chapter 14 oscillations
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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
S I M P L E H A R M O N I C
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.
S I M P L E H A R M O N I C
M OT I O N
S I M P L E H A R M O N I C
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.
S I M P L E H A R M O N I C
M OT I O N
S I M P L E H A R M O N I C
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.
S I M P L E H A R M O N I C
M OT I O N
S I M P L E H A R M O N I C
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
S I M P L E H A R M O N I C
M OT I O N
S I M P L E H A R M O N I C
M OT I O N
��
Image: http://clas.mq.edu.au/acoustics/waveforms/shm.gif
Dis
lace
me
nt
S I M P L E H A R M O N I C
M OT I O N
S I M P L E H A R M O N I C
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?
S I M P L E H A R M O N I C
M OT I O N
S I M P L E H A R M O N I C
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
S I M P L E H A R M O N I C
M OT I O N
S I M P L E H A R M O N I C
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,
United Kingdom, 2008.
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,
United Kingdom, 2008.
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.
E N E R G Y C O N S I D E R AT I O N S
• Diagram 14.13, page 445,
Sear’s and Zemansky’s
University Physics, Young
and Freedman, 13th edition,
Pearson Education, San
Francisco, 2012.
E N E R G Y C O N S I D E R AT I O N S
• 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.
E N E R G Y C O N S I D E R AT I O N S
• 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.
E N E R G Y C O N S I D E R AT I O N S
• Diagram 14.14, page 447, Sear’s and Zemansky’s University Physics,
Young and Freedman, 13th edition, Pearson Education, San Francisco,
2012.
E N E R G Y C O N S I D E R AT I O N S
• Diagram 14.15, page 447, Sear’s and Zemansky’s University Physics,
Young and Freedman, 13th edition, Pearson Education, San Francisco,
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,
United Kingdom, 2008.
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,
United Kingdom, 2008.
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,
United Kingdom, 2008.
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
Page 275, Chapter 10: Circular Motion and Oscillations , International A/AS Level
Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United Kingdom,
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
A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United
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
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
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
A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United
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
A/AS Level Physics, by Mee, Crundle, Arnold and Brown, Hodder Education, United
Kingdom, 2008.
RESONANCE
• Diagram 14.28, page 460, Sear’s and Zemansky’s University Physics,
Young and Freedman, 13th edition, Pearson Education, San Francisco,
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 2008, Paper 4, question 3 (cont’d).
EXAMPLESMay/Jun 2008, Paper 4, question 3 (cont’d).
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).
EXAMPLESMay/Jun 2010, Paper 42, question 2 (cont’d).
EXAMPLESMay/Jun 2010, Paper 42, question 2 (cont’d).
EXAMPLESMay/Jun 2010, Paper 42, question 2 (cont’d).
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.
4. Oct/Nov 2009, Paper 42, question 3.
5. May/June 2010, Paper 41, question 3.
6. Oct/Nov 2010, Paper 41, question 3.
7. Oct/Nov 2010, Paper 43, question 3.
8. May/June 2011, Paper 41, question 3.
9. Oct/Nov 2011, Paper 43, question 3.
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