the delivery guide
- Thinking Conceptually: Expert guidance on the key concepts involved, common difficulties students may have, approaches to teaching that can help students understand these concepts and how this topic links conceptually to other areas of the subject
- Thinking Contextually: A range of suggested teaching activities using a variety of themes so that different activities can be selected which best suit particular classes, learning styles or teaching approaches.

## Curriculum Content

### Overview

**Content (from A Level)**

**5.2.1 Kinematics of circular motion**

*Learners should be able to demonstrate and apply their knowledge and understanding of:*

the radian as a measure of angle

period and frequency of an object in circular motion

angular velocity \(\displaystyle\omega\); \(\displaystyle\omega = \frac{2\pi}{T} \text{ }\) or \(\displaystyle\text{ }\omega=2\pi f\)

**5.2.2 Centripetal force**

*Learners should be able to demonstrate and apply their knowledge and understanding of:*

a constant net force perpendicular to the velocity of an object causes it to travel in a circular path

constant speed in a circle; \(\displaystyle v=\omega r\)

centripetal acceleration; \(\displaystyle a=\frac{v^2}{r}\) ; \(\displaystyle a=\omega^2r\)

**(i)** centripetal force; \(\displaystyle F=\frac{mv^2}{r}\) ; \(\displaystyle F=m\omega^2r\)

**(ii)** techniques and procedures used to investigate circular motion using a whirling bung.

**5.3 Oscillations**

Oscillatory motion is all around us, with examples including atoms vibrating in a solid, a bridge swaying in the wind, the motion of pistons of a car and the motion of tides (HSW1,2,3,5,6,8,9,10,12).

This section provides knowledge and understanding of simple harmonic motion, forced oscillations and resonance.

**5.3.1 Simple harmonic oscillations**

*Learners should be able to demonstrate and apply their knowledge and understanding of:*

displacement, amplitude, period, frequency, angular frequency and phase difference

angular frequency \(\displaystyle \omega\); \(\displaystyle \omega=\frac{2\pi}{T}\) or \(\displaystyle \omega=2\pi f\)

**(i)** simple harmonic motion; defining equation \(\displaystyle a=\textbf{- }\omega^2x\)

**(ii)** techniques and procedures used to determine the period/frequency of simple harmonic oscillations

5.3.1(d) solutions to the equation \(\displaystyle a=\textbf{- }\omega^2x\)

*e.g.* \(\displaystyle x=A\text{ cos }\omega t\) or \(\displaystyle x=A\text{ sin }\omega t\)

velocity \(\displaystyle v=\pm \text{ }\omega\sqrt{A^2 \textbf{- } x^2}\) hence \(\displaystyle v_{max}=\omega A\)

the period of a simple harmonic oscillator is independent of its amplitude (isochronous oscillator)

graphical methods to relate the changes in displacement, velocity and acceleration during simple harmonic motion.

**5.3.2 Energy of a simple harmonic oscillator**

*Learners should be able to demonstrate and apply their knowledge and understanding of:*

interchange between kinetic and potential energy during simple harmonic motion

energy-displacement graphs for a simple harmonic oscillator

**5.3.3 Damping**

*Learners should be able to demonstrate and apply their knowledge and understanding of:*

free and forced oscillations

**(i)** the effects of damping on an oscillatory system

**(ii)** observe forced and damped oscillations for a range of systems.

resonance; natural frequency

amplitude-driving frequency graphs for forced oscillators

practical examples of forced oscillations and resonance.

### Learner Resource 1

Students can familiarise themselves with the key terms involved in circular motion using the definitions exercise in Learner Resource 1.

### Radians calculations

**Teaching Advanced Physics (TAP):**

Learners can become more familiar with the use of radians using the exercises (see link 'Radians and angular speed').

### Circular motion animation

**School Physics:**

Some useful introductory animations to circular motion.

### Whirling bungs experiments

Experiments that involve whirling bungs can be found at both:

**Nuffield Foundation:**

Whirling a rubber bung on a string

**Teaching Advanced Physics (TAP):**

Verification of the equation of centripetal force using a whirling bung

**Nuffield Foundation:**

What happens when the bung is let go can be investigated using the experiments described (see link 'Whirling a rubber bung and letting go').

### Introduction to SHM

An introduction to oscillatory motion and SHM can be found at both:

**Teaching Advanced Physics (TAP):**

Recognisng simple harmoninc motion

**Nuffield Foundation:**

Introduction to oscillations

A practical introduction.

### Link between SHM and circular motion

**School Physics:**

Animation showing the link between circular motion and SHM

**PhET Interactive Simulations:**

A useful simulation of a pendulum experiment.

### Graphing SHM

**Nuffield Foundation:**

A consideration of the energy transfers in SHM

**comPADRE Physics and Astronomy Education Communities:**

Horizontal SHM can be investigated using a linear air track. This video clip is particularly good, as it contains graphs of acceleration, velocity and displacement against time as well.

### Damped oscillations and resonance animation

**Teaching Advanced Physics (TAP):**

Damping in oscillatory systems is shown in the activities

**Learners TV:**

An animation on damped simple harmonic motion

**Science Animations:**

Animation showing damping and resonance.

### Energy changes in SHM

Forced oscillations and resonance are introduced in the activities at both:

**Teaching Advanced Physics (TAP):**

Resonance

**Physclips:**

Resonance laboratory.

## Thinking Conceptually

### Overview

**Approaches to teaching the content**

There is a lot of new material in this topic, but it is an area rich with opportunity for learners to develop their knowledge through practical investigation. This area has also proved a rich source of practical examination questions in the past, and there is ample opportunity to develop examination technique as well as skills in obtaining and analysing experimental data.

Circular motion is easy to demonstrate (for example, using a bung on a string or, for the more intrepid, swinging a bucket containing some water in the vertical plane), but a proper definition of it in terms of forces and motion is likely to be a significant challenge for most students. Once a proper definition is made, its utility across many areas of Physics can then be appreciated. Studies of moving charges in electric and magnetic fields, of moving masses in a gravitational field (although Kepler showed that planetary motion is not quite circular, the orbit of a satellite around the Earth may be treated as such), the reasons why banked corners on racing tracks can be negotiated at higher velocities than flat ones and in-depth studies of the elegance of simple harmonic motion are made considerably easier if the students understand the simplicity of circular motion in terms of the directions of the force causing the motion and the direction of the motion.

The starting place is the definition of circular motion – the force acting on an object is perpendicular to its motion. Once this is understood, the student can go on to describe circular motion in terms of its velocity and frequency. Introduction of the concept of angular velocity is important, and some time may need to be spent on developing an understanding of the radian and its use.

Once the simple definition of the force acting on an object in circular motion is made, then \(\displaystyle F = \frac{mv^2}{r}\) and \(\displaystyle a = \frac{v^2}{r}\) are easily derived. The angular equivalents of these equations (\(\displaystyle F = m \text{ } \omega ^2 r\) and \(\displaystyle a = \omega ^2 r\)) should be introduced at the same time. There are plenty of ways of illustrating circular motion in the laboratory, and measurement of such motion will allow students to further develop their practical skills, and in particular their knowledge of experimental uncertainties.

The proper use of the terms ‘centripetal force’ and ‘centripetal acceleration’ is key to an understanding of circular motion, and students should avoid the term ‘centrifugal’. ‘Centripetal’ means ‘towards the centre’, which is an excellent summary of the forces acting on an object in circular motion. The circular motion of a bung on a string illustrates this well (the force can even be measured using a Newton meter), but when the student lets go of the string, the subsequent motion of the bung clearly shows the effect of the absence of the centripetal force, and possibly the reaction speed of any students in the path of the bung. The equations of motion could then be used to show where the bung will fall to earth, which is a good example of the synoptic nature of Physics.

Once circular motion is studied and understood, then learners can progress to a study of oscillatory motion. Oscillations are all around us, with examples ranging from the very rapid oscillations in a transmitter of electromagnetic waves and in atomic physics, through piston motion in engines, the motion of springs in suspension systems, to the motion of structures swaying in the wind to the relatively slow oscillations seen in tidal motion. Advanced study of electromagnetic waves, sound and alternating current also depends on a firm knowledge of oscillatory motion. The term ‘simple harmonic motion’ covers all of these and more, and should be introduced early in the study of oscillations.

A proper understanding of oscillatory motion requires a firm knowledge of several definitions, and it is crucial that such terms as displacement (\(\displaystyle d\)), amplitude (\(\displaystyle A\)), period (\(\displaystyle t\)), frequency (\(\displaystyle f\)), angular frequency (\(\displaystyle \omega\)) and phase difference (in radians) are understood.

Oscillatory motion has a point of equilibrium, and simple harmonic motion can be understood as motion where there is a restoring force towards the point of equilibrium (zero displacement) and inversely proportional to the displacement. The restoring force can be explained in terms of Hooke’s Law for simple spring systems, and this is a good starting point for the study of oscillatory motion.

The mathematical model of simple harmonic motion applies to most oscillatory systems but not all, but as long as the displacement is sufficiently small, then the restoring force is proportional to the displacement from the equilibrium position. Learners should be made aware of this, since it will help them prepare for study at university level. Advanced learners could investigate simple harmonic motion as the projection of uniform circular motion onto a diameter.

The definitions of the basic terms are contained in Learning Resource 1, and they can be investigated using simple experiments. Measurements of oscillatory systems are reasonably simple to arrange and set up, and can be easily used to further the students’ understanding of uncertainties in measurement of physical quantities. It is especially important for the student to start counting at zero when measuring oscillations!

More detail of some experiments to determine the period and frequency of simple harmonic oscillations are given elsewhere in this guide, and the measurements used can be used to determine solutions to the equations \(\displaystyle a = -\omega^2x\), where \(\displaystyle x = A \text{ cos} \text{ }\omega t\) or \(\displaystyle x = A \text{ sin} \text{ } \omega t\). Acceleration can be difficult to measure directly in the school laboratory, but it can be determined using these equations. Video recording of small oscillations using a background scale could be used to gain further insight into simple harmonic motion, particularly when the experimental data obtained can be compared with graphical methods that are used to demonstrate displacement, velocity and acceleration during SHM. Graphical illustrations of the phase difference between these quantities enables the student to better understand the relationship between these quantities and how they change over time.

Similar techniques could also be used to verify that velocity \(\displaystyle v=\pm \text{ }\omega\sqrt{A^2 \textbf{- } x^2}\), and so \(\displaystyle v_{max}=\omega A\).

The independence of the period t of a simple harmonic oscillator from the amplitude A may seem surprising to the student at first, but it is simple to demonstrate using a pendulum.

The motion of a simple harmonic oscillator can be elegantly described in mathematical form using relatively simple equations, but an appreciation of the energy changes (between kinetic and potential energy) is also very important. It is essential that the learner is able to describe the energy changes during a full oscillation (not just half of one), as well as to appreciate that potential energy comes in both the gravitational and elastic kinds – demonstration of SHM using a linear air track and springs is a clear way of demonstrating that not all potential energy is gravitational. Energy-displacement graphs for kinetic and potential energy during full oscillations are a useful way of visualising the energy changes.

The effect of changing pendulum length on the frequency of oscillation should be investigated, and the derivation of the equation \(\displaystyle T= 2\pi \text{ }\sqrt{\frac{l}{g}}\) should be examined. The use of this method to determine g is a classical Physics experiment, and it yields very good results.

Similarly, for mass-spring systems, the derivation of \(\displaystyle T= 2\pi \text{ }\sqrt{\frac{m}{k}}\) should be undertaken, and the value of \(\displaystyle k\) can be compared against the value obtained using simple Hooke’s Law experiments.

An examination of the simple pendulum (all mass concentrated at a single point) is all that is required in this specification, but extension to the real pendulum could be undertaken by more advanced students.

Oscillations can be affected by external forces, and the effects of outside influences on oscillation amplitude and frequency can lead to some of the most memorable demonstrations of Physics. The application of a periodic force to an oscillating system is often used to overcome damping effects and maintain a constant amplitude, and this specification enables study of both freely oscillating systems and forced oscillations.

There are always energy losses in oscillating systems, which are usually due to friction in the system and/or air resistance. The corresponding loss in amplitude is called ‘damping’, and can be easily experimentally observed by students. The utility of damping in oscillating systems, such as suspension springs, can then be discussed, along with a separation into overdamping (system no longer oscillates, and slowly returns to equilibrium), underdamping (oscillations continue with steadily decreasing amplitude) and critical damping (system no longer oscillates and returns to equilibrium more quickly than with overdamping). It is important that students can differentiate between these systems – a mathematical separation is however not required in this specification.

Application of a periodic driving force to a naturally oscillating system can have surprising results, especially when the driving frequency approaches, and then equals, the natural frequency of oscillation. A good demonstration of the resonance condition is memorable for learners, and the graphical representation of such resonance effects can be developed after observing such demonstrations. Application of their knowledge to practical examples leads to a better understanding of structural design.

**Common misconceptions or difficulties students may have**

The definition of circular motion in terms of the force acting being perpendicular to the motion of the object can be confusing at first, and needs to be clearly demonstrated to the learner. The term ‘centripetal’ should be used rather than ‘centrifugal’, as clarity can avoid confusion over the direction of the force. The introduction of the concept of angular velocity in terms of radians (rather than degrees) per second can be difficult, and time will be needed for the learner to gain confidence in using the concept of \(\displaystyle \omega\).

The definition of oscillatory motion needs to be clear in the minds of the learner before in-depth study is undertaken, and the positions of maximum acceleration and velocity need to be very clear. The learner should be confident in the use of angular frequency, and be able to easily manipulate the equations for \(\displaystyle \omega\) in terms of \(\displaystyle T\) and \(\displaystyle f\). The defining equation for SHM (\(\displaystyle a=\textbf{- }\omega^2x\)) needs to be emphasised (particularly the presence of the negative sign).

The concept of an isochronous oscillator can seem counter-intuitive at first, but experimental verification should address any issues. The concept of phase difference often causes uncertainty amongst less confident learners, and a simple graphical approach to visualising the relationship between acceleration, velocity and displacement will be needed to assist learning.

Learners should be confident with the concept of a vertical simple pendulum undergoing SHM, but such motion can also be shown in mass-spring systems, both vertical and horizontal (such as a linear air track). The concept of potential energy as a stored energy should not be an issue, but a full description of the energy changes during a complete oscillation (rather than just half of one) is required. The application of the Principle of Conservation of Energy to oscillating systems should be straightforward, but students can often omit to mention all the forms of energy that may be involved in an oscillatory system.

The difference between critical damping, overdamping and underdamping needs to be made clear, ideally in graphical form, and the learner needs to be able to explain examples of the application of damping, e.g. car suspension systems.

A clear description of resonance effects when the driving frequency equals the natural frequency of oscillation can pose problems, and it is important that learners observe some practical examples alongside a learning of amplitude-driving frequency graphs.

A common misconception when measuring oscillations is not to start at zero when counting them. Fudicial markers are very useful in the study of oscillations, and should not be forgotten when describing experiments. The likely sources of error in the study of oscillatory motion should be fully described – ‘parallax error’ needs to be more fully described (i.e. in measuring which quantity?) to gain credit.

**Conceptual links to other areas of the specification – useful ways to approach this topic to set students up for topics later in the course.**

An understanding of circular motion is required for study of motion of satellites around planets. While the motions of planets around the Sun is governed by Kepler’s Laws, an understanding of circular motion is required as an introduction to Kepler’s work. The concepts behind circular motion are also a key part of the study of the motion of charged particles in a magnetic field (through combining \(\displaystyle F=\frac{mv^2}{r}\) and \(\displaystyle F = BQv\), resulting in \(\displaystyle r = \frac {mv}{Bq}\), which is very useful in mass spectrometers).

An understanding of oscillations is very important for successful study across many areas of Physics. The production of electromagnetic waves using alternating electrical currents, alternating currents themselves, the production of sound waves and the motion of mass-spring systems are just a few areas where the study of oscillating systems leads to a greater understanding of the particular subject area.

### Concepts of circular motion

**Teaching Advanced Physics (TAP):**

Some demonstrations that serve as an introduction to the concept of circular motion.

###
F=mv^{2}/r and loop the loop

**Nuffield Foundation:**

Students can verify F = mv^{2} / r using experiments

Students can further investigate circular motion using a ‘loop the loop’ track.

### The concept of SHM as circular motion

The concept of SHM as circular motion projected onto a diameter may be seen using the material at both:

**School Physics:**

Simple harmonic motion as projected circular motion

**Nuffield Foundation:**

S.H.M. and circular motion.

### Satellites in circular motion

Good starting points for extending ideas of circular motion to satellites can be found at both:

**The Physics Classroom:**

Circular motion and satellite motion

**YouTube:**

Satellites in circular orbit.

### Electron beams in circular motion

The operation of a fine beam tube is shown at:

**YouTube:**

This is a useful clip for learners to view as preparation for carrying out the experiment.

## Thinking Contextually

### Overview

### Electron beam in a magnetic field

Students can observe circular motion through the effect of a magnetic field on a beam of electrons using a fine beam tube, such as that described at:

**Nuffield Foundation**

Fine beam tube.

### Circular motion

The physics of fairground rides can be investigated first-hand at local fun parks, but a useful introductory summary of the physics of such rides can be found at:

**School Physics:**

Fairground rides

### Foucault’s pendulum

Foucault’s pendulum can be studied using a number of resources, such as:

**Smithsonian Information:**

The idea of circular motion and the motion of a pendulum were combined to show that the Earth rotates around its axis.

### Time period of a pendulum

The construction of a pendulum for use in timing experiments (as used prior to the introduction of reliable chronometers) can be a useful exercise for the learner – source material can be used, such as:

**YouTube:**

Time period of a pendulum depends on it's length.

### Investigating factors affecting simple harmonic motion

### Observing forced and damped oscillations

### Comparing static and dynamic methods of determining spring stiffness

## Acknowledgements

### Overview

OCR’s resources are provided to support the teaching of OCR specifications, but in no way constitute an endorsed teaching method that is required by the Board and the decision to use them lies with the individual teacher. Whilst every effort is made to ensure the accuracy of the content, OCR cannot be held responsible for any errors or omissions within these resources. We update our resources on a regular basis, so please check the OCR website to ensure you have the most up to date version.

© OCR 2015 - This resource may be freely copied and distributed, as long as the OCR logo and this message remain intact and OCR is acknowledged as the originator of this work.