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# SHM - Simple Harmonic Motion

OCR A2 G484 Newtonian World
by

## E Allcoat

on 22 September 2015

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#### Transcript of SHM - Simple Harmonic Motion

where A is the amplitude of oscillation.
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

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
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.

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

Mass oscillating on a spring
Pendulum

What is the time period
in each instance?

SHM
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
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

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,

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
is set in RADIANS not degrees!
You will learn to...
Energy
& 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
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)

Full transcript