chapter 14 oscillations

CAMBRIDGE A – LEVEL

PHYSICS

OSCILLATIONS

LEARNING OUTCOMESNUMBER LEARNING OUTCOME

i U n d e r s t a n d t h e c o n c e p t o f o s c i l l a t i o n s

a n d b e a b l e t o d i s t i n g u i s h b e t w e e n f r e e

a n d f o r c e d o s c i l l a t i o n s .

ii L e a r n w h a t i s m e a n t b y s i m p l e h a r m o n i c

m o t i o n ( S H M ) a n d a n a l y s e t h e

d i s p l a c e m e n t s , v e l o c i t y a n d a c c e l e r a t i o n

f o r S H M .

iii A n a l y s e t h e i n t e r c h a n g e b e t w e e n k i n e t i c

a n d p o t e n t i a l e n e r g y o f a s y s t e m i n S H M

iv W h a t i s d a m p i n g ?

v U n d e r s t a n d t h e r e l a t i o s h i p b e t w e e n

f o r c e d o s c i l l a t i o n s a n d r e s o n a n c e .

vi L e a r n s i t u a t i o n s w h e n r e s o n a n c e i s

u s e f u l a n d w h e n r e s o n a n c e s h o u l d b e

a v o i d e d .

OSCILLATIONS• An oscillation is defined as a repeated

back and forth motion on either side of

a fixed position made by an object.

• Examples of objects undergoing

oscillation: a swinging pendulum, a

beating heart, a vibrating guitar string,

etc.

T Y P E S O F O S C I L L AT I O N S

• Oscillations can be divided into:

I. free oscillations:

� Occur when the only force acting on the object is the

restoring force.

� Oscillations that occur when an rigid body is given an

initial disturbance and allowed to come to rest after

undergoing oscillations on its own.

� This object will oscillate at its natural frequency. The

natural frequency of a oscillating system is the frequency

at which the system will oscillate when given an initial

disturbance.

T Y P E S O F O S C I L L AT I O N S

II. forced oscillationsII. forced oscillations

� forced oscillations occur when anexternal driving force is used to get adamped system to continue itsoscillations.

� The system undergoing forcedoscillations will vibrate at a frequencyequal to the frequency of the drivingforce.

S I M P L E H A R M O N I C

M OT I O N


M OT I O N• Simple harmonic motion is defined as the motion of a

particle about a fixed point such that its acceleration, a is

proportional to its displacement, x from the fixed point,

and is directed towards the fixed point.

• Its instantaneous displacement from the undisturbed

position, � and acceleration, � can be related by the

equation � � ��, where � �angular frequency of

the object and � � �� , where = frequency of

oscillations of the system, in Hz.

• This equation can only be solved by using differential

equations.


M OT I O N


M OT I O N• A few notes about the equation

� � ��

I.I.I.I. �

• A few notes about the equation� � ��:

I.I.I.I. is directly proportional to � . Thisimplies a will be maximum when x is thelargest, and a = 0 when x = 0;

II. The minus sign indicates thatacceleration is due to a restoring force;i.e. the acceleration and instantaneousdisplacement will be opposite indirection to each other.


M OT I O N


M OT I O N

Figure 10.15; Page 265, Chapter 10: Circular Motion and Oscillations , International

A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education,

United Kingdom, 2008.


M OT I O N


M OT I O N• One possible solution for the differential equation

� � �� • One possible solution for the differential equation

for SHM is � � �� , where �� amplitude, or maximum possible displacementfrom undisturbed position, and � = angularfrequency, in rad s-1. This equation is used iftiming is started (t = 0) when object is atundisturbed position.

• Another possible solution is � � ��. Thisequation is used if timing is started when objectis at maximum displacement.

• View: http://www.showme.com/sh/?h=U7u1mLI


M OT I O N


M OT I O N

��

Image: http://clas.mq.edu.au/acoustics/waveforms/shm.gif

Dis

lace

me

nt


M OT I O N


M OT I O N• If we plot displacement, x as a function of• If we plot displacement, x as a function of

time, t, we obtain the graph in theprevious slide.

• We always set a direction for positive andnegative displacements.

• At which point(s) are acceleration:I. maximum,

II. minimum?

• How do we obtain the relationship betweenvelocity and time?


M OT I O N


M OT I O N• Differentiating �� w.r.t � , we can obtain��

� � � ��

• Differentiating �� w.r.t � , we can obtain�� , by also noting that

��

�� . �� represents the

maximum velocity.

• We can also use � � �� , where

�� amplitude, and � = instantaneousdisplacement.

• View: http://www.showme.com/sh/?h=Ys8EMfA,http://www.showme.com/sh/?h=qxnP6K8 andhttp://www.showme.com/sh/?h=6Hlg5ui


M OT I O N


M OT I O N

�� • If we look at �� , we

see that:

I. Speed is maximum when x = 0 (at undisturbedposition),

II. Speed is zero when � � ��,

III.The � indicates that velocity can be positive ornegative for the same displacement becausethe object could be moving away or towardsthe undisturbed position,

E X A M P L E S

Example; Page 267, Chapter 10: Circular Motion and Oscillations , International

A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education,


E X A M P L E S

Questions from Section 10.2 – 10.4; Page 271, Chapter 10: Circular Motion and

Oscillations , International A/AS Level Physics, by Mee, Crundle, Arnold and Brown,

Hodder Education, United Kingdom, 2008.

E N E R G Y C O N S I D E R AT I O N S

• The speed of an object undergoing SHM will

is always changing w.r.t time.

• This means that the kinetic energy of the

object is also changing w.r.t time.

• What happens to the kinetic energy?

• To understand, let us consider a mass –

spring system that is oscillating.


• Diagram 14.13, page 445,

Sear’s and Zemansky’s

University Physics, Young

and Freedman, 13th edition,

Pearson Education, San

Francisco, 2012.


• From the diagram above, when the extension of the

spring is maximum (maximum elastic potential

energy), the speed is zero (minimum kinetic

energy).

• Conversely, when the extension of the spring is zero

(minimum elastic potential energy), the speed of

the object is maximum (maximum kinetic energy).

• As the displacement of the mass decreases, its

kinetic energy increases, while as its displacement

increases, its kinetic energy decreases.


• This shows that there is an interchange between

kinetic energy and stored elastic potential energy.

• ! �"

#$%# �

&

�'��

� � ��)

• ( �&

�'��

• Adding yields ( ) ! �&

�'��

� �constant

• This means that the total energy of a system in SHM

is constant; i.e. energy is interchanged between

kinetic energy and elastic potential energy.


• Diagram 14.14, page 447, Sear’s and Zemansky’s University Physics,

Young and Freedman, 13th edition, Pearson Education, San Francisco,

2012.




2012.

E X A M P L E S

Example from Section 10.5; Page 273, Chapter 10: Circular Motion and Oscillations ,

International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education,


E X A M P L E S

Questions from Section 10.5; Page 274, Chapter 10: Circular Motion and Oscillations ,

International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education,


DAMPING• So far, we have seen systems oscillating only under the

actionof the restoring force.

• What happens when there exists dissipative forces (forces

that do work on the system to remove its energy)?

Examplesof dissipativeforces: friction,air resistance.

• Dissipative forces reduce the amplitude of the oscillations

becausethe totalenergyof the systemwould be lower.

• Damping is defined as the reduction of amplitude of

oscillation.

DAMPING• So far, we have seen systems oscillating only under the

actionof the restoring force.

• What happens when there exists dissipative forces (forces

that do work on the system to remove its energy)?

Examplesof dissipativeforces: friction,air resistance.

• Dissipative forces reduce the amplitude of the oscillations

becausethe totalenergyof the systemwould be lower.

• Pleaseview:http://www.showme.com/sh/?h=j4FjpYm

DAMPING

III. Criticaldamping.

• Damping is defined as the reduction of amplitude

of oscillation due to the action of dissipative

forces on the system.

• The oscillations that continue after damping are

known as damped oscillations.

• There are three degrees of damping that exist:

I. Light - damping or under - damping,

II. Over - damping,and

III. Criticaldamping.

DAMPING• Damping is said to be light if the amplitude of oscillation

decreases gradually with time.

• The decrease in amplitudeobeys an exponentialenvelope.

Figure 10.25; Page 275, Chapter 10: Circular Motion and Oscillations , International A/AS

Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United Kingdom,

2008.

DAMPING• The effects of damping can be increased to a point damping

is said to be critical.

• Damping is said to be critical if the system returns to its

equilibrium without any further oscillations over a short

time period.

• Further increase in damping would cause the system to be

overdamped. A system is overdamped if it returns to the

equilibrium position without any further oscillations but

over a time period longer than that required for critical

damping.

DAMPING

Figure 10.26; Page 275, Chapter 10: Circular Motion and Oscillations , International

A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United

Kingdom, 2008.

DAMPING• Examples of damped oscillating systems:

Page 275, Chapter 10: Circular Motion and Oscillations , International A/AS Level

Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United Kingdom,

2008.

DAMPING• Examples of damped oscillating systems:

• View http://www.showme.com/sh/?h=T6z1Uwq



2008.

FORCED OSCILLATIONS

• We can use an external driving force to maintain

the amplitude of the natural oscillation of a system.

• When we use the external driving force to cause a

system to oscillate, we are causing the system to

undergo forced oscillations.

• A system undergoing forced oscillations will

oscillate at a frequency equal to the frequency of

the driving force.

FORCED OSCILLATIONS

Figure 10.28, Page 276, Chapter 10: Circular Motion and Oscillations , International


Kingdom, 2008.

• The diagram above shows mass – spring system that is set up

to undergo forced oscillations.

RESONANCE• If the frequency of forced oscillations is increased

until the frequency of the driving force is equal /

matches the natural frequency of the system, the

system will undergo resonance.

• During resonance, maximum energy transfer

occurs resulting in maximum amplitude of

vibration.

• View: http://www.showme.com/sh/?h=eID4Gqu



2008.

RESONANCE

Definition of resonance from Page 276, Chapter 10: Circular Motion and Oscillations ,

International A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder

Education, United Kingdom, 2008.

RESONANCE

Figure 10.29: Page 277, Chapter 10: Circular Motion and Oscillations , International


Kingdom, 2008.

• The diagram on the left slide

shows how the amplitude of

the driven system varies as

the frequency of the driving

force increases.

• Amplitude peaks when the

frequencies arematched.

• View:

http://www.showme.com/sh/

?h=tb6bIVU

RESONANCE

Figure 10.30: Page 277, Chapter 10: Circular Motion and Oscillations , International


Kingdom, 2008.

RESONANCE



2012.

USEFUL RESONANCE

• Please also view:

http://www.showme.com/sh/?h=ihn8AUa

From Page 278, Chapter 10: Circular Motion and Oscillations , International A/AS

Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United

Kingdom, 2008.

U N D E S I R E D R E S O N A N C E

• Please also view:

http://www.showme.com/sh/?h=rl2DJZY and

http://www.showme.com/sh/?h=kyO1Wng

From Page 278, Chapter 10: Circular Motion and Oscillations , International A/AS

Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United

Kingdom, 2008.

EXAMPLESMay/Jun 2008, Paper 4, question 3.

EXAMPLESMay/Jun 2008, Paper 4, question 3 (cont’d).

EXAMPLESMay/Jun 2010, Paper 42, question 2.

EXAMPLESMay/Jun 2010, Paper 42, question 2 (cont’d).

HOMEWORK1. Oct/Nov 2008, Paper 4, question 3.1. Oct/Nov 2008, Paper 4, question 3.

2. May/June 2009, Paper 4 , question 4.

3. Oct/Nov 2009, Paper 41, question 4.


5. May/June 2010, Paper 41, question 3.



8. May/June 2011, Paper 41, question 3.


chapter 14 oscillations

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