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Oscillations and Waves
Oscillations
Examples of oscillating systems
• a boat at anchor at sea
• the human vocal chords
• an oscillating cantilever
• the Earth’s atmosphere after a large explosion.
• a mass at the end of the spring
• a tuning fork
• a pendulum
• the strings of a guitar or piano
• bridge can vibrate when heavy truck passes
• electromagnetic waves – light waves, radar, radio waves
• atoms vibrate within molecule
• molecules of the solid oscillate about their equilibrium
positions.
In a solid, the molecules are bond together as if
they are connected by springs. The molecules
are in random vibration and the temperature of
the solid is a measure of the average kinetic
energy of the molecules.
The particles in a solid vibrate more
when it is heated, and take up more
room.
What all of them have in common? To make a mathematical
model of oscillatory motion, we will analyze two different
oscillations and see if there is any similarity.
Analyzing two examples of oscillations:
pendulum and a mass on a spring
The pendulum - The “restoring” force
• Point O is the equilibrium
position of the pendulum
• To start the pendulum, you
displace it from point O to
point A and let it go!
• The mass is falling from
point A to point O then rises
from O to B. Why?
• It is not a constant force. It varies with the object’s
displacement from its equilibrium position; greater
displacement, greater force;
zero displacement, zero force – equilibrium position.
It always points toward equilibrium acting to bring
(RESTORE) the object back to equilibrium, point O.
• from A to O the restoring force accelerates the
pendulum down
• It gets smaller and smaller and at point O is zero, so
what keeps the pendulum going??? It is inertia!!!
• as it moves from O to B, restoring force slows it down
(as it would any object that is moving up), until at B it
momentarily comes to rest.
let’s look at energy
 to start the pendulum, we move it from O to A. At point A it
has only potential energy due to gravity (GPE)
 from A to O, its GPE is converted to kinetic energy, which
is maximum at O (its speed is maximum at O)
 from O to B, it uses its kinetic energy to climb up the hill,
converting its KE back to GPE
 at B it has just as much GPE as it did at A.
springs  amazing devices!
 To start the oscillations,
you pull the mass and let it
go!
 The spring force (result of
intermolecular forces of the
spring) always acts to
restore the spring back to
equilibrium. In doing so it
pulls the mass toward the
equilibrium.
 The greatest force is at
the maximum distance. As
the distance decreases, the
force decreases. EPE is
being converted into KE of
the ball.
 Spring force is zero at equilibrium (unstretched) position.
 Once the mass passes equilibrium position (because of
inertia) the spring force will act in opposite direction of the
motion slowing down the mass.
 After the mass has come momentarily to rest, the spring
pulls it back toward equilibrium and the process continues
- oscillations.
 The spring force is restoring force.
Energy in the spring oscillations
• The net force is the vector sum of grav. and tension
force: red vector
• This is the force responsible for the motion of the
pendulum and it is called “restoring force”
The restoring force is the key to understanding all
systems that oscillate or repeat a motion over and over.
 a compressed or stretched spring has elastic potential
energy ( = work done on it to stretch it or compress it)
 this elastic potential energy is what drives the system
(PEmax = ½ k x02 = total energy)
 if you let the mass go, this elastic PE changes into KE.
 when the mass passes the equilibrium point, the KE goes
back into PE
 if there is no friction the energy keeps sloshing back and
forth but it never decreases
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Formal definition of SHM.
Some terminology
Cycle – one complete oscilation
Equilibrium position – position where an object would
rest if not disturbed.
If the acceleration a of a system is directly proportional to its
displacement x from its equilibrium position and is directed
towards the equilibrium position, then the system will execute
SHM.
Displacement x, θ – displacement from equilibrium position.
Amplitude x0, θ0 – the maximum displacement of an
oscillating object from equilibrium.
Period T – the time it takes an oscillating system to make
one complete oscillation. (A  O  B  O  A )
Frequency f – the number of complete oscillations made by
the system in one second.
Frequency = 1/period
f=
Another definition:
Whenever the force acting on a particle is linearly
proportional to the displacement and directed toward
equilibrium, the particle undergoes simple harmonic motion.
Such a force is called a linear restoring force.
Properties of SHM
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T
(f) = 1/s = s-1 = 1 Hz (hertz)
Example:
A weight suspended from a spring is seen to bob up
and down over a distance of 20 cm, twice each second. What
is its frequency? Its period? Its amplitude?
Frequency = 2 per second = 2 Hz
Period = 1/frequency = ½ s
Amplitude = 10cm
Graphical treatment and math
 To analyse these oscillations further, we can plot graphs
for these motions.
 You can plot a displacement – time graph by attaching a
pen to a pendulum and moving paper beneath it at a
constant velocity, or by shining the light on and oscillating
spring.
 When x is max. or min. the velocity is zero, and
acceleration and force are maximum in the direction
opposite to displacement.
 When x = 0, object is at equilibrium position, a = 0, F = 0
and v is maximum.
Period is CONSTANT and does NOT depend on amplitude.
Examples:
1. A pendulum completes 20 cycles in 12s. What is
(a) frequency?
(b) the angular frequency?
a. f = 20/12 s = 1.7 Hz
b. ω = 2πf = 10.5 Hz
the ink trace
should look like
this graph
The shape of this displacement – time graph is cosine or sin
curve depending on when you start counting the time.
2. A steel ball is dropped onto a concrete floor. Over and
over again, it rebounds to its original height. Is this SHM?
o Look for equilibrium position in the middle and a force that
is directed toward it from both sides of the equilibrium. If
that force changes as the distance changes you found
restoring force!!!
o During the time when the ball is in the air, either falling
down or rebounding up, the only force acting on the ball is
its weight, which is constant.
o There is no equilibrium position about which oscillations
occur. Thus, the motion of the bouncing ball is not simple
harmonic motion.
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5. The graph in next figure shows the variation with time t of
the displacement x of a system executing SHM.
Examples of the ones that are:
- Pendulum: if the amplitude is less then 150,
period doesn’t depend either on amplitude, or on mass but
only on length.
- Spring: if the amplitude is small compared to the
length of the spring, oscillations are SHM.
Even if friction or air resistance decreases the
amplitude, the period remains the same.
What is meant by damping?
Use the graph to determine the
(i) period of oscillation
(ii) amplitude of oscillation
A: (i) T = 2.0 s
(ii) x0 = 8.0 cm
Period of oscillations for two simple harmonic
oscillators
L
g
period is independent of amplitude
period only depends on its length
period does not depend on the mass
period depends on the value of g e.g.. the same
pendulum oscillates slower on the moon than on earth.
Simple pendulum
o
o
o
o
T=2π
Mass-spring system T = 2π
m
k
o the period gets smaller if a stronger spring (larger k) is
used
o the period of oscillation is longer if a bigger mass (m) is
used
 Heavier kids can not swing faster.
 When walking, we allow our legs to
swing with the help of gravity, like a
pendulum. In the same way that a long
pendulum has a greater period, a
person with long legs tends to walk with
a slower stride than a person with short legs.
 This is most noticeable in long-legged animals such as
giraffes, horses, and ostriches, which run with a slower
gait than do short-legged animals such as dachshunds,
hamsters and mice.
 Example: Is the time required to swing to and fro on a
playground swing longer or shorter when you stand rather
than sit?
 When you stand, the pendulum is effectively shorter,
because the center of mass of the pendulum (you) is
raised and closer to the pivot.
So period is less – it takes a shorter time.
◘
 "to damp" is to decrease the amplitude of a wave
 When deriving equations for PE and KE for an oscillating
system we assumed that no energy is lost.
 In real system there is always friction at the support and
sometimes air resistance. The work the system has to do
against these forces results in loss of energy as it
oscillates.
 The amplitude of the oscillations gradually decreases with
 time Oscillations, the amplitude of which decrease with
time, are called damped oscillations, whereas the effect is
called damping.
 All oscillating systems are subject to damping as it is
impossible to completely remove friction.
 Because of this, oscillating systems are often
classified by the degree of damping.
Not every periodic motion produced by restoring
force is SHM.
Light damping
 If the opposing forces are small, the result is gradual loss
in total energy. The oscillations are said to be lightly
damped.
 The decay in amplitude is relatively slow and the
pendulum will make quite a few oscillations before finally
coming to rest.
 Example: spring in air would have a little damping due to
air resistance.
 Frequency of damped harmonic motion
You can see from the graph that the frequency does not
change as the amplitude gets less. As the motion slows
down, the distance travelled gets less, so the time for each
cycle remains the same.
Heavily damped oscillations
 The amplitude of the heavily damped oscillations decay
very rapidly and the system quickly comes to rest. Such
oscillations are said to be heavily damped.
 Example: If the mass is suspended in water, the damping
is greater, resulting in a more rapid energy loss.
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Natural frequency
If the spring is pulled down and released it will oscillate.
The frequency of this oscillation is called natural frequency.
Definition: Natural frequency is the frequency an object will
vibrate with after an external disturbance.
 All objects have a natural frequency or set of frequencies at which they vibrate freely. These frequencies depend
only on the system itself. (imagine jello)
Critical damping
Critical damping occurs if the resistive force is so big that the
system returns to its equilibrium position without passing
through it. The mass comes to rest at its equilibrium position
without oscillating. The friction forces acting are such that
they prevent oscillations.
Example: This would be the case if the mass were
suspended in a thicker liquid such as honey.
 The mass/spring system oscillates at a certain frequency
determined by its mass, m and the spring stiffness
constant, k
 A pendulum always oscillates at the same frequency,
determined by length, when set in motion.
 More complicated systems, such as bridges, also vibrate
with a fixed natural frequency. A glass and stone too.
Your heart too. And spleen.
Forced oscillations
 If the support of the spring is oscillated, then the system
will be forced to vibrate at another frequency.
 If a system is forced to oscillate at a frequency other than
the natural frequency, this is called a forced oscillation.
 If the driving force has the same frequency as the natural
frequency, the resonance occur.
Definition of phenomenon known as resonance:
The increase in amplitude of oscillation of a system
exposed to a periodic driving force with a frequency
equal to the natural frequency of the system.
 The oscillations (vibrations) can produce undesirable and
sometimes, dangerous effects.
 For example, when a ball strikes the strings of a tennis
racquet, it sets the racquet vibrating and these vibrations
will cause the player to lose some control over his or her
shot. For this reason, some players fix a “damper” to the
springs. If placed on the strings in the correct position, this
has the effect of producing critically damped oscillations
and as a result the struck tennis racquet moves smoothly
back to equilibrium.
 In addition, vibrations caused by the impact of the ball with
the strings of a racquet normally are transmitted through
the handle of the racquet and the hand and wrist of the
player to the forearm where it may cause a tennis elbow.
Examples:
Identify which of the following oscillatory systems are likely to
be lightly damped and which are likely to be heavily damped.
1.
2.
3.
4.
5.
6.
7.
8.
Atoms in a solid.
Car suspension
Guitar string
Harmonic oscillator under water.
Quartz crystal.
A cantilever that is not firmly clamped.
Oil in a U-tube
Water in a U-tube
Lightly damped
Heavily damped
1, 3, 5, 8
2, 4, 6, 7
Resonance examples
 The resonance can result in a quite
dramatic increase in amplitude that
sometimes can be very unfortunate
 A lazy monkey gives a single push to
a
swing. The swing oscillates at its
natural
frequency. With no further pushes (no energy input), the
oscillations of the swing will die out and the swing will
eventually come to rest. This is an example of damped
harmonic motion.
 A busy monkey, each time the swing returns to him, gives
it another push. The amplitude of the
swing gets larger and larger and if not
careful he’ll end up with the swing doing
the work on his face. Driving force has
the same frequency as the natural
frequency of the swing. Resonance occurs.
Those of you who have siblings
might have had an unpleasant
knock down after being so good. Or
you might have been on the swing.
When you push a child on a swing
you are using resonance to make the child go higher and
higher.
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Using Resonance to shatter a Kidney stone.
 By tuning ultra sound waves to the natural frequency of a
kidney stone, we can rely on resonance to pulverize the
stone.
 The result can be dramatic increase in amplitude that
sometimes is very unfortunate
 MOVIE – TACHOMA BRIDGE
Resonance curve
Forced frequency and amplitude
What is of particular interest is when the forced frequency
is close to and when it equals the natural frequency.
We now look to see how the amplitude of an oscillating
system varies with the frequency of the driving force
(resonance curve).
The graph shows the variation with frequency f of the driving
force of the amplitude of three different systems to which the
force is applied.
 Enrico Caruso's voice possessed a richness of sound that
was said to be able to shatter a crystal goblet by singing a
note of the right frequency at full voice. Sound waves
emitted by the voice act as forced vibration on the glass.
At resonance, the resulting vibration may be large enough
in amplitude that the glass exceeds its elastic limit and
breaks.
 two tuning forks
 A structure such as bridge has natural frequency and can
be set into resonance by an appropriate driving force. It
has been reported that a railway train has collapsed
because a nick in one of the wheels of a passing train set
up a resonant vibration in the bridge.
 Marching soldiers break step when crossing the bridge to
avoid the possibility of similar catastrophe.
 Resonant vibrations due to the wind turbulences that
matched the natural frequency of the bridge destroyed
Tacoma Narrows.
 Have you ever had a strange feeling while listening to loud
music in a car (apart from that you are going deaf). Like
something is shaking inside you.
 Some infrasound frequencies (the ones you can’t hear)
can actually have the same frequency as natural
frequency of some of yours internal organs.
 And yes, that’s what you are feeling.
 It’s shaking.
How resonance works energy wise
 resonance is a way of pumping energy into a system to
make it vibrate
 in order to make it work the energy must be pumped in at
a rate (frequency) that matches one of the natural
frequencies that the system likes to vibrate at.
 you pump energy into the child on the swing by pushing
once per cycle
The
sharpness of
the peak is
affected by
the amount of
damping in
the system.