Everything oscillate
even stars – but it takes some time –
1 oscillation per 70 million years
Oscillations

most solids are elastic - most material objects vibrate when
given an impulse
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

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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? 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 - a closer look.
The “restoring” force
q
L

𝐹𝑐𝑝

T
T



mg
mg

A
O
Everything cancels except for the red arrows
Force 𝒎𝒈 𝒔𝒊𝒏 𝜽 is responsible for changing the speed
Force 𝑭𝒄𝒑 acts as the centripetal force. It is responsible
for changing the direction of velocity only.
restoring force
B
To start the pendulum, you displace it from point O to
point A and let it go!
If no string – no tension - the mass falls down
Tension in the string T forces it to move in a circle
Resolve mg into components:
𝑚𝑔 cos 𝜃 𝑎𝑛𝑑 𝑚𝑔 sin 𝜃




At point O 𝒎𝒈 𝒔𝒊𝒏 𝜽 vanishes. There is no tangential
acceleration that changes speed.
Point O is translational equilibrium.
On either side of point O, 𝒎𝒈 𝒔𝒊𝒏 𝜽 act to bring
(restore) the pendulum back to point O.
That’s why we call it “restoring” force.
This is the force responsible for the motion of the
pendulum.
the role of the restoring force

the restoring force is the key to understanding all systems that
oscillate or repeat a motion over and over.

the restoring force always points in just the right direction to bring
the object back to translational equilibrium

from A to O the restoring force accelerates the pendulum down

from O to B it slows the pendulum down so that at point B it can
turn around

It is not constant force. It varies with the object’s distance from its
equilibrium position; greater distance, greater force; zero distance,
zero force – translational equilibrium position.

notice that at the very bottom of the pendulum’s swing (at O ) the
restoring force is ZERO, so what keeps it going?

It is inertia!!!

even though the restoring force is zero at the bottom of the
pendulum swing, the ball is moving and since it has inertia it keeps
moving to the left.

as it moves from O to B, gravity 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
ME

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

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

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
Some terminology
Cycle – one complete oscilation
Equilibrium position – position where an object would rest if not
disturbed.
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= 1
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.

the shadow should
look like this graph
The shape of this displacement – time graph is cosine curve.
The amplitude is x0 is initial displacement
displacement x = x0 cos ωt
where angular frequency is: ω = 2πf = 2π/T

velocity is a slope (derivative) of displacement

acceleration is a slope (derivative) of velocity
x = x0 cos ωt
v = - ω x0 sin ωt
v = - v0 sin ωt
a = - ω2 x0 cos ωt
a = - a0 cos ωt
a = – ω2 x
x = x0 cos ωt
v = - v0 sin ωt
a = – ω2 x
a = - a0 cos ωt
F = – m ω2 x
Formal definition of Simple Harmonic Motion – SHM.
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.
a = - ω2 x is the mathematical definition of SHM.
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
a = – ω2 x

The negative sign indicates that the
acceleration is directed towards equilibrium,
as x is directed away from equilibrium.

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.

for any x,
acceleration is a = – ω2 x
and velocity is v = ± ω (x02 – x2)1/2
v = - ωx0 sin ωt
= - ωx0 1-cos2ωt
= - ωx0
Period is CONSTANT and does
NOT depend on amplitude.
x2
1- 2
x0
common equations for SHM
x = x0 cos ωt
v = - v0 sin ωt
v0 = ωx0
a = - a0 cos ωt
a0 = ω2x0
a = – ω2 x
v = ± ω (x02 – x2)1/2
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
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.
The graph in next figure shows the variation with time t of the displacement x of a system
executing SHM.
Use the graph to determine the
(i) period of oscillation
(ii) amplitude of oscillation
(iii) maximum speed(iv) the speed at t = 1.3 s
(v) maximum acceleration
(i) T = 2.0 s
(ii) x0 = 8.0 cm
(iii) v0 = ωx0 = (2π/2) (8.) = 25 cm s-1
(iv) v = −v0sinωt = −25sin (1.3π).
1.3π = 1.3 π × 1800 /π = 2340
v = −25sin
(2340
) = +20 cm
s-1.
Or we can solve using v = ω x02 -x2
from the graph
at t = 1.3 s, x = − 4.8 cm ,
and ω = 2π/T = π s-1
v = π (8.0)2 -(4.8)2 = 20 cm s-1
(v) a0 = ω2x0 = π2 × 8.0 = 79 m s-2
Answer the same questions (a)(i) to (a)(iv) in the above example for the system oscillating
with SHM as described by the graph in the next figure.
Use the graph to determine the
(i) period of oscillation
(ii) amplitude of oscillation
(iii) maximum speed(iv) the speed at t = 1.3 s
(v) maximum acceleration
Also state two values of t at which the magnitude of the velocity is a maximum and two
values of t at which the magnitude of the acceleration is a maximum.
(i)
T = 2.4 s
(ii) x0 = 6.2 cm
(iii) v0 = 16 cm s-1
(iv) 42 cm s-2, 0 and 1.2 s, 0.6 s and 1.8 s
Restoring force: F = - mg sin θ
for small angles: sin θ ≈ θ
Restoring force: F = - kx
true for small x compared to length
F = - mg θ


F = - kx
restoring force is proportional to displacement
The negative sign indicates that the restoring force
is directed towards equilibrium, as displacement (θ,x)
is directed away from equilibrium.
Period of oscillations for two simple harmonic oscillators
Simple pendulum
T=2π
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.

Mass-spring system
T = 2π
m
k
the period gets smaller if a
stronger spring (larger k) is used
 the period of oscillation is longer
if a bigger mass (m) is used

To find out more about
Spring/Tension force – Hooke’s law
Click on the spring – it will take you to the end of power
point
• 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.
Not every periodic motion produced by restoring force is SHM.
Examples 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.
Which one is SHM ? Why?
Energy changes during simple harmonic motion

velocity of a mass undergoing SHM is given by equation
v = ± ω (x02 – x2)1/2
→
KE =
1
mω2  x 02 -x 2 
2
When x = 0, object is at equilibrium position, F = 0, a = 0, and v is
maximum, therefore KE is maximum, and PE zero (EPE or GPE).
KEmax =
1
mω2 x 02
2
Since no external work is done on the system, according to the law
of conseravtion of energy mechanical energy is conserved. As the
system oscillates there is a continual interchange between kinetic
energy and potential energy such that the loss in kinetic energy
equals the gain in potential energy and
ME = PE + KE =
1
mω2 x 02
2
Potential energy at any moment = total energy - KE
PE =
1
mω2 x 2
2
1
mω2 x 02 cos2ωt
2
1
KE = mω02 x 02 sin2ωt
2
PE =
At intermediate points, the energy
is part kinetic and part potential
ME = KE + PE.
ME =
1
mω2 x 02 = const. for given oscillations
2
What is meant by damping?
"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.
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.
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.
All objects have a natural frequency or set of
frequencies - at which they vibrate freely. These
frequencies depend only on the system itself.




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.
free
oscillation
Mass oscillates at
frequency f0
forced
oscillation
System driven at
frequency f caused
mass to oscillate
resonance
System driven at
frequency f0 caused
large amplitude
oscillations.
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.
When you push a child on
a swing you are using
resonance to make the
child go higher and higher.
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.
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.

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.


yours – old one
yours – new one
mine
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
• 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 result can be dramatic increase in amplitude that
sometimes is very unfortunate
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.
The sharpness of
the peak is affected
by the amount of
damping in the
system.
Each system has the same frequency of natural oscillation, f0 .
They each have a different degree of damping.
For the heavily damped system the amplitude
stays very small but starts to increase as the
frequency approaches f0 and reaches a maximum
at f = f0; it then starts to fall away again with
increasing frequency.
For the medium damped system, as f approaches
f0, the amplitude again starts to increase but at a
greater rate than for the heavily damped system.
The amplitude is again a maximum at f = f0 and is
greater than that of the maximum of the heavily
damped system.
For the lightly damped system, again the amplitude starts to increase as f
approaches f0, but at a very much greater rate than for the other two systems;
the maximum value is also considerably larger and much more well-defined i.e.
it is much easier to see that the maximum value is in fact at f = f0.
Clearly this is when the amplitude is a maximum the system
receives maximum energy input from the driver.
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.
Spring/Tension force – Hooke’s law
 Holding one end and pulling the other produces a tension
(spring) force in the spring.
 You’ll notice as you pull the spring, that the further you extend
the spring, then the greater the force that you have to exert in
order to extend it even further.
 As long as the spring is not streched beyond a certain
extension, called elastic limit, the force is directly proportional
to the extension.
 Beyond this point the proportionality is lost.
 If you stretch it more, the spring can become permanently
deformed in such a way that when you stop pulling, the spring
will not go back to its original length.
Unstretched
spring
spring
force
𝐹𝑎𝑝𝑝𝑙𝑖𝑒𝑑
Stretched
spring
Hooke’s Law:
In the region of proportionality
we can write
𝐹𝑠 = − 𝑘𝑥
L
x
x
spring
force
Compressed
spring
𝐹𝑠
𝐹𝑠
𝐹𝑎𝑝𝑝𝑙𝑖𝑒𝑑
𝐹𝑠 = 𝐹𝑜𝑟𝑐𝑒 𝑒𝑥𝑒𝑟𝑡𝑒𝑑 𝑏𝑦 𝑡ℎ𝑒 𝑠𝑝𝑟𝑖𝑛𝑔. 𝑆𝐼: 𝑁
𝑘 = 𝑆𝑝𝑟𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. 𝑆𝐼: 𝑁/𝑚
k measures stiffness
𝑘 𝑙𝑎𝑟𝑔𝑒 ∙ 𝑠𝑡𝑖𝑓𝑓 𝑠𝑝𝑟𝑖𝑛𝑔
𝑘 𝑠𝑚𝑎𝑙𝑙 ∙ 𝑠𝑜𝑓𝑡 𝑠𝑝𝑟𝑖𝑛𝑔
𝑥 = 𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡 𝑓𝑟𝑜𝑚 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛. 𝑆𝐼: 𝑚
Graph of applied force vs. extension: slope is the spring
constant
k=
F
x
Example:
k = 20 N/m
L = 10 cm
How much force do you have to exert
if you want to extend the spring for
a. 1cm
b. 2 cm
c. 3 cm
a. F = kx = (20 N/m)(0.01 m) = 0.2 N
b. F = kx = (20 N/m)(0.02 m) = 0.4 N proportionality
c. F = kx = (20 N/m)(0.03 m) = 0.6 N
Note: The spring force has the same value, but
it is in the opposite direction !!!!!
Example:
The graph shows how the length of a spring varies with applied
force.
i. State the value of the unstretched length of the spring.
ii. Use data from the graph to plot another graph of force against
extension and from this graph determine the spring constant.
i. when the applied force is zero
the length of the spring is 20 cm,
therefore unstretched length is 20
cm.
ii. F/N x/cm
5
6
10
12
k=
F
5
10
=
=
= 0.83 N/cm = 83 N/m
x
6
12
The work done by a non-constant applied force on a Hooke’s
spring is found from the area under the graph F vs. x
Work done by a force F = kx when
extending a spring from extension x1 to
x2 is:
F
–
kx2
W
kx1
x1
x2 extension
Work done by a force F = kx when
extending a spring from extension 0 to x
is:
W = ½ kx2
1
1
(kx2  x2 )  (kx1  x1 )
2
2
1
W  k ( x22  x12 )
2
Energy, E
In physics energy and work are very closed linked; in some
senses they are the same thing. If an object has energy it can
do a work. On the other hand the work done on an object is
converted into energy.
work done = change in energy
W=∆E
Elastic potential energy, EPE
 If some force stretches a spring by extension x, the work
done by that force is ½ kx2. Since work is the transfer of
energy, we say that the energy was transferred into the
spring, and that work is now stored in stretched spring as
elastic potential energy.
 EPE = ½ kx2
 A spring can be stretched or compressed. The same
mathematics holds for stretching as for compressing
springs.
Just imagine how
much energy is stored
in the springs of this
scale.
To continue with power point
Click on the spring – it will take you where you left off