Everything oscillate

even stars – but it takes some time –1 oscillation per 70 million years

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

Oscillations

atoms vibrate within molecule

molecules of the solid oscillate about their equilibrium positions

H2O

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.

cold hot

The particles in a solid vibrate more when it is heated, and take up more room.

What do all of them have in common? 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 mass is falling from point A to point O then rises from O to B. Why?

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 pendulum - The “restoring” force

L

mg

T

A

O

B

q0 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!

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!!!

The restoring force is the key to understanding all systems that oscillate or repeat a motion over and over.

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.

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

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 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 A, θ0 – the maximum displacement of an oscillating object from equilibrium.

Period T – the time it takes an oscillating system to makeone 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/s = s-1 = 1 Hz (hertz)1f = T

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 HzPeriod = 1/frequency = ½ sAmplitude = 10cm

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 by looking at an oscillating spring.

Graphical treatment and math

it looks like this graph

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.

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.

Formal definition of Simple Harmonic Motion – SHM.

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.

Properties of SHM

Period is CONSTANT and does NOT depend on amplitude.

A pendulum completes 20 cycles in 12s. What is (a) frequency?

a. f = 20/12 s = 1.7 Hz

QUESTION: A steel ball is dropped onto a concrete floor. Over and over again, it rebounds to its original height. Is this SHM?

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.

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!!!

Restoring force:component of the gravitational

force along the circle

Restoring force: spring force

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 oscillatorsSimple pendulum Mass-spring system

LT=2πg

mT = 2πk

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.

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

period is independent of amplitude

• Heavier kids can not swing faster.

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.

• 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.

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.

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.

Not every periodic motion produced by restoring force is SHM.

Even if friction or air resistance decreases the amplitude, the period remains the same.

Which one is SHM ? Why?

What is meant by damping?

The amplitude of the oscillations gradually decreases withtime. Oscillations, the amplitude of which decrease with time, are called damped oscillations, whereas the effect is called damping.

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.

"to damp" is to decrease the amplitude of a wave

All oscillating systems are subject to damping as it isimpossible to completely remove friction.

Because of this, oscillating systems are often classified by the

degree of damping.

Light damping

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.

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.

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 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.

This would be the case if the mass were suspended in a thicker liquid such as honey.

Examples of damping

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)

Damper is a fluid. The more viscous a fluid is, the more resistant it is to flow.Damper in suspension system is oil.

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 playersfix 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.

Examples:

Identify which of the following oscillatory systems are likely to be lightly damped and which are likely to be heavily damped.

1. Atoms in a solid.2. Car suspension3. Guitar string4. Harmonic oscillator under water.5. Quartz crystal.6. A cantilever that is not firmly clamped.7. Oil in a U-tube8. Water in a U-tube

Lightly damped 1, 3, 5, 8Heavily damped 2, 4, 6, 7

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)

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 a system is forced to oscillate at a frequency other than the natural frequency, this is called a forced oscillation.

If the support of the spring is oscillated, then the system will be forced to vibrate at another frequency.

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.

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.

Jaime Vendera glass-shattering vocal coach

My dear parents, physics teacher NEVER told your student to try

this.

First it was this

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.

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.

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

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.