91 Vibrations n Waves Gtb

8/3/2019 91 Vibrations n Waves Gtb

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Vibrations & Waves

Learning outcomes: Oscillations and types of oscillators

Related definitions and terms

Free oscillations, forced & damped oscillations & resonant frequency

Simple harmonic motion

Kinematics and graphs of shm

Types of waves

Properties of waves

Principle of superposition


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Oscillations & types of oscillatorymotions

Vibrations and oscillations occur all the time and are

everywhere.

Vibrations are physical evidence of waves, such as a loud

stereo shaking a table, sound waves cause vibrations Oscillations are a repetitive vibrations, typically in time.

Oscillation occurs when a system is a disturbed from

position of stable equilibrium.

One complete movement from the starting point and backto the starting point is known as an oscillation


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The displacement of oscillation from equilibrium changes periodically over time.

For a system to be oscillating, the shape of displacement - time graph does not matter.The only property that matters is that the motion is periodic.


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Basic properties of OscillatingSystems

The first of these properties we must understand is the Amplitude of the

oscillation. The amplitude of the oscillation is the parameter that varies withtime and this resides on the y-axis of the oscillation graphs.

Another important property of an oscillation system is the Time Period (T) ofthe oscillation. The time period of the oscillation is simply the time taken forthe oscillation to repeat itself. That is, it is the time between successiveoscillations of the system.

The other basic property of an oscillating system is the frequency, which is closelyrelated to the time period. As we know, one complete oscillation of the system isdefined by the time period, T and is known as 1 cycle. The frequency of the oscillatingsystem is simply the amount of cycles that happen in 1 second.


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Examples of oscillatory motion

beating of a heart

a simple pendulum

a vibrating guitar string

vibrating tuning fork

atoms in solids

air molecules oscillate when sound waves travel through

air. oscillations in electromagnetic waves such as light and

radio waves

oscillations in alternating current and voltage.


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A free oscillation is one where an object or system oscillates inthe absence of any damping forces, and it is said to oscillatingin its natural frequency

When one object vibrates at the same frequency as another it is

said to be in resonance

The swing of a frictionless pendulum is an example of a freeoscillation.

The displacement-time graph of a free oscillation is sinusoidal in

nature and the amplitude is constant with time


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Amplitude, Period, Frequency,and Phase Shift

The maximum displacement is called amplitude, A.

The time interval for the block to complete a full cycle is called theperiod, T.

The inverse of the period which represents the number of oscillationsper unit time is called its frequency, f.


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Damped Oscillations

A damped oscillation is one where frictional forcespresent gradually slow down the oscillation andthe amplitude decreases with time ie decreasingenergy

Damped oscillations are divided into under-damped,critically damped and over-damped oscillations.


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Under-damped oscillations

An under-damped oscillation is one where the

amplitude of oscillation or displacement

decreases with time

Example: oscillation of a simple pendulum with

the damping or dissipative force as air

resistance.


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Critically-damped oscillations

In a critically damped system, oscillations are

reduced to naught in the shortest possible time

examples: moving coil ammeter or volt meter,

shock absorber, door closer.


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Over-damped oscillations

In an over-damped system, a displacement

from its equilibrium position takes a long time

for the displacement to be reduced to zero.


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Resonance & effects

In the absence of external forces to an oscillating system, the

system oscillates at its natural frequency f0.

When an external force is applied to an oscillating system, the

system is under forced oscillations

Resonance occurs when a system is force to oscillate at its

natural frequency

When resonance occurs, the system oscillates with maximum

amplitude as maximum energy is transferred from the forcingsystem


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Eg Barton's pendulum only the pendulum with the same length as

the original will oscillate with the biggest amplitude

Applications wind instruments, excessive noise from a movingbus, radio & tv tuning

The Tacoma Narrows suspension bridge in USA in 1940 collapsed

due to a moderate gale (of same frequency as natural frequency of

bridge) setting the bridge into resonance until the main span broke

up


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Simple Harmonic Motion

A simple harmonic motion (S.H.M.) is themotion exhibited by an object or a system such

that the force F acting on it is directly

proportional to the displacement y from a fixedpoint of equilibrium and is always directedtowards that point


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This means that in shm, acceleration is directlyproportional to the displacement/distance fromthe fixed point and is always directed to thatpoint

Acceleration is always opposite to thedisplacement since the force is also opposite tothe displacement


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Comparisons

In linear motion acceleration is constant inmagnitude and direction

In circular motion acceleration is constant in

magnitude but not direction In shm the acceleration changes periodically in

magnitude and direction


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Any system which obeys Hooke's Law exhibitsshm

The force exerted by a spring is given by

Hooke's Law,F = - kx

where k is the spring constant and x is thedistance from the equilibrium position

and since F = ma, ma = - kx

hence a = -(k/m)x = dv/dt = d2x/dt2


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Radians

Angles can be measured in radians as well as in degrees

The angle in radians = s/r where s = length of arc and r = radius of circle

If s = r, then = 1rad, ie 1 radian is the angle subtended at the centre of acircle by an arc equal in length to the radius

When s = 2r (circumference of a circle), then

= 2 radians = 3600

Therefore 1 radian = 3600/2 = 570

Therefore s = r


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Angular velocity

The speed of a body moving in a circle can be specified

either by its speed along the tangent at any instant ie by its

linear speed, or by its angular velocity measured in radians

Hence angular velocity = /t

But v = s/t, and since s = r , v = r /t, hence v = r

Since a = v2 /r, therefore a = 2r


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Example

If r = 3m and = 1 rev per second = 2 rad s -1 , the linearspeed v = 6 m s -1


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Waves

. Waves are everywhere in nature.

. A wave is a disturbance that transfers energy between 2

points through vibrations in a medium, without transfering

matter between the 2 points

. 2 most common types of waves - sound waves and light

waves.


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A wave transports energy and notmatter

When a wave is present in a medium (that is,when there is a disturbance moving through amedium), the individual particles of the mediumare only temporarily displaced from their restposition.

There is always a force acting upon theparticles which restores them to their original

position

E i f Di l


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Equation for Displacement The general equation is

Wherexis the displacement at time, tandA is the amplitude of the motion.The angle (t + )is known as the phase angle and is a constant knownas the initial phase determined by the initial condition ofxwhen t=0.

The work done on the system is stored as the potential energy of the system.Thus we may deduce that

x=Acos t

U=1

2m

2x

2

or U=1

2m

2A

2cos

2 t

y=A sin 2 f t

f= 1T

phase angle ,=180[sin1 y0A ]


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While in motion, the system must also have a kinetic energy which is given by

Therefore the total energy of the system is

This result shows that the total energy of the system is always a constant.

K=1

m v2

K=1

m

2A

2sin

2 t

=1

2m

2A

2x

2

E=UK

that isE=1

2m

2A

2cos

2 t

1

m

2A

2sin

2 t

=1

2

m2A

2


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Behaviour and characteristics

In a slinky spring, a person does work by moving his hand and hencetransfering kinetic energy to the first coil.

The first coil receives a large amount of energy which it subsequentlytransfers to the second coil.

When the first coil returns to its original position, it possesses thesame amount of energy as it had before it was displaced. The firstcoil transferred its energy to the second coil.

The second coil then has a large amount of energy which itsubsequently transfers to the third coil.

In this manner, energy is transported from one end of the slinky to theother, from its source to another location


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Categories of Waves

Waves come in many shapes and forms

One way to categorize waves is on the basis ofthe direction of movement of the individual

particles of the medium relative to the directionwhich the waves travel.

Categorizing waves on this basis leads to threenotable categories: transverse waves,

longitudinal waves, and surface waves.


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Transverse waves

A transverse wave is a wave in which particlesof the medium move in a directionperpendicular to the direction of travel of the

wave

Examples are water waves, light waves, X-rays


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Longitudinal waves

A longitudinal wave is a wave in which particlesof the medium move in a direction parallel tothe direction in which the wave moves.

Example is a sound wave which vibratesforwards and backwards


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Compression maximum density

Rarefaction - minimum density


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Surface waves

While waves which travel within the depths ofthe ocean are longitudinal waves, the waves

which travel along the surface of the oceansare referred to as surface waves.

A surface wave is a wave in which particles ofthe medium undergo a circular motion. Surface

waves are neither longitudinal nor transverse.

In a surface wave, it is only the particles at thesurface of the medium which undergo thecircular motion

Electromagnetic versus Mechanical


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Electromagnetic versus MechanicalWaves

Another way to categorize waves is on thebasis of their ability or inability to transmitenergy through a vacuum (i.e., empty space).

Categorizing waves on this basis leads to twonotable categories: electromagnetic waves andmechanical waves.


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Behaviour and characteristics

An electromagnetic wave is a wave which is capableof transmitting its energy through a vacuum (i.e.,empty space).

Electromagnetic waves are produced by the vibration

of charged particles A mechanical wave is a wave which is not capable of

transmitting its energy through a vacuum.

Mechanical waves require a medium in order totransport their energy from one location to another.

Anatomy of a wave


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Anatomy of a wave

The peak or crest/compression of a wave is the point on the medium which exhibits

the maximum amount of positive or upwards displacement from the rest position.Points C and J on the diagram represent the troughs of this wave.

The trough/rarefaction of a wave is the point on the medium which exhibits themaximum amount of negative or downwards displacement from the rest position.

Wavelength, , is the distance between 2 successive crests or troughs, eg A-E or E-H

A high energy wave is characterized by a high amplitude; a low energy wave ischaracterized by a low amplitude.

Putting a lot of energy into a transverse pulse will not effect the wavelength, thefrequency or the speed of the pulse. The energy imparted to a pulse will only effectthe amplitude of that pulse.


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Related definitions & terms

For an object or system that oscillates:

The amplitude (r) is the maximum displacement from its equilibrium (or mean or rest orundisturbed) position.

The period (T) is the time it takes to go through one complete cycle or oscillation or wavecycle. T = 2r/v or T = 2/

The frequency (f) is the number of complete cycles an oscillating object makes in onesecond. f= 1/T is in Hz

The distance from the equilibrium position is known as the displacement

The angular frequency () is the change in angle per unit time. = 2 /Tand = 2 f.

The phase difference is the difference in step of vibration for two points along theoscillating path

The energy transported by a wave is directly proportional to the square of the amplitudeof the wave. E = kA2


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The wave equation and principle

Speed = distance/time Speed = wavelength/period and since

frequency = 1/period

Speed = wavelength x frequency, ie v = f the principle is that wave speed is dependent

upon medium properties and independent of

wave properties.


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Boundary behaviour of waves

Waves exhibit reflection, refraction and diffraction at boundariesfrom one medium to another associated with the bending of thepath of a wave.

reflection involves a change in direction of waves when theybounce off a barrier

refraction of waves involves a change in the direction of wavesas they pass from one medium to another; and

diffraction involves a change in direction of waves as they passthrough an opening or around a barrier in their path


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Reflection, refraction & diffraction

Reflection of waves off straight barriers follows the law of reflection.

Reflection of waves off parabolic barriers results in the convergence of thewaves at a focal point.

Refraction is the change in direction of waves which occurs when wavestravel from one medium to another.

Refraction is always accompanied by a wavelength and speed change.

Diffraction is the bending of waves around obstacles and openings. Theamount of diffraction increases with increasing wavelength.


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Echos and speed of sound

Echo phenomenon are commonly observed with waves eg Noah stands 170 meters away from a steep canyon wall. He shouts and hears the

echo of his voice one second later. What is the speed of the wave?


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When there is a reflection, the wave doubles itsdistance. In other words, the distance travelled by thesound wave in 1 second is equivalent to the 170meters down to the canyon wall plus the 170 meters

back from the canyon wall. Sound waves travel at 340 meters in 1 second, so the

speed of the wave is 340 m/s.

W i f


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Wave interference

Wave interference is the phenomenon which occurs whentwo waves from 2 coherent sources meet while travellingalong the same medium.

2 waves are said to be coherent if

They produce waves of the same frequency

They produce waves of the same phase


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The interference of waves causes the medium to take on ashape which results from the net effect of the twoindividual waves upon the particles of the medium

When waves are produced on the surface of water, the

wave crests will act like a convex lens while a trough willact like a concave lens causing bright and dark fringes

Waves interference can be constructive or destructive

A wave-front is a line that joins all the points vibrating in-

phase and is represented by the bright and dark fringes

P i i l f S iti


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Principle of Superposition

The task of determining the shape of the resultant demands that theprinciple of superposition is applied. The principle of superposition issometimes stated as follows:

When two waves interfere ie meet at the same point, the resultingdisplacement of the medium at any location is the algebraic sum of thedisplacements of the individual waves at that same location

C t ti I t f


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Constructive Interference

This is the superposition of 2 waves which are in phase to produce aresultant wave of maximum amplitude


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91 Vibrations n Waves Gtb