The maximum speed of a simple harmonic oscillator is v = (2 πf) A

max

Oscillatory motion in which acceleration, a, is directly proportional to displacement, x, and acts in the opposite direction, i.e. a a – x

a =-(2 f) x

Time period, T, is independent of the amplitude of the oscillation.

T=1/f

**Simple**

Harmonic

Motion

Harmonic

Motion

One complete cycle of oscillation can be compared with one rotation.

Angular frequency ω = 2 πf is a scalar measure of rotation rate.

Phase difference is the relationship between the pattern of vibration at two points.

Oscillations &

Circular motion

**Experiment!**

Simple Harmonic Motion

Angular frequency

Relationship between the pattern of vibration at two points.

2 points that have exactly the same pattern of oscillation are said to be in phase – zero phase difference.

2 points exactly opposite to one another – waves in antiphase.

Phase difference is the angle in radians between 2 oscillations.

Phase difference

Displacement is the distance an object has moved from the mean (or rest) position.

Amplitude is the maximum displacement.

Frequency is the number of oscillations per unit time at any point.

The period is the time for one complete oscillation to take place at any point.

Key terms

Describe simple examples of free oscillations;

Define and use the terms displacement, amplitude, period, frequency, angular frequency and phase difference;

Select and use the equation period = 1/frequency;

Define simple harmonic motion;

Select and apply the equation a = – (2 πf)2 x as the defining equation of simple harmonic motion;

Explain that the period of an object with simple harmonic motion is independent of its amplitude;

You will learn to…

Banging a drum

Hitting a nail with a hammer

Knocking on a door

Light

Warming effect of the Sun

Radio waves

X-rays

Microwaves

Examples of free oscillations

Refers to motion involving

a body that oscillates.

Smooth oscillations with no variation in amplitude.

The period is independent

of the amplitude.

e.g. a swinging pendulum

T, f & are also used,

but has no direct physical meaning.

Particle P moving at a steady speed in a circle.

T = period

f = frequency

ω = angular velocity

Periodic motion

A free oscillation has no driving mechanism and zero friction

and thus would oscillate for ever.

In practice, this does not happen

and so no oscillation is truly free.

Sketch a graph showing a more realistic oscillation

Free oscillation

Simple Harmonic Motion

**SHM**

Circular motion

Oscillatory motion

Displacement

Amplitude

Frequency

Period

Key terms

You will learn to...

-1

Select and use x = Acos(2π ft) or x = Asin(2π ft) as solutions to the equation a = – (2π f)2 x .

Select and apply the equation vmax = (2π f)A for the maximum speed of a simple harmonic oscillator.

Describe, with graphical illustrations, the changes in displacement, velocity and acceleration during simple harmonic motion.

A displacement time graph for a pendulum

Displacement follows a sine

or a cosine curve,

x= Asin(wt) or x = Acos(wt)

where A is the amplitude of oscillation.

2π f in units of radians per second (rads ).

One complete cycle of oscillation can be compared with one rotation.

One oscillation = one rotation = angle of 2 π radians

f = oscillations per unit time

2π f radians per unit time

Velocity &

Acceleration

Sketch the graphs for velocity & acceleration

Sketch the displacement graph as a sine curve

a very long pendulum

a trolley or other mass

tethered horizontally

between springs

a mass on a long vertical spring

Can show this using a motion sensor placed underneath the oscillating mass

Whether the equation is expressed as a sine or a cosine function depends on the phase of the oscillation at the time designated as zero.

x = Asin(2 ft)

x = Acos(2 ft)

Use differentiation to find equations for velocity and acceleration, as well as respective graphs.

Acceleration a =-(2 f) x

2

2

Questions

Textbook page 47 questions 1 & 2

Qu 1:

a) S - greatest negative a

b) R - greatest positive a

Qu 2:

a) (i) A = 0.11 m (ii) f = 1.43 Hz (iii) v = 0.99 m/s

you know the traces are sinusoidal

use the equations to find the max & min values

Hints:

Obtain a value for g by timing the oscillation of a pendulum

**Practical task**

Mass oscillating on a spring

Pendulum

What is the time period

in each instance?

**SHM**

in specific systems

in specific systems

The system reaches equilibrium and comes to rest in the shortest time possible, about T/4.

With any less damping, the motion would be under-damped.

Critical damping

Energy changes

No loss of energy to surroundings,

but energy changes take place.

When an object on a spring undergoes simple harmonic motion, the system’s potential energy and kinetic energy vary with time.

The sum of PE & KE is constant.

Energy in SHM

Describe and explain the interchange between kinetic and potential energy during simple harmonic motion.

Describe the effects of damping on an oscillatory system.

Light damping reduces oscillations slowly.

Heavy damping reduces oscillations quickly.

Critical damping stops the oscillation within one cycle.

The way in which the amplitude of the motion decays depends upon the nature of the damping forces.

If the energy lost depends only upon the speed of the oscillating system (for example if it is due to drag), the decay in amplitude will be exponential.

i.e. oscillations die down quickly at the beginning and less dramatically at the end

Effect of damping

Amplitude gradually decreases over time.

Also known as under-damped.

Light damping

Left to itself, a spring or a pendulum eventually stops oscillating because the mechanical energy is dissipated by

frictional forces.

This leads to a decrease in the amplitude of oscillation - such motion is said to be damped.

Damped oscillations

In reality energy is lost.

If energy is not supplied to an oscillation, it will die away.

Energy changes

Energy changes

Light damping: Only a small damping force.

Period of oscillation almost unchanged.

Heavy damping: Very strong damping force.

No oscillations occur and the body returns to its equilibrium position.

Critical damping: The cross over point between oscillation and no oscillation. (Before heavy damping).

Types of damping

The system takes a long time to return to its equilibrium point.

Also called over-damped, these systems do not oscillate.

Heavy damping

What are the

energy changes

taking place

during

simple

harmonic

motion?

Light damping

Heavy damping

Very heavy damping

E = 1/2 k x

Graph showing how the total energy E of an undamped oscillator of mass m varies with displacement x.

A curve can be added to show how the K.E. of the oscillator varies with displacement.

For a mass on a spring, the potential energy can be calculated using:

Mass

oscillating

on a

spring

What is the frequency and the angular frequency of the oscillation?

What is the phase difference between points i) V and Y and ii) Y and Z?

Displacement-time graph. Which points are:

i) at the amplitude of the oscillation,

ii) one period apart,

iii) in phase,

iv) in antiphase?

As you cycle,

the vertical displacement of your foot

is plotted against time

A displacement-time graph for a pendulum with no driving mechanism

A displacement-time graph for a simple harmonic oscillator.

What is the amplitude, the frequency, the maximum velocity?

A mass rests on a smooth surface.

It is attached by an open-wound spring that can be stretched or compressed to a fixed support.

Damping

For an oscillating system with no damping:

applied frequency natural frequency

of periodic force of the system at resonance

Resonance

Barton’s pendulums

One vibrating object can make another vibrate.

When these two frequencies are the same, a large amplitude is built up and this effect is called resonance.

Resonance

Describe practical examples

of forced oscillations and resonance

Describe graphically how the amplitude of a forced oscillation changes with frequency near to the natural frequency of the system

Describe examples where resonance is useful

and other examples where resonance

should be avoided

Tuning a radio or television receiver

Magnetic Resonance Imaging (MRI)

Microwave ovens

Uses of resonance

The Millennium Bridge over the Thames

“Typical English. Just got too much natural rhythm.”

Toronto CN tower is designed to be able to vibrate with a amplitude of over a metre!

Car designers try to eliminate resonance

Takoma Narrows Bridge

Examples of

unwanted resonance

When the amount of damping applied to a resonating object is varied it has two effects:

Damping effects

When resonance occurs,

there is a /2 phase difference

between the driver and the driven,

with the driver leading.

Phase difference

in resonance

The effect of damping on

the resonant response of an oscillator

A graph showing how the amplitude of a driven object varies with

the frequency of the driver

Barton’s pendulums:

One will oscillate with a much larger amplitude than the others as the driver pendulum’s frequency matches its natural frequency

A singer shattering a wine glass

Examples of resonance

All mechanical systems are damped since energy is continually being lost.

To maintain constant amplitude, energy must be supplied at the same rate that it is being lost.

A force must be applied to oppose

the damping force.

Free and forced

oscillations

the oscillator is more heavily damped

there is very little damping

The peak in (b) is broader.

The width of the peak gives us a measure of damping.

The resonance is less sharp and shifted to

a lower frequency.

Resonance

(Same diagram simplified)

acceleration, a

displacement, x

Make sure your calculator

is set in RADIANS not degrees!

You will learn to...

**Energy**

& Damping

& Damping

What values of PE & KE will the pendulum have at points x, y & z?

Sketch a graph of energy against time

to show the PE, KE

and total energy

Example:

Pendulum clocks used to have to be wound daily. Nowadays they use electric motors.

Examples:

A note being played on a piano getting quieter.

A child on a swing slowing down, etc.

Damping is often deliberate

e.g. car suspension system.

Example:

A child on a swing who stops being pushed by their parent.

Example:

The return spring on a door.

Example:

Car suspension system.

As the damping forces increase,

there is a slight increase

in the time period of the oscillation.

Graph of displacement x against time t

Graph of acceleration a against time t

Graph of velocity against time t

You will learn to...

=

Think about pushing a child on a swing...

Increasing the amount of damping reduces the amplitude of the driven oscillation.

Increasing the damping also slightly reduces the frequency of the driven oscillation.

2

p

Damping also refers to the process of deliberately reducing the amplitude of an oscillation.

The forces causing the amplitude to decrease are called dissipative forces.

The amplitude gradually decreases, reducing by the same fraction each cycle.

The effects of damping on

an oscillatory system

Textbook page 45 question 1 to 3

Simple Harmonic Motion Questions

Questions

Make sure your calculator

is set in RADIANS not degrees!

A displacement-time graph showing an oscillation.

At which point does the oscillator have:

a) the greatest negative acceleration,

b) the greatest positive velocity?

An object oscillates vertically in simple harmonic motion

along the line between points A and B.

At which point(s) is it when it:

is stationary,

has maximum velocity upwards,

has zero acceleration,

has maximum kinetic energy,

has maximum acceleration downwards?

The object causing this effect has a driver frequency, whilst the driven object oscillates at its natural frequency.

A displacement-time graph -

this time there is a driving force to keep the amplitude constant

-1

Reminder:

A graph showing

light damping is

the same as a graph with

no driving force

There IS friction and

there IS energy loss...

Questions

Textbook page 49 Qu 2, 3 & 4

Textbook page 51 Qu 1 & 2

Springs

worksheet

Questions

Textbook page 55 Qu 4

Textbook page 57 Qu 4

The mass is pulled to the right and released at t = 0. The equation for its

Displacement is x = 0.050 cos t. The spring constant is 10Nm .

Calculate the potential and KE of the system at t=0, 0.5s, 0.75s and 1.25s.

Hint: remember KE + PE is constant, and KE = 0 at t = 0.

**Resonance**

Time period, T, is independent of the amplitude of the oscillation.

T=1/f

Displacement follows a sin or a cosine curve,

x=Asin(2 ft) or x=Acos(2 ft)