Everything oscillate
even stars – but it takes some time –
1 oscillation per 70 million years
Oscillations
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most solids are elastic - most material objects vibrate when
given an impulse
a boat at anchor at sea
the human vocal chords
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
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molecules of the solid oscillate about their equilibrium positions
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atoms vibrate within molecule
H 2O
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.
cold
hot
What do all of them have in common?
Restoring force !!!!
And the same math -
Oscillations
Oscillations are vibrations which repeat themselves.
EXAMPLE: Oscillations
can be driven externally,
like a pendulum in a
gravitational field
EXAMPLE: Oscillations can
be driven internally, like a
mass on a spring.
Time period, amplitude and displacement
 Consider a mass on a spring that is displaced
4 meters to the right and then released.
 Amplitude is the maximum
distance x0 from
equilibrium position.
In this example x0 = 4 m.
 Equilibrium position is the point of zero displacement.
 Displacement x is measured from equilibrium.
 The period T is the time it takes for the mass to make one
full oscillation or cycle.
For this particular oscillation, the period T is about 24 s.
x
x0
Time period and frequency
 The frequency f (measured in Hz or cycles / s) is defined as how
many cycles (oscillations, repetitions) occur each second.
 Since period T is seconds per cycle, frequency must be 1 / T
1
𝑓 =
𝑇
1
𝑇 =
𝑓
relation between T and f
EXAMPLE: The cycle of the previous example repeated each 24 s.
What are the period and the frequency of the oscillation?
SOLUTION:
The period is T = 24 s.
The frequency is f = 1 / T = 1 / 24 = 0.042 Hz
1
(𝑓) = = 𝑠 −1 = 1 𝐻𝑧 (ℎ𝑒𝑟𝑡𝑧)
𝑠
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
A pendulum completes 20 cycles in 12s. What is
(a) frequency?
a.
f = 20/12 s = 1.7 Hz
The pendulum - a closer look.
The “restoring” force
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To start the pendulum, you displace it
from point O to point B and let it go!
The mass is falling from point B to
point O then rises to the other side.
Why?
Two forces: tension and grav. force
When added up we get net force
Net force has two components: one in
the direction tangential to the circle
speeding it up on the way down and
slowing it down on the way up and the
one toward the pivot: centripetal force
responsible for motion in the circle
At point O the tangential net force 𝑚𝑔 sin 𝜃 vanishes. Point O is the equilibrium
position of the pendulum
On either side of point O, 𝑚𝑔 sin 𝜃 always act to bring (restore) the pendulum
back to equilibrium, point O.
That’s why we call it “restoring” force.
This is the force responsible for the oscillatory motion of the pendulum.
Mass on the spring - a closer look.
The “restoring” force

To start the oscillations, you pull the mass and let
it go!
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The spring force F= – kx always acts to restore
the spring back to equilibrium. In doing so it pulls
the mass toward the equilibrium.
𝐹 = −𝑚𝑔 𝑠𝑖𝑛𝜃
The role of the restoring force
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the restoring force is always in opposite
direction from displacement,
pointing toward equilibrium to bring the
object back to equilibrium,
It is not constant force. It varies with the
object’s displacement from its
equilibrium position; greater
displacement, greater force; greater
acceleration.
On the way toward equilibrium restoring
force is speeding up the object
On the way away from equilibrium, force
is slowing it down.
as it moves toward the end points,
gravity/spring force slows it down,
until at the ends it momentarily comes
to rest and starts to return back
v=0
v=0
v = vmax
𝑎 = 𝑎𝑚𝑎𝑥 →
𝐹 = 𝐹𝑚𝑎𝑥 →
v=0
𝑎=0
𝐹=0
v = vmax
𝑎 = 𝑎𝑚𝑎𝑥 ←
𝐹 = 𝐹𝑚𝑎𝑥 ←
v=0
x
𝐹 = −𝑘𝑥
Qualitatively describing the energy changes
taking place during one cycle of an oscillation
 Work done on pendulum in lifting it ⇾ grav. potential energy
 Let the green rectangle represent
the potential energy of the system.
 When we release it, the kinetic energy
(red part) will grow as the potential energy
diminishes.
 A continuous exchange between EK
and EP occurs.
 If friction and drag are both zero,
EK + EP = ET = CONST
Qualitatively describing the energy changes
taking place during one cycle of an oscillation
 Consider the mass-spring system. The mass is pulled to the right
and held in place
x
x0
 A continuous exchange between EK and EP occurs.
 If friction and drag are both zero,
EK + EP = ET = CONST
Qualitatively describing the energy changes
taking place during one cycle of an oscillation
EK + EP = ET = CONST
relation between EK and EP
Energy
 If we plot both kinetic energy
and potential energy vs. time
for either system we would
get the following graph:
time
x
Phase difference
 We can pull the mass to the right and then release it to begin its motion:
Start
stretched
x
The two motions are half a cycle out of phase.
Start
compressed
x
 Or we could push it to the left and release it:
 Both motions would have the same values for T and f.
 However, the resulting motion will have a phase difference of half a cycle.
𝜋
Phase difference
PRACTICE: Two identical mass-spring systems are started in two
different ways. What is their phase difference?
Start stretched and
then release
x
Start unstretched with
a push left
x
SOLUTION:
The phase difference is one-quarter of a cycle
𝜋
2
.
Phase difference
PRACTICE: Two identical mass-spring systems are started in two
different ways. What is their phase difference?
Start stretched and
then release
x
Start unstretched with a
push right
x
SOLUTION:
 The phase difference is three-quarters of a cycle
3𝜋
2
Conditions for simple harmonic motion
EXAMPLE: A spring having a
spring constant of 125 N m-1 is
attached to a 5.0-kg mass, stretched
+4.0 m as shown, and then released from rest.
(a) Using Hooke’s law, show that the
acceleration a of a mass-spring system
is related to the spring’s displacement
x by the proportion a  -x.
x

F = -kx & F = ma
ma = -kx
a = -(k / m) x.

∴ a  -x.


a = -(k / m) x
 a = -25x
 a = -25(-2.0) = +50. ms-2.
(b) Tailor your equation to this
example, and find the acceleration
of the mass when x = -2.0 m.

(c) What is the displacement of the
mass when the acceleration is
-42 ms-2?

a = -25x
 x = -a/25
 x = +1.7 m
F
 The minus sign in Hooke’s law, F = -kx, tells us
that if the displacement x is positive (right),
the spring force F is negative (left).
F
x x
0
F and x oppose
each other.
 It also tells us that if the displacement x is negative (left), the
spring force F is positive (right).
 Any force that acts in the direction opposite to displacement ,
pointing always toward equilibrium, trying to restore system back
to equilibrium is called a restoring force. In addition to that,
restoring force varies in magnitude. It varies with the object’s
displacement from its equilibrium position; greater displacement,
greater force; greater acceleration.
But only if force is linearly dependent on displacement, then we have
x
Simple harmonic motion
 A very special kind of oscillation that shows up often in the
physical world is called simple harmonic motion.
 In simple harmonic motion (SHM), a and x are related in a very
precise way: Namely, a  -x.
a  -x
definition of SHM
Formal definition of Simple Harmonic Motion – SHM.
If the acceleration/force a/F of a system is directly/linearly proportional
to displacement x from equilibrium position and directed
towards the equilibrium position, then the system undergoes SHM.
PRACTICE: Show that a mass oscillating on a spring executes SHM.
x
SOLUTION:
 We did when we showed that a = -(k / m)x, since this means that a  -x.
QUESTION:
A steel ball is dropped onto a concrete floor. Over and over again, it
rebounds to its original height. Is this SHM?
 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!!!
1. 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.
2. There is no equilibrium position about which oscillations occur. Thus,
the motion of the bouncing ball is not simple harmonic motion.
Not every periodic motion produced by restoring force is SHM.
Examples of SHM 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.
Period is CONSTANT and does
NOT depend on amplitude.
Which one is SHM ? Why?
Graphical treatment and math
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One can plot a displacement – time graph by
attaching a pen to a pendulum and moving
paper beneath it at a constant speed.
or by shining the light on an oscillating spring, The
shadow will trace out the displacement vs. time
graph of SHM.
 SHM traces out perfect sinusoidal waveforms
Properties of SHM
for any x, acceleration is 𝑎 ∝ −𝑥
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The negative sign indicates that the
acceleration is directed towards equilibrium,
as x is directed away from equilibrium.
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When x is max. or min. the velocity is zero, and acceleration and force
are maximum in the direction opposite to displacement.
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When x = 0, object is at equilibrium position, a = 0, F = 0 and v is
maximum.
Sketching and interpreting graphs of SHM
The graph is a sine curve (but if we chose to start
measuring the displacement from any other time it
could just as easily be a cosine, or a negative sine, or
any other sinusoidally shaped graph).
If we want to find the velocity at any particular time
we simply need to find the gradient of the
displacement–time graph at that time.
With a curved graph we must draw a tangent to the
curve (at our chosen time) in order to find gradient.
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velocity is a slope (derivative) of displacement
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acceleration is a slope (derivative) of velocity
Since 𝑎 ∝ −𝑥, 𝑎 is just a reflection of 𝑥
v=0
-2.0
v = vMAX
v=0
0.0
2.0
x-black
v-red
t
a-blue
(different scale)
𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑣𝑠. 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑔𝑟𝑎𝑝ℎ 𝑎 ∝ −𝑥
x
Sketching and interpreting graphs of SHM examples
EXAMPLE: The displacement x vs. t
for a 2.5-kg mass on a spring having
spring constant k = 4.0 Nm-1 is
shown in the sinusoidal graph.
(a) Find the period and frequency
of the motion.
SOLUTION: The period is the time for a complete cycle.
 From the graph it is T = 6.0 ms = 6.010-3 s.
 Then f = 1 / T = 1 / 0.006 = 170 Hz.
(b) Find the amplitude of the motion.
SOLUTION: The amplitude is the maximum displacement.
 From the graph it is xMAX = 2.0 mm = 2.010-3 m.
(c) Sketch the graph of x vs. t for the situation where the amplitude is cut in half.
SOLUTION: For SHM, the period is independent of the amplitude.
(c) The blue graph shows an equivalent system in SHM. What is the phase
difference between the red and blue?
SOLUTION:
 We see that it is T / 6 (= 360/ 6 = 60 = 2/ 6 rad).
Sketching and interpreting graphs of SHM examples
EXAMPLE: The kinetic energy vs. displacement for
a system undergoing SHM is shown.
The system consists of a 0.125-kg mass on a spring.
(a) Determine the maximum velocity of the mass.
SOLUTION: When the Ek is max, the velocity is also max.
1
𝐸𝑘 𝑚𝑎𝑥 = 2 𝑚 𝑣𝑚𝑎𝑥
vMAX = 8.0 ms-1.
2
1
→ 4.0 = 2 0.125 𝑣𝑚𝑎𝑥
2
ET
(b) Sketch EP and determine the total energy of the system.
SOLUTION: Since EK + EP = ET = CONST,
EP = 0 when EK = 𝐸𝑘 𝑚𝑎𝑥 ,
it must be that ET = 𝐸𝑘 𝑚𝑎𝑥 = 4.0 J
Thus the EP graph will be the “inverted” EK graph
EK
EP
(c) Determine the spring constant k of the spring.
SOLUTION: EP = (1/2)kx2.
 EK = 0 at x = xMAX = 2.0 cm. EK + EP = ET = CONST
ET = 0 + (1/ 2)kxMAX2  4.0 = (1/ 2)k 0.0202
k = 20000 Nm-1.
(d) Determine the acceleration of the mass at x = 1.0 cm.
SOLUTION:
 From Hooke’s law, F = -kx
F = -20000(0.01) = -200 N.
 From F = ma
-200 = 0.125a
a = -1600 ms-2.
Sketching and interpreting graphs of SHM examples
EXAMPLE: A 4.0-kg mass is placed on a spring’s end
and displaced 2.0 m to the right.
The spring force F vs. its displacement x from
equilibrium is shown in the graph.
F = -5.0 N
x = 1.0 m
(a) How do you know that the mass is undergoing SHM?
SOLUTION: In SHM, a  -x. Since F = ma, then F  -x also.
 The graph shows that F  -x. Thus we have SHM.
(b) Find the spring constant of the spring.
SOLUTION: Hooke’s law: F = -kx.
 Pick any F and any x. Use k = -F / x.
 k = -(-5.0 N) / 1.0 m = 5.0 Nm-1.
(c) Find the total energy of the system.
SOLUTION: ET = (1/2)kxMAX2. Then
(d) Find the maximum speed of the mass.
SOLUTION: ET = (1/2)mvMAX2.
10. = (1/2)  4.0  vMAX2
vMAX = 2.2 ms-1
ET = (1/2)kxMAX2 = (1/2)  5.0  2.02 = 10. J.
(e) Find the speed of the mass
when its displacement is 1.0 m.
SOLUTION: ET = (1/2)mv 2 + (1/2)kx 2.
Then 10. = (1/2)(4)v 2 + (1/2)(5)12
v = 1.9 ms-1.
What is meant by damping?
"to damp" is to decrease
the amplitude of a wave
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.
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.
If the mass is
suspended in water, the
damping is greater,
resulting in a more
rapid energy loss.
Critical damping
Critical damping ccurs 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.
This would be the
case if the mass were
suspended in a
thicker liquid such as
honey.
Examples of damping
Damper is a fluid. The
more viscous a fluid is, the
more resistant it is to flow.
Damper in suspension
system is oil.
A car suspension system has many springs
between the body and the wheels. Their
purpose is to absorb shock caused by
bumps in the road.
The car is therefore an oscillating system
that would oscillate up and down every time
the car went over the bump. As this would
be rather unpleasant for the passengers,
the oscillations are damped by dampers
(wrongly known as shock absorbers)
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.
Another example is one that involves vibrations that may be set up
in buildings when there is an earthquake. For this reason, in regions
prone to earthquakes, the foundations of some buildings are fitted
with damping mechanisms.
These mechanisms insure any oscillations set up in the building are
critically damped.
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.
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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)
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 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 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 money, 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.
• you pump energy into the child on the swing
by pushing once per cycle
• The result can be dramatic increase in
amplitude that sometimes is very unfortunate
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.
• 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.
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.
First it was
this
Jaime Vendera
glass-shattering vocal coach
My dear parents,
physics teacher NEVER
told your student to try
this.
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two tuning forks
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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.
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yours – old one
yours – new one
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.
The radio tuner
When you tune your radio, you are adjusting an electric circuit
so that it resonates with the signal of a particular frequency. If
the resonance curve for the circuit were not sharp, you would
be able to tune into the station over a wide range of
frequencies,and would be likely to get interference from other
stations.