Seeing and Hearing
Dr. L. Bradley
Course 1004A
• 15 lectures and 3 tutorials
• Full syllabus on the web:
http://www.tcd.ie/Physics/Local/Students/JF/
Seeing_Hearing.html
or
http://www.tcd.ie/Physics/People/Louise.Bradle
y/Lecture/seeing%20and%20hearing.htm
Course 1004A
•Textbooks
Sears and Zemansky’s University Physics
with modern physics (10th edition – Addison
Wesley) by Young and Freedman
How things work – the physics of everyday
life (2nd edition – Wiley) by Bloomfield
•Read recommended sections before the lectures
Topics
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Oscillations
Waves
Sound
Properties of Light
Geometrical Optics
Interference
Diffraction
Oscillations/Vibrations
Most things vibrate/oscillate when given an impulse
Forces can be different but observed behaviour can be
described in a common way.
Examples: Vibration of quartz crystal in a watch, pendulum in
grandfather clock, sound from vibration of wind instrument,
motion of piston in a car engine, child on a swing
http://home.a-city.de/walter.fendt/phe/pendulum.htm
Periodic Motion: motion that repeats itself over and over
Oscillation will always occur if there is a force trying to return
the body to its equilibrium position – Restoring force
Displacement: change in position relative to equilibrium
Amplitude: maximum magnitude of the displacement from the
equilibrium position (e.g. unit metres)
Cycle: one round trip
Oscillations 2
Period: the time for one cycle (unit: seconds)
Frequency: number of cycles per second (unit: Hertz)
1
1
f = , T=
T
f
Angular Frequency
2π
ω = 2πf =
T
Another example: the spring
http://home.a-city.de/walter.fendt/phe/springpendulum.htm
Simple Harmonic Motion (SHM)
• Simplest kind of oscillation
Restoring Force exerted by the
spring on the body is
proportional to the displacement
k = force constant, always
positive (N/m)
• Body undergoing SHM is a
harmonic oscillator
F∝x
Hooke ' s
F = −kx F = ma
−k
a=
x ( SHM )
m
Law
University Physics
Figure 13-1
Page 393
SHM 2
• SHM is the projection of uniform circular
motion onto a diameter
http://surendranath.tripod.com/Shm/Shm01.html
Q
O
P
Acceleration proportional to displacement as before
=> SHM
Projection of Circular Motion
University Physics
Figure 13-3
Page 395
SHM 3
Result: angular frequency of mass m acted on
by a restoring force k
ω determined by k and m
ω=
k
m
ω
1
f =
=
2π 2π
1
m
T = = 2π
f
k
k
m
Larger m => less acceleration, moves
more slowly, longer T
Larger k => stiffer spring,
greater acceleration, higher speeds,
shorter T
NB in SHM f and T do not depend on
the amplitude A
SHM 4- General Description
http://home.acity.de/walter.fendt/phe/springpendulum.htm
Consequences
At equilibrium position x = 0
=> v = vmax or –vmax, a = 0
At max displacement x = A
or
x = -A
=> v = 0,
i.e. body instantaneously at
rest and a has maximum
magnitudes,
i.e. restoring force
maximum
http://home.acity.de/walter.fendt/phe/springpendulum.htm
University Physics
Figure 13-9
Page 399
SHM 5 – Determine φ and A
Given xo and vo
To find A:
To find φ:
xo = A Cos φ
2
vo − ωA sin φ
=
= −ω tan φ
x0
A cos φ
 − v0 

⇒ φ = arctan
 ωx 0 
2
2
2
vo
2
2
=
A
Sin
φ
2
ω
2
0
2
v
A= x +
ω
2
0
SHM 6 - Energy
• Closed/Isolated System (frictionless)
• Total mechanical energy is conserved = constant
• Spring mass negligible
Total mechanical energy
is related to the amplitude A
Energy is continuously transformed from kinetic to potential and back
http://home.a-city.de/walter.fendt/phe/pendulum.htm
Question
A block with mass 200g is connected to a light
horizontal spring of force constant 5.00 N/m and
is free to oscillate on a horizontal frictionless surface.
a) If the block is displaced 5.00 cm from equilibrium
and released from rest, find the period of the motion?
b) What is φ?
c) Determine the maximum speed and maximum acceleration
of the block.
The Vertical Spring
Is this an example of SHM?
• A mass suspended from the
end of a spring
Equilibrium x = 0, forces balanced k∆l
Extension = ∆l − x
Net force spring exerts on body =
= mg
University Physics
Figure 13-14
Page 404
k (∆l − x )
Look familiar?
How about if the weight is placed on top of the spring?
Same physics describes the vibration of molecules !
No springs – The Pendulum?
• Is this an example of SHM?
Can take simple pendulum, idealised, mass all concentrated at
one point.
Easily extended to finite bodies
To have SHM must have restoring force α displacement
Find Restoring force is proportional to gravity
F = − mgSinθ
F ∝ Sinθ
But for small angles sin θ ≅ θ
x
L
k
g
=
m
L
⇒ F = −mg
Longer pendulum =>
longer period lower
frequency
ω=
1
⇒ f =
2π
g
L
Question
A 40.0 N force stretches a vertical spring 0.250 m.
What mass must be suspended from the spring so that
the system will oscillate with a period of 1.00 s?
If the amplitude of the motion is 0.050 m and the
period is 1.00 s, where is the object and in what
direction is it moving after 0.35 s?
(Assume object at equilibrium at time zero).
What force (magnitude and direction) does the spring
exert on the object when it is 0.030 m below the
equilibrium position moving upward?
Resonance
University Physics
Figure 13-22
Page 412
Natural frequency: frequency body
oscillates at when given a single impulse
Damped oscillations: amplitude of
displacement decreases over time
e.g. due to air resistance
Forced oscillations: providing an impulse
at some frequency, a driving force
Resonance: at some frequency of the
driving force the largest displacement
is achieved, i.e. when the driving force
frequency matches the natural
frequency
http://home.a-city.de/walter.fendt/phe/resonance.htm