Angular phase and angular frequency of a simple harmonic motionFrom figure 1.2, the angular ωt +ϕ specifies the position of point P at the time t, it is called the phase or angular phase
Trang 1MINITRY OF EDUCATION AND TRANNING
PHYSICS
12
2000
Trang 2TABLE OF CONTENTS
Part I OSCILLATIONS AND WAVES 8
Chapter I – Mechanical Oscillations 8
§1 Periodic and simple harmonic motions Oscillation of a mass-spring system .8
1 Oscillations 8
2 Periodic motion 8
3 Mass-spring system Simple harmonic motion 8
§2 Exploring a simple harmonic motion 10
Uniform circular motion and simple harmonic motion 10
2 Angular phase and angular frequency of a simple harmonic motion 11
3 Free motion 11
4 Velocity and acceleration in a simple harmonic motion 11
5 Oscillation of a simple pendulum 12
§3 Energy in a simple harmonic motion 13
1 Energy changes during oscillation 13
2 Conservation of mechanical energy during oscillation 14
§4 - §5 The combination of oscillations 15
1 Examples of the combination of oscillations 15
2 Phase-differences between oscillations 15
3 Vector-diagram method 16
4 The combination of two oscillations of same directions and frequencies 16
5 Amplitude and initial phase of the combinatorial oscillation 17
§6 - §7 Underdamped and forced oscillations 18
1 Underdamped oscillation 18
2 Forced oscillation 18
3 Resonance 19
4 Applying and surmounting resonant phenomenon 19
5 Self-oscillation 20
Summary of Chapter I 20
Chapter II – Mechanical wave Acoustics 22
§8 Wave in mechanics 22
1 Natural mechanical waves 22
2 Oscillation phase transmission Wavelength .22
3 Period, frequency and velocity of waves 23
4 Amplitude and energy of waves 23
§9 - §10 Sound wave 24
Sound wave and the sensation of sound 24
2 Sound transmission Speed of sound 25
3 Sound altitude 25
Timbre 25
5 Sound energy 26
6 Sound loudness 26
7 Sound source – Resonant box 27
§11 Wave interference 28
1 Interferential phenomenon 28
Trang 32 Theory of interference 28
3 Standing wave 29
Summary of Chapter II 31
Chapter III – Electric oscillation, Alternating current 32
§12 Harmonic oscillation voltage Alternating current 32
1 Harmonic oscillation voltage 32
2 Alternating current 32
3 Root mean square (rms) value of intensity and voltage 33
§13 - §14 Alternating current in a circuit containing only resistance, inductance or capacitance 34
1 Relation between current and voltage 34
2 Ohm’s law for an AC circuit containing only resistance 34
1 Effect of capacitors to the alternating current 34
2 Relation between current and voltage 35
3 Ohm’s law for an AC circuit containing only capacitance 35
1 Effects of an inductor to the alternating current 36
2 Relation between current and voltage 36
3 Ohm’s Law for an AC circuit with inductors 36
§15 Alternating current in an RLC circuit 37
Electric current and voltage in an RLC circuit 37
Relation between current and voltage in an RLC circuit 38
3 Ohm’s Law for an RLC circuit 38
4 Resonance in an RLC circuit 39
§16 Power of the alternating current 39
1 Power of the alternating current 39
2 Significance of the power coefficient 40
§17 Problems on AC circuits 41
Problem 1 41
Problem 2 41
§18 Single-phase AC generator 42
1 Operational principle of single-phase AC generators 42
2 Structure of an AC generator 42
§19 Three-phase alternating current 43
1 Operational principle of three-phase AC power generators 43
2 WYE connection 44
3 Delta connection 45
§20 Asynchronous three-phase motors 45
Operational principle of asynchronous three-phase motors 45
Rotating magnetic field of three-phase current 46
3 Structure of an asynchronous three-phase motor 46
§21 Transformers Electricity transmission 47
1 Operational principle and structure of transformers 47
2 Transformation of current and voltage via transformer 47
3 Transmission of power 48
§22 Generation of direct current 49
1 Benefits of direct current 49
Trang 43 Two half-cycle rectifying method 49
4 Operational principle of DC power generators 50
Summary of Chapter III 50
Chapter IV – Electromagnetic oscillation Electromagnetic wave 52
§23 Oscillation circuits Electromagnetic oscillation 52
1 Fluctuation of charges in an oscillation circuit 52
2 Electromagnetic oscillation in an oscillation circuit 53
§24 Alternating current, high-frequency electromagnetic oscillation, and mechanical oscillation 54
1 Electric oscillation in an alternating current 54
2 High-frequency electromagnetic oscillation 54
3 Electromagnetic oscillation and mechanical oscillation 54
§25 Electromagnetic field 57
1 Fluctuated electric field and fluctuated magnetic field 57
2 Electromagnetic field 57
3 Transmission of electromagnetic interaction 57
§26 Electromagnetic waves 58
1 Electromagnetic waves 58
2 Properties of electromagnetic waves 58
3 Electromagnetic waves and wireless communication 58
§27 Transmitting and receiving electromagnetic waves 59
Periodic-oscillation transmitters using transistors 59
2 Open oscillation circuit Antenna 60
Principles of transmitting and receiving electromagnetic waves 60
§28 - §29 A glance at radio transmitters and receivers 61
1 Principle of oscillation amplification 61
2 Principle of amplitude modulation 62
3 Operational principle of radio transmitters 62
4 Operational principle of radio receivers 63
Summary of Chapter IV 64
Part II OPTICS 66
Chapter V – Light reflection and refraction 66
§30 Light transmission Light reflection Plane mirror 66
1 Light propagation 66
2 Light reflection 67
3 Plane mirror 67
§31 Concave spherical mirrors 68
Definitions 68
2 Reflection of a light in a concave spherical mirror 69
3 Formation of images by concave spherical mirror 69
4 Main focal point Focal length 70
5 Method to draw an object’s image obtaining from a concave spherical mirror 70
§32 Convex spherical mirrors Convex spherical mirror equations Applications of convex spherical mirrors 72
1 Convex spherical mirror 72
2 Convex spherical mirror equations 72
3 Applications of convex spherical mirrors 74
Trang 5§33 Light refraction 75
Light refraction phenomenon 75
2 The law of light refraction 75
3 Index of refraction (refractive index) 76
§34 Total internal reflection 77
Total internal reflection 77
2 Conditions to achieve total internal reflection 78
3 Critical angle 78
4 Applications of total internal reflection 79
§35 Prism 80
1 Definition 80
2 Path of a monochromatic ray through a prism Angle of deviation 80
3 Prism equations 80
4 Minimum deviation angle 80
§36 Thin lenses 82
1 Definition 82
2 Main focal point Optical center Focal length 82
3 Supplemental focal points Focal plane 83
4 Lens power 84
§37 Image of an object through lenses Lenses equations 85
1 Observing an object’s image through a lens 85
2 Method to draw an object’s image through a lens 85
3 Lens equation 86
4 Lateral magnification 87
The human eye and optical instruments 91
§38 Camera and the human eye 91
1 Camera 91
2 The human eye 91
§39 Eye’s defects and correcting methods 94
1 Near-sightedness (myopia) 94
Farsightedness (hyperopia) 95
1.State the characteristics of near-sighted eye and the correcting method .95
§40 Magnifying glass 95
1 Definition 95
2 Near point and infinite point 96
3 Angular magnification 96
§41 Microscope and telescope 98
1 Microscope 98
2 Telescope 99
Chapter VII – The wave-nature of light 103
§42 Light dispersion phenomenon 103
1 Experiment on light dispersion phenomenon 103
2 Experiment on monochromatic light 103
3 Synthesizing white light 104
4 Dependence of the index of refraction of a transparent medium on the color of the light 104
Trang 61 Young's experiment on light interference phenomenon 105
2 Explanation of the phenomenon 105
3 Conclusion 106
1 Describe the experiment on the interference of light? 106
§44 Measuring the wavelength of the light The wavelength and the color of the light 106
1 Interference fringe distance 106
2 The wavelength and the color of the light 108
§45 Spectrometer Continuous spectrum 108
1 Relation between the index of refraction of a medium and the wavelength of the light 108
2 Spectrometer 109
3 Continuous spectrums: 109
§46 Line spectrum 110
1 Emission line spectrum 110
2 Absorption line spectrum 111
3 The spectroscopic analysis approach and its advantages 112
§47 Infrared and ultraviolet rays 112
1 Experiments to discover infrared and ultraviolet rays 112
2 The infrared ray 113
3 The ultraviolet ray 113
§48 X-rays 114
1 X-ray tube 114
2 The nature of X-rays 114
3 Properties and uses of X-rays 114
4 Electromagnetic waves scale 115
Chapter VIII – Light quantum 118
§49 The photoelectric effect 118
1 Hertz’s experiment 118
The experiment with a photocell 118
§50 The quantum hypothesis and photoelectric laws 120
1 Photoelectric laws 120
2 The quantum hypothesis 120
3 Explaining photoelectric laws by using the quantum hypothesis 121
4 Wave-particle duality of the light 122
§51 Light dependant resistor and photoelectric battery 123
1 The photoconduction phenomenon 123
2 Light dependant resistor (LDR) 123
3 The photoelectric battery 124
§52 Optical phenomena relating to the quantum property of the light 125
1 The luminescence 125
2 Photochemical reactions 125
§53 Application of the quantum hypothesis to hydrogen atom 126
1 The Bohr model of the atom 126
2 Using the Bohr model to explain hydrogenous line spectrum 127
Summary of Chapter VIII 128
Part III NUCLEAR PHYSICS 130
Chapter IX – Basic knowledge on the atomic nucleus 130
Trang 7§54 Structure of the nucleus The unit for atomic 130
1 Structure of the nucleus 130
2 Nuclear forces 130
3 Isotopes 130
4 The unified atomic mass unit 131
§55 Radioactivity 132
Radioactivity 132
2 The radioactive decay law 133
§56 Nuclear reactions 134
1 Nuclear reactions 134
2 Conservation laws in nuclear reactions 134
3 Application of conservative laws to radioactivity Transmutation rules 135
§57 Artificial nuclear reactions Applications of isotopes 136
1 Artificial nuclear reactions 136
2 Particle accelerators 136
§58 Einstein’s relation between mass and energy 138
1 Einstein’s axioms 138
§59 THE LOSS OF MASS NUCLEAR ENERGY 140
1 The loss of mass and binding energy 140
Exothermic and endothermic nuclear reactions 140
3 Two exothermic nuclear reactions 141
§60 Nuclear fission Nuclear reaction plants 142
1 Chain nuclear reactions 142
Nuclear reaction plants 143
§61 THERMONUCLEAR REACTION 144
Supplemental reading: Primary Particles 145
1 Properties of the primary particles: 145
2 Antiparticles Antimatter .146
3 Fundamental interactions Classification of primary particles .146
4 Quarks 147
SUMMARY of chapter IX 147
Part IV PRACTICAL EXPERIMENTAL EXERCISES 150
Experimental exercise 1 – Clarification of the law on the simple pendulum’s oscillation Determination of the gravity acceleration 150
Experimental exercise 2 – Determination of Sound wavelengths and frequencies 152
Experimental exercise 3 – The alternating current circuit with R, L, C 154
Experimental exercise 4 – The refraction index of glass 156
Experimental exercise 5 – Observation of light dispersion and interference phenomena 158
COMBINED EXPERIMENTAL EXERCISES 160
Experimental exercise A – Determination of Capacitance and inductance (2 sections) 160
Experimental exercise B – Characteristics and applications of transistors (2 sections) 163
Experimental exercise C – Determination of focal length of lenses (2 sections) 166
Trang 8Part I OSCILLATIONS AND WAVES
Chapter I – MECHANICAL OSCILLATIONS
§1 P ERIODIC AND SIMPLE HARMONIC MOTIONS O SCILLATION OF A MASS - SPRING SYSTEM
1 Oscillations
Flower stirs in the branch as the wind breezes Pendulum of the clock swings to the left and right On therippled lake, a small piece of wood bobs and rolls The string of the guitar vibrates when it is played
In the examples above, things move in a small space, not too far away from a certain equilibrium
position The movement likes that is called the oscillation An oscillation, or vibration, is a limited
motion on a space, repeating back and forth many times around an equilibrium position.
That position often is where thing is at rest (does not move): when there is no wind, a clock does notwork, a smooth lake, non vibrating guitar’s strings
2 Periodic motion
Observing the oscillation of a pendulum of the clock, after a certain period of time of 0.5s, it passes
through a lowest position from the left to the right The oscillation like that is called the periodic
oscillation Periodic oscillation is the oscillation whose state is repeated as it was after a constant period
of time The smallest period of time of T after that states of oscillation are repeated as they were is called
the period of periodic oscillation.
The quantity f =
T
1
showing the number of oscillations (i.e how many times a state of oscillation is
repeated as it were) per unit of time is called the frequency Frequency is usually specified in hertz (Hz).
In the example above, the period of the pendulum is T = 0.5s so its frequency is f =
5 0
1
= 2Hz, it meansthat the pendulum carries out 2 oscillations in a second
The vibration of the guitar’s strings do not permanently maintained It is damped then ended But if it is
observed in a very small period of time, it is approximately a harmonic oscillation.
3 Mass-spring system Simple harmonic motion
Considering a mass-spring system consisted of a small ball of m kg attached rigidly to a spring of
negligible mass, put in the horizontal plane as shown (figure 1.1a) There is a small hole through the ball
so it can be translated along a fixed rod in the same plane
We choose a datum axis that coincides with the rod, is directed from the left to the right, and the origin O
is the equilibrium position of the ball (position where the ball is at rest) A ball is deflected to the right by
a force F then released (figure 1.1b; the spring is not shown) It is observed that the ball moves toward
Trang 9the point O, passes through O This translation is repeated many times, i.e the ball oscillates around theequilibrium position O.
This phenomenon is analyzed as following: when the ball is pulled to an ordinate x, forces exert into it
consist of the pulling force F’, the elastic force F of the spring, the gravity force and the reacting force of
the rod to the ball (these two forces are not shown in the figure) The gravity and reacting force are in thevertical plane, equal to each other and opposite in the direction so they have no affect on the horizontaltranslation of the ball At the time the ball is released, there is only an elastic force exert on it
Within the limitation of elasticity of the spring, the force F is always proportional with the displacement
x of the ball from the equilibrium position (is also the deflection of the spring), and directs toward the
point O Since F is along the coordinate axis, it can be written as:
Here k is the spring constant (stiffness) of the spring, and the minus sign indicates that the force F is
acting in opposite direction compared with the deflection x of the ball
According to Newton’s second law, it can be written as F = ma, or ma = - kx Thus a = x
where A and ϕ are constants and ω =
Replacing the value of x into (1-5) we get: x!! = - ω 2 x (1-6)
(1-6) has the same format as (1-2a), it shows that (1-3) is the solution of (1-2a), in another way, the equation of the oscillated ball is x = Asin( ω t + ϕ ).
Since sine function is a periodic function, it is said that the oscillation of the ball (i.e the oscillation of
the mass-spring system) is a simple harmonic motion (SHM) Note that a cosine expression can be
transformed to a sine expression such a way that: Acos(ωt+ϕ) = Asin(ωt+ϕ+π/2)
Therefore, it can be defined that a SHM is an oscillation that can be described by a sinusoidal (or
cosinusoidal) function, where A, ω, ϕ are constants
In the equation (1-3), x is the displacement of the oscillation, showings precisely the deflection of the ball from the equilibrium position A is the amplitude of the oscillation It is the maximum value of
Trang 10displacement, occurred when sin(ωt+ϕ) has the maximum value of 1 The meanings of ω, ϕ and ωt+ϕwill be clarified at §2.
It is known that sine function is a periodic function with the period of 2π Thus, it can be written as
x = Asin(ωt +ϕ) = Asin(ωt +ϕ +2π), or x = Asin[ω(t +2π
Questions
1 Make statement about the definitions of oscillation, periodic oscillation and harmonic oscillation?
2 Differentiate between periodic and general oscillation, between periodic and harmonic oscillation?
3 Make statement about the definitions of time constant, frequency, displacement, amplitude of
harmonic oscillation?
4 Give more example about oscillation and harmonic oscillation?
§2 E XPLORING A SIMPLE HARMONIC MOTION
1 Uniform circular motion and simple harmonic motion
Let consider a point M moves in a circle of central point O and radius
A (figure 1.2) The angular velocity of point M is ω (measured in
rad/s) A point C in the circle is chosen to be an origin At the initial
time t = 0, the position of the moving point is M0, is specified by an
angle of ϕ At an arbitrary time, the position of the moving point is Mt
specified by an angle of (ωt + ϕ)
We project the path of point M onto an axis x’x pass through point O
and perpendicular to OC At time t, the projection of point M onto x’x
axis is point P which has the ordinate of x = OP Since OP is the
projection of OMt onto the x’x axis so we have:
x = OMtsin(ωt +ϕ)
(1.8) has the same format as (1.3) so we can conclude that the motion of point P on the x’x axis is a
SHM In the other way, a simple harmonic oscillation can be considered as the projection of an uniform
circular motion onto any straight line in the same plane.
Trang 112 Angular phase and angular frequency of a simple harmonic motion
From figure 1.2, the angular (ωt +ϕ) specifies the position of point P at the time t, it is called the phase
(or angular phase) of the oscillation at the time t The angle of ϕ specifies the position of P at the initial
time t = 0, and is called the initial phase (or initial angular phase) of the oscillation The angular velocity
ω allows us to determine f =
π
ω
2 , which is the number of circle of M in a unit of time, and is also the
number of oscillation of P in a unit of time We know that f is the frequency of the oscillation, therefore
ω is called the angular frequency (circular frequency) of the oscillation Here ϕ, ω and (ωt +ϕ) arespecified angles and can be measured directly
In equation (1-3) for the mass-spring system, the quantities ϕ, ω and (ωt+ϕ) have the same names butthey are not the real angles which can be experimentally measured They are intermediary quantitieswhich allows determining the frequency and states of the oscillation
3 Free motion
Let’s analyze more detail the motion of the mass-spring system described in §1 (figure 1.1)
The maximum displacement the ball can reach is the amplitude A The time when the ball is released andstart to move is chosen to be the initial time t = 0 At that time x = A In order to have the equation
x = Asin(ωt +ϕ) satisfied, we must have sin(ωt +ϕ) = 1, and since ωt =0 so ϕ = π/2
Therefore, the oscillation equation of the ball is x = Asin(ωt +π/2) (1-9)
So we have determined the amplitude, initial phase and the cycle of the oscillation The amplitude andinitial phase depends on the initial conditions, i.e the way to excite the oscillation and the way to choosethe space and temporal coordinate The period depends only on the mass of the ball and the spring
constant, not on other factors If the initial conditions are changed then the amplitude A and initial phase
ϕ will be changed as well but ω, T are constant
An oscillation the period of which depends only on the system’s characteristics (here is a mass-spring
system), and not on other stimulating factors, is called a free oscillation A system that can implement a
free oscillation by itself is called a self-oscillation system After being stimulated, a self-oscillation system will proceed with its own frequency The oscillation of a mass-spring system is a free oscillation.
4 Velocity and acceleration in a simple harmonic motion
acceleration The phase of the oscillation determine the state of the oscillation Similarly, the initial
phase ϕ specifies the initial state of the oscillation.
When the ball is in the harmonic oscillation, its velocity and acceleration fluctuate following a sinusoidal
or cosinusoidal function, i.e they fluctuate harmonically with the same frequency of the ball.
Trang 125 Oscillation of a simple pendulum
A simple pendulum consists of a ball attached at one end of a string The ball has a mass of m and itssize is very small in comparison with the length of the string The string is inelastic (constant length) andhas a negligible mass The ball can be seen as a point of mass m attached to a no-mass string When it ishung at point Q, its equilibrium position is QO (figure 1.4) The ball is pushed follow an arc from O to Pcorresponding to a deflection angle of α We only investigate the case in which α is small enough to havethe arc "OP coincide with the chord OP and sinα can be approximated as α (in radian) If α≤10o then theerror is not greater than 6/1000
Thus we have: sinα≈α =
From Newton’s second law, it can be written as :
ma = F (1-13) Point O is chosen to be the origin while the chord OP is taken as coordinate axis Since a and F are in OP axis and the direction of F is opposite with the ordinate s = OP so
l g
Trang 13
The time constant of the pendulum is:
For small oscillation, i.e with α≤10o, the cycle of a simple pendulum is not dependent on the oscillation
amplitude All the discussion have been made for a mass-spring system in §2 can be applied to the single
pendulum as well
The period of the single pendulum depends on the gravity constant g At a specified location to the earth(g is constant), the oscillation of the single pendulum can be regarded as free oscillation
Calculations in details have proved that when the ball is moving, the tend force T has a magnitude T > F’ The
result is that the ball is exerted by a force of (T - F’) directed to Q This force cause a centripetal acceleration so that the ball travels in a circle path while the acceleration in the OP direction maintains a =- gs/l.
In the calculation above, the change of force T is not taken in to account, but the result is still valid.
Questions
1 Make statement about definition of phase and initial phase of periodic oscillation?
2 What is angular frequency? What is the relationship between angular frequency ω and the frequency f?
3 What kind of oscillation that can be called free oscillation?
4 Why is the formula (1.15) valid only for small oscillations?
5 The displacement of an object (measured in cm) fluctuates is described by x = 4cos4πt Calculate thefrequency of this oscillation Determine the displacement and velocity after it starts to oscillate in 5seconds?
6 A single pendulum has a period of 1.5s when it oscillates at a place where the gravity constant is9.8m/s2 Determine the length if the string?
7 Determine the time constant of the pendulum in exercise 6 when it is brought to the Moon, knowingthat the gravity constant in the Moon is 5.9 times smaller than the Earth
Hints: 5) 2Hz; 4cm; 0cm/s; 6) 0.56m; 7) 3.6s
§3 E NERGY IN A SIMPLE HARMONIC MOTION
1 Energy changes during oscillation
When the ball of the mass-spring system is pulled from point O to point P (figure1.5, the spring is notshown), the force has done a work to elongate the spring, this work is passed to the ball as potentialenergy At that time, the elastic force of the spring has a maximum value so the potential energy also has
a maximum magnitude
When the force is not exerted, the spring compressed, the
elastic force directs the ball toward point O Its velocity is
increasing, the kinetic energy is increasing while the
potential energy is decreasing
When the ball is at the equilibrium position O, the elastic force and the potential energy are zero, itsvelocity and kinetic energy reach maximum value The ball continues to move due to the inertial motion,the spring is contracted, the elastic force appears in opposite direction and grows up, and the velocity isdecreasing
When the ball reaches to point P’, the spring is contracted to the shortest length, the elastic force reachesthe maximum value and the ball is stopped Its kinetic energy is zero, the potential energy has maximummagnitude and stop increasing
After that, the spring is stretching, the elastic force is decreasing, the ball is pushed to point O Its kineticenergy is increasing while the potential energy is decreasing
Trang 14During the oscillation process of the mass-spring system, there is always a transformation between
kinetic and potential energy: when the kinetic energy is increasing then the other is decreasing and viceversa
2 Conservation of mechanical energy during oscillation
We will analyze quantitatively a process of energetic transformation of a mass-spring system
The kinetic energy of the ball is Ed =1
Note that the kinetic energy of a single pendulum is dependent on the initial excitation If it is excited by
a powerful force to make a bigger displacement then the amplitude is bigger hence the total energy is alsobigger Certainly, we just can increase the amplitude to a limited value within elastic limitation of thespring
Questions
1 Describe quantitatively the process of transformation of total energy of a single pendulum?
2 How to increase the total energy of single pendulum and at what value it can be increase?
3 How many time is the total energy of a pendulum changed if its frequency is increased 3 times whilethe amplitude is reduced 2 times?
Trang 15§4 - §5 T HE COMBINATION OF OSCILLATIONS
1 Examples of the combination of oscillations
In the real life as well as in science and technology, there are cases in which the oscillation of an object is
a combination of many different oscillations When the hammock is hung on a ship, it will swing with itsown frequency However, the ship is also oscillated due to wave Finally, the oscillation of the hammock
is a combination of two components: its own oscillation and the oscillation of the ship
Generally, partial oscillations can have different directions, amplitudes, frequencies and phases
Therefore, it is very complicated and difficult to determine the combined oscillation We will only dealwith simple situations that are usually encountered in science and technology
2 Phase-differences between oscillations
Two oscillations which have the same frequency, generally can have different phases For example, twoidentical mass-spring systems are hung next to each other, they have the same angular frequency of ω.The balls are pulled to displacements x1 = A1 and x2 = A2 respectively At the time t = 0, ball 1 is released
to start moving At the time when ball 1 passes through its equilibrium position, ball 2 is released andstart traveling
It takes a quarter of period for ball 1 to travel from position x1 = A1 to the equilibrium position So thatthe oscillation of ball 2 is retarded a mount of
4
T
compared with ball 1
We find the oscillating equations of two ball in the form of
The phase difference is a constant quantity and equal to the difference between the initial phases It is
called the phase difference ∆ϕ and these two oscillation are called phase-different oscillations When
∆ϕ = ϕ1 - ϕ2 > 0, it is said that oscillation 1 is a lead in phase to oscillation 2 or oscillation 2 is a lag inphase to oscillation1 When ∆ϕ = ϕ1 - ϕ2 < 0, it is said in a contrast way
In this example, it is said that ball 1 leads ball2 by an angle of
Trang 163 Vector-diagram method
In order to combine two harmonic oscillations with the same directions, frequencies but different
amplitudes and phases, it is usually used a very convenient method called Fresnel’s vector diagram Thismethod is based on a property having been discussed in §2: a simple harmonic oscillation can be treated
as the projection of a uniform circular motion on to a straight line in the same plane
According to this method, each oscillation can be represented by a
vector Suppose that an oscillation x = A sin(ωt+ϕ) need to be
represented A horizontal axis (∆) and a vertical axis x’x that
intersects (∆) at point O are built (figure 1.6) A vector A whose
origin is at point O, magnitude is proportional with amplitude A and
makes with axis (∆) an angle of initial phase ϕ At the time t = 0,
vector A (its head is M0) is rotated in positive direction
(conventionally is counter-clockwise) with angular velocity of ω
When the head M of vector A is projected on to x’x axis then the
motion of the projection P on x’x is a harmonic oscillation At any
time t, the head of A is M, its projection on x’x is P and we have:
x = OP = Asin(ωt + ϕ)That is the simple harmonic oscillation necessary to express It is said that a simple harmonic oscillation
x = Asin(ωt + ϕ) is represented by a vector A.
4 The combination of two oscillations of same directions and frequencies
Suppose that one object (e.g a mass-spring system hung in a moving train) simultaneously takes part intwo oscillations of the same directions and the same frequency ω, but they have different amplitudes A1,
A2 as well as initial phases ϕ1 , ϕ2
The resultant motion is the combination of two components (1-22) and (1-23) The Fresnel’s vectordiagram method will be applied to find the equation of resultant motion
Two axes (∆) and x’x are drawn as in figure 1.7 Draw a vector,
namely A1, whose magnitude is proportional with amplitude A1
makes an angle of ϕ1 with (∆) (figure 1.7) Similarly, draw
vector A2 whose magnitude is proportional with amplitude A2
makes an angle of ϕ2 with (∆) Draw vector A which is the
resultant vector of A1 and A2, this vector makes an angle of ϕ
with (∆)
In figure 1.7, the angle between A1 and A2 is (ϕ2 - ϕ1) (the phase
difference of two components x1 and x2) Since ϕ1 and ϕ2 are
constant then (ϕ2 - ϕ1)is also constant
Now rotate A1 and A2 around point O in positive direction with
the same frequency of ω Then a trapezoid OM1MM2 is not
deformed since both sides OM1, OM2 and the angle "M OM2 1 are unchanged Therefore A has a constant
magnitude and rotates around O in positive direction with angular velocity ω of A1 and A2
Since the resultant of projections of components onto an axis is the projection of the resultant vectorprojected on that axis So motion of P (projection of M) on x’x is the combination of P1 (projection of
M1) and P2 (projection of M2) on x’x axis, it is also a harmonic oscillation A is the resultant vector of A1and A2, and it also represents the combined oscillation and its initial phase is ϕ (figure 1.7)
Similarly, if it is necessary to combine various oscillations x1, x2, x3… it is recommended to draw the
resultant vector A of A1, A2, A3
Trang 17Figure 1.7 is called a vector diagram.
5 Amplitude and initial phase of the combinatorial oscillation
The equation of resultant motion is x = x1 + x2 = Asin(ωt +ϕ) (1-24)
where A is proportional with the magnitude of amplitude vector A.
It is necessary to evaluate A and ϕ in (1-24) For triangle OMM2 in figure 1.7, we have:
cos "OM M = - cos "2 M OM = - cos(2 ϕ2 - ϕ1) (1-25)From figure1.7 we can deduce that
1.What is the phase difference?
2.How are in phase oscillations, out of phase oscillation, leading phase oscillation, lagging phase
oscillation?
3.From figure 1.3 in §2, compared with the oscillation of a mass-spring system, are the velocity andacceleration of ball lagging or leading and how much are they?
4.Briefly state the Fresnel’s vector-diagram method?
5.Two harmonic oscillations have the same direction and the same frequency f = 50Hz, and have theamplitudes A1 = 2a, A2 = a and the initial phases ϕ1 =
3
π , ϕ
2 = π.a) Write the equations of these two oscillations
b) Draw in the same diagram vectors A1, A2 , A, of these two components and of the resultant oscillation.
c) Calculate the initial phase of the resultant
Hints: 5) ϕ = π/2
Trang 18§6 - §7 U NDERDAMPED AND FORCED OSCILLATIONS
1 Underdamped oscillation
In research of harmonic oscillation of mass-spring system, simple pendulum and other things, it is
observed that their frequencies and amplitudes are time-independent quantities It means that they willrepeats back and forth forever, never ended But in fact, amplitudes of free oscillations will be dampedthen ended because generally they move in a certain medium and are effected by frictions Depending onhow much the friction is, the oscillation will be damped fast or slowly Such oscillations are called
underdamped oscillations An underdamped oscillation does not have harmonic properties, therefore
when talking about the amplitude, frequency, or cycle of an underdamped oscillation, it implies
approximation
When a mass-spring system oscillates in the air, the air friction make it be damped
But since it is small so it takes quite a long time to ended Therefore, if this
oscillation is examined in a short time, the damping is negligible and it can be seen
as a harmonic oscillation
Let a pendulum fluctuates in a container of water (figure 1.8) The friction of water
is strong enough so it will be damped fairly quickly and it will stop at the
equilibrium position (figure 1.9a)
Replacing with a container of lubricating oil, if its friction is large enough there is
no fluctuation The ball passes through an equilibrium position (one time only) then
returns and stops there (figure 1.9b)
If the friction of lubricating oil is much stronger, the
ball even can not passes through the equilibrium
position and stops immediately (figure 1.9c)
In the real life and technology, in some cases the
underdamping is harmful and it is required to
overcome this phenomenon (i.e for clock pendulums)
In contrast, in some cases it is useful and needed so
people make use of that For example, we all know
that the surface of the road is not fairly flat, and the
faster the vehicle travels, the more vibrative it is,
hence vehicles and motorcycles must have buffer
springs When there is a gap, the spring is compressed
or stretched After that, the frame continues vibrating
like a spring, makes travelers tired and uncomfortable
In order to make it damped faster, vehicles are
equipped with a special equipment It consist of a
piston that can travel in a vertical cylinder contain
lubricating oil, piston is assembled to the frame and
the cylinder is mounted to a shaft of wheel When the
frame vibrates on buffer springs, the piston is also
fluctuated inside the cylinder The lubricating oil
make the vibration damped faster and so the vibration
of the frames does
Trang 19Fn = Hsin(ωt +ϕ)where H is the amplitude and ω is the frequency of the forced force Generally, the frequency
of the external excitation f =
2
ω
π is different from that of the free oscillation f0 of the ball.
Theoretical calculations resulted in: during a period of initial time ∆t, total vibration of the system is acomplicated, a combination of many free vibrations as well as external excitation After that, free
vibrations are ended, the ball oscillates due to the external excitation Its frequency is the frequency of theexternal force and the amplitude is dependent on a relationship between the externally excited frequency
f and free frequency f0 That is why a vibration after along time is a forced oscillation If the excitation ismaintained for a long time then the forced vibration is also maintained during that time
The complicated oscillation time ∆t is always very small compared with the forced oscillating timeafterward It can be said that after ∆t, the system ‘forgot’ its free vibration Therefore, in fact, it is usuallystudied the forced oscillation after ∆t and it is unnecessary to care about a complicated vibrations during
∆t
3 Resonance
This phenomenon can be examined by an experiment (figure 1.10)
A is a pendulum consisted of a mass of m fixed on to the metal rod N is a
light and thin slab by assemble composite The frequency f0 of the pendulum
when it does not assembled slab N is directly determined by a chronometer
B is another pendulum consisted of a mass of M >> m that can easily slide on
to a thin calibrated metal rod The frequency f is determined corresponding
to each position of pendulum B on the rod by a chronometer
These two pendulums, A (that is not assembled to slab N) and B, are hung
next to each other, two rod are joined by a light spring L Pendulum B is
allowed to swing in the plane that is perpendicular to the plane of paper The
frequency f of pendulum B is transferred to pendulum A as a forced
excitation by the spring This force makes pendulum A vibrating, and after a
time, pendulum A has forced oscillation with the frequency of f Changing in
position of pendulum B results in the change in frequency f as well, and it is
observed that the vibration of pendulum A reaches the maximum value when
f ≈ f0, but when f is smaller or greater than f0 then the amplitude of pendulum
A is decreasing dramatically
The phenomenon of the amplitude of forced oscillation is increased dramatically to a maximum value
when the frequency of forced excitation is equal to the free frequency of system is called the resonance.
Now, slab N is assembled to pendulum A to increase the atmospheric friction Repeating the wholeprocess above, it is shown that pendulum A has a resonant oscillation at f = f0, but its amplitude is
smaller than that in the case of no assembly with slab N In this case, because energy provided by forcedexcitation is mainly used to compensate frictional losses, thus it does not make the amplitude increasesignificantly The resonance exhibits clearest when the friction is insignificant
4 Applying and surmounting resonant phenomenon
Resonance is the most encountered phenomenon in life and technology, it can be harmful and useful forpeople
A child can use a small force to swing an adult’s hammock when the hammock reaches the highest level.Continuing swing like that after certain time, i.e exerting on the hammock a periodic force whose
frequency is the same as hammock’s frequency, a child can swing it to higher level, i.e hammock’samplitude is bigger It is impossible for the child to push a hammock to that level
In many cases, resonance is harmful and need to be overcome Every elastic thing are oscillations and
Trang 20resonantly with the other (e.g a big electric generator), and they will vibrate dramatically and can bebroken, collapsed that is the concern of engineers.
At the middle of XIX century, there was a troop paraded on to a bridge, it was vibrated dramatically andbroken, make a lot of human losses That is because the parading frequency of the troop coincided
accidental with the free oscillating frequency of the bridge and it made resonance After this accident,army regulation of some countries do not allow parade on the bridge
frequency and amplitude In the watch and table clock, spiral pendulum plays the role of the pendulumclock
The vibration that can be maintained without external excitation is called self-oscillation A system, such
as a pendulum clock, consists of oscillating mass, energy source and energy transfer mechanism is calledself oscillation system
Note that in the forced oscillation, oscillating frequency is the externally excited force and the amplitudedepends on the external amplitude But in the self oscillation, the frequency and amplitude is unchangedfrom their original value as well as free oscillation
Questions
1 In what conditions the oscillation is underdamped oscillation? How does the amplitude of
underdamped oscillation change?
2 How to make a forced oscillation? Why it has the name like that?
3 What is the resonant phenomenon? When does it happen?
4 A motorcycle is traveling on the road, consequently there is a small gap after a distance of 9m Thefree frequency of the frame of the motorcycle is on the buffer springs is 1.5s At what speed, the
motorcycle is vibrated most dramatically?
SUMMARY OF CHAPTER I
1.Oscillation is a space limited motion, repeat back and forth around an equilibrium position
In all kind of oscillations, periodic oscillation is a kind in which the state of motion is repeated as it wasafter a certain period of time
Cycle T is the smallest time after that the state of motion is repeated as it was Frequency 1
f T
= is thenumber of oscillation in a unit of time Its unit is Hertz (Hz)
In all kind of periodic oscillations, harmonic oscillation is described by a sinusoidal or cosinusoidal law:
x = Asin(ωt+ϕ) or x = Acos(ωt+ϕ) The displacement x is the deflection from an equilibrium position.The amplitude is the maximum displacement The angular frequency ω is a quantity to specify thefrequency
Trang 21A harmonic oscillation can be regarded as the projection of a uniform circle motion in the projectileplane The angular velocity of the circle motion is the angular frequency of harmonic oscillation In thevector diagram method of Fresnel, each oscillation is represented by a vector rotates in the datum plane
in the positive direction, and the total oscillation is the projection of the motion of the head of resultantvector on a straight line in the same plane
2 The cycle of a mass-spring system is T = 2π
3 The phase difference of two oscillation is the difference of initial phase and is call the phase difference
∆ϕ= ϕ1 - ϕ2 Two oscillations are in phase if ∆ϕ = 2nπ, are out of phase if ∆ϕ = (2n+1)π
The combination of two harmonic oscillations with the same directions and frequencies but differentamplitudes is a harmonic oscillation with the same frequency However, the amplitude and initial phase
of total oscillation is dependent on the phase difference of two components If two components are inphase then the total amplitude is maximum of A=A1 + A2 if they are out of phase then it is minimum ofA=| A1 – A2 |
4.In fact, every vibration is underdamped vibration, because the environmental frictions dissipate theoscillating energy In order for an oscillating system which has free frequency f0 is not damped, it isapplied a externally harmonic force of frequency f, called forced excitation Forced oscillation has thesame frequency f with external excitation The resonance happens when f = f0, the amplitude of forcedoscillation is increased dramatically to the maximum value Bigger maximum amplitude is, smallerenvironmental friction is
In life and science, technology, the resonance can be either harmful or useful
Trang 22Chapter II – MECHANICAL WAVE ACOUSTICS
§8 W AVE IN MECHANICS
1 Natural mechanical waves
When we throw a rock into a still water surface, we can observe a number of circular water wavesspreading out to every direction from the place where the stone is thrown If we drop a cork or a leafdown to the water surface, it will also rise up and down in response to the stimulated water waves But itonly fluctuate in one vertical direction, instead of moving horizontally with the circular water wave
We can explain the observation as follows Among the water molecules, there is a coalescent force thatmake them united together When a water molecule, say A, rises up, the coalescent force makes thenearby molecules to go up also, but a few time later It is also these forces that helps to draw the watermolecule A back to its previously resting place These forces acting very much the same role as theelastic force does in an elastic pendulum In conclusion, each molecule oscillating in a vertical directionwill tend to make the nearby molecules to oscillate in the vertical mode likewise and this mechanicmakes the oscillation to spread faring away
Mechanical waves are mechanical oscillations that spread out with time in a material medium.
Note that in mechanical waves only the oscillation states, i.e the phases of the oscillation, is spreadingaway, while the medium’s small mediums are only fluctuating around its original resting balance place.The water wave is one type of waves that can be observed by normal eye In reality, using appropriateequipment, scientists can observe waves in all other types of material – say it in solid, liquid or in gasform For example, if dropping some grains of sand into the surface of a wide big iron board, then using ahammer to smash hard in one far end of the iron surface, we can still see the grains of sand bumping up.This is because of the waves spreading through the iron board Unfortunately, we cannot see this type ofwaves with bared-eyes
In the example of the water waves, the direction of the oscillations of the mediums’ elements is
perpendicular to the direction in which the waves travel Such a wave is called a transverse waves There exists another type of wave, known as a longitudinal wave, in which the oscillation of particles of the
medium is along the same direction as the motion of the wave Longitudinal waves will be discussed indetails in this chapter
2 Oscillation phase transmission Wavelength.
A stone thrown into a water surface can create only a few
small waves, the oscillation will soon die out To make
better research in mechanical waves, a small equipment is
created to help making the waves last longer Using a thin
pieces of elastic metal, at one end sticking in a small ball or
needle Place the metal piece so that the marble slightly
touches the surface of a large water tray (figure 2.1) Then
we just need to flip on the right end of the metal piece to
make the ball harmonically vibrate with period T Then all
water molecules contacting with the ball will also vibrate
with period T in a relatively long time On the water
surface, a number of circular waves will start to spread in
Trang 23upwards At the time of t = T/4 (figure 2.2b) the oscillating phase of point A transmitted to point B, and
at different times of t = T/2; (figure 2.2c), t = 3T/4 (figure 2.2d) and t = T (figure 2.2e) the phasetransmitted to points C, D and E respectively It should be noted that the oscillating phase is transmitted
in a horizontal direction, while water elements only fluctuating vertically
In figure 2.2 we can see that points A, E and I are always in phase with each other The distance between
two successive in-phase points along the direction of wave transmission is called the wavelength,
denoted by λ (the Greek letter lambda) In general, those points the distance between which is a multiple
of the wavelength will oscillate in phase.
The distance between points I and G is a half of the wavelength and they are in opposite phase of each
other In general, those points the distance between which is an odd multiple of a half of the wavelength
will oscillating out of phase.
3 Period, frequency and velocity of waves
At every points through which the mechanical waves go, the medium elements oscillate with the same
period, which equals to the period T of the wave source This period is called the period of wave The reciprocal of the period, f = 1/T, is called the frequency of wave.
In the above example, we can see that after each period T the oscillation phase travels through a distance
equal to the wavelength Thus, we can also say that: the wavelength is the distance the wave travels in a
period T.
The speed of wave transmission is called the wave velocity Since in a period T the wave travels through
a distance equal to the wavelength λ, we have the following relation:
4 Amplitude and energy of waves
Once the wave reached a point, it makes the medium elements at that point oscillate with a particular
amplitude This amplitude is called the wave amplitude at the specific point in question.
We have known that the energy of a harmonic oscillation is proportional to the square of its amplitude.The wave makes the elements oscillating, thus provides them an energy We say: the process of wave
Trang 24For waves that originating from one point and spreading out in a surface, the wave energy is alsostretched in to a circle that keep expanding Since the length of the circle is proportional to its diameter,when the wave spreads far away its energy also decreases proportional to the traveling distance Forwaves that originating from one point and spreading out in a space, the wave energy reduces proportional
to the square of the traveling distance
In an ideal case when the wave is transmitted in one straight line, the wave energy will not be reducedalong the direction of wave propagation and the wave amplitude is the same at every point the wave goesthrough
Questions
1 What is a wave?
2 For a wave, what is transmitted, what is not?
3 Define the transverse wave and the longitudinal wave?
4 State two definitions of wavelength If the wave velocity is constant, then what is the relationshipbetween the wavelength and the wave frequency?
5 In figure 2.2, which points oscillate in phase and out of phase to point H?
§9 - §10 S OUND WAVE
1 Sound wave and the sensation of sound
Take a thin and long iron bar, then firmly clamp the below end of it (figure 2.3a)
Flip on the other end of the bar, we can see the iron bar to swinging back and
forth Lowering the clipped end of the bar, so that the swinging part is shortened
Again we flip on the free end of the bar We can observe that the bar also
oscillating but with a faster frequency than before When the swinging part of the
iron bar is shortened to some extend (that means the iron bar oscillating frequency
has been raised to some specific degree) we start hearing the small little noise
“uh”, i.e the vibration of the iron bar starts creating sounds Thus, we conclude,
the oscillating of the iron bar do sometime create sounds, and sometimes do not
This phenomenon can be explained as follows: When the iron bar swing into one side, it makes the airlayer right before it was compressed, and it also makes the air layer right after it being relaxed Thiscompressing and relaxing of the air happens periodically, and has created in the air a mechanical wavewhich have a frequency equals to that of the oscillation of the iron bar This wave spread towards ourears, and forced our eardrums starting to be compressed and relaxed with the same frequency It is thefluctuating of our eardrums at a certain range of frequency that helped us to recognize sounds
The human ears can detect only oscillations having frequencies in the range from 16Hz to 20,000Hz Any
oscillation with the frequency in the range of 16Hz – 20,000Hz is called the sound wave, or shortly the
sound The science that study sounds is called acoustics.
Sounds can spread in any medium, say it solid, liquid or gas/air For example: if we have someexperiences, we can here the sounds of a horse herd trotting or the sounds of a train running far away,which we cannot hear from the sound spreading through the air The reason is that, sound wavesspreading in the air was barred by a lot of barricade and was faded away very rapidly
Mechanical waves with frequency greater than 20,000Hz is called ultrasonic waves Some natural
species can emit and detect ultrasonic Those waves which has its frequency smaller than 16Hz is called
infrasonic waves Human by using appropriate equipment can also create and detect these sounds, and
these has a lot of applications in science and technology
In terms of physic nature, sound waves, ultrasonic and infrasonic sounds are not different to each otherand to other mechanical waves The classification is made based on the sensation ability of the human ear
in detecting different waves This is determined by the physiological characteristics of the human ear.Thus, in acoustics, scientists do distinguish the physical and physiological characteristics of sounds
Trang 252 Sound transmission Speed of sound
Sound waves can spread in any medium, but its sound-transmission speed (the speed of sound) depends
on the elasticity and density of the medium
In general, the speed of sound in a solid medium is greater than that in the liquid, and spreading speeds in
a liquid medium is greater than that of a gas/air medium The speeds also change in response to thechange in the temperature of the medium
Below are speeds of sound in some substances:
Solid and liquid (t = 20 o C) Gas (in atmospheric pressure)
3 Sound altitude
Among the sounds that we can detect, there are sounds that its frequency is specified, e.g the sound of a
singer singing a song, or the sounds of a musical instrument This is called musical sounds There are
also sounds that do not have specific frequency, like the sound of a diesel engine, or the sound of a herd
of horse running These sounds are called interference In its nature, these sounds are the combination of
a number of oscillations that having very different frequency and amplitude We’ll only study the musicalsounds
With only one tune of a song, but if it was song by either soprano or tenor, can give us very differentexperiences Sounds with different frequencies present us different sound senses Those sounds that have
a high frequency is called high-pitch sounds or treble Those sounds that have lower frequency is called
low-pitch sounds or bass The pitch of a sound is a physiological characteristic of a sound, it is based
upon a physical characteristic: the sound frequency
4 Timbre
Even when two singers sings the same tune in the same
pitch, we can distinguish the sound of each singer Or when
the guitar, the flute, the clarinet is playing the same musical
tune, we can still make distinctions between those different
musical instruments Each person, each musical equipment
create sounds that have different characteristics that we, by
our hearing senses, can distinguish These characteristics of
sounds is called timbre.
Timbre is a physiological characteristics of sounds This
characteristic is based on physical characteristics of sound:
the frequency and the amplitude Experiments have proved
that when a man, or a musical instrument, produces a sound
wave with frequency f1, he or it also produces other sound
waves of frequencies f2 = 2f1, f3 = 3f1, f4 = 4f1, etc
The sound waves with frequency f1 is called the
Trang 26having frequency of f2, f3, f4, … are called the second harmonic, the third harmonic, the forth harmonic…Depending on the structure of the music instrument, and the structure of the human mouth and throat,then the relative amplitude of the different harmonics is different Because of this, the sound emitted is acombination of the different harmonics It still has frequency f1 but is no longer a sinusoidal curve,instead it becomes a complicated periodic curve Each form of the curve represents a different timbre Infigure 2.4, the curves represent timbres of a piano and a clarinet, corresponding to the same fundamentalsound They have the same period, but the shape of curves is different.
Depending on the sensitivity of the ears, we can distinguish different singing voices, or different sounds
of different music instruments
5 Sound energy
Like any other mechanical waves, sound wave retains in it an energy that proportional to the square ofthe amplitude of the wave This energy is transmitted from the source of the sound to our ears
The intensity of sound is the energy that a sound transports per unit time across unit area The unit of the
sound intensity is watt per square meter (W/m2)
To the human ear, the absolute value of a sound intensity I is not as important as its relative value incomparison with a certain value I0 which is selected as the standard sound intensity The intensity level,
L, of any sound is defined as the (decimal) logarithm of the ratio I/I0
0
I
I lg ) B (
The unit of the intensity level is bel (B) Thus if L = 1, 2, 3, 4B… then the sound intensity I is 10, 102,
103, 104… times the standard sound intensity I0
Actually, the unit of decibel (dB) is usually used A decibel is equal to 1/10 of bel, i.e a measurement indecibel would be ten times the measurement in bel Thus, equation (2-3) would become
0
I
I lg 10 ) dB (
When L = 1dB, then I = 1.26 I0 It is the smallest sound intensity that the human ear can detect
6 Sound loudness
To create the feeling of a sound, the sound intensity must greater than a certain minimum value which is
called the threshold of hearing Due to the physiological characteristics of human ears, the threshold of
hearing changes according to sound frequencies With the sound frequencies in the range of 1000 –5000Hz, the threshold of hearing is about 10-12 W/m2 With the frequency of 50Hz, the threshold ofhearing is 105 times bigger
Thus, a sound wave of frequency 1000Hz and of intensity 10-7 W/m2 is considered ‘very loud’ to thehuman ear, while another sound of frequency 50Hz is considered as a ‘whispering’ sound Hence, theloudness of a sound, or volume, is not coincide with the intensity of sound
Human ears is most sensitive with sounds in the range of 1000 – 5000Hz, and normally more sensitivewith high-pitch than low-pitch Therefore, broadcasting announcers are often female than male Andthat’s the reason why when we lower the volume of a radio we cannot hear the low-pitch sounds
anymore
If the sound intensity increase to as high as 10W/m2, the human ear will be suffering from stinging nomatter which frequency is, and the sound is no longer considered normal That highest level of the sound
intensity is called the threshold of hurting The range between the threshold of hearing and the
threshold of hurting is called the audible range of the sound.
While calculating the intensity level by using equation (2-4), the standard intensity I0 is selected as thethreshold of hearing of a sound wave of frequency 1000Hz
Below are some sound’s intensity that should be noted:
Trang 27The threshold of hearing 0dB
Noises in a busy supermarket 60dB
The sound of a big lightning 120dB
Sounds with high intensity make stresses and tiredness to human Living or working long time in a place
of high sound-intensity reduces the sharpness of the ears, and badly affects the human health and mind.The performances of rock music with the speaker’s volume of around 100dB also bring bad
consequences to the audiences
7 Sound source – Resonant box
In our surroundings, there are many sources of sound The sound source can be an interference/noise one,e.g the car engine on the nearby streets, the wind squeezing through the leaves, etc.; or it can be a
musical sound source They distinguish two main types of musical sound sources: producing sounds
using strings (stringed instruments), and producing sounds using air columns (wind instruments).
When a stretched string is stimulated by flipping on it, it vibrates with a specified frequency, which isdetermined by the length, the section area, the strain and the material of string Musical strings have verysmall section areas so it can only make whirling oscillations in the nearby air space, and can hardlyproduce a sound A musical string is normally stretched over a wooden or leather surface, when it
oscillating it also makes the surface vibrate with the same frequency Since the surface has a large area, itcan create considerable compressive and expanding air regions and therefore produce sound waves
We also know that when a string vibrates and produces a fundamental sound, it also produces otherharmonics Each type of musical instruments has a gourd with a certain shape, acting as a resonant box,i.e an empty object that has the capability of resonance to a number of different frequencies, and thus itcan intensify sound waves of those frequency Depending on the shape and the material of the gourd,each type of musical instruments can intensify certain harmonics, and therefore can produce its owncharacteristic timbre
The human vocalization organ operates in the same way as a string instrument The vocal cords act like musical strings, while the larynx, the mouth space and the nose space play the role of a resonant box Especially, by adjusting the positions of the lower jaw, lips and the tongue, it is possible to change the shape of the mouth space and therefore change the timbre accordingly Hence, the voice of a person is very rich in timbre, and one can imitate voices of others or voices of musical instruments quite
successfully.
For flutes and clarinets, when we are blowing the air into these instruments, the air columns inside theseinstruments oscillate with fundamental and harmonic frequencies The bodies of these instruments havedifferent shapes and materials, they act as a resonant box and produce timbres characterizing each type ofinstruments
Questions
1 What is sound waves, and how it created sound senses? Sound waves can travel in which medium?
2 What is the timbre? And why do we have timbres?
3 What is the audible range and what are its thresholds?
4 Distinguish the sound intensity and the sound loudness
5 How the stringed instrument created sounds?
6 One person does hammering heavily onto the train rails At a distance of 1,090m away, another personhears the hammering sound through the rails 3s before hearing it from the air Calculate the sound speed
in the rails if we know that the sound speed in air is 340m/s
Trang 287 One metal surface vibrates with a frequency of 200Hz It produces a sound wave of wavelength 7.17m
in water Calculate the sound speed in water
Hints: 6) ≈5300m/s; 7) 1434m/s
§11 W AVE INTERFERENCE
1 Interferential phenomenon
In real life, sometimes there are two or more waves
originating from different sources and interfere each other
at a specific point Such a situation creates a characteristic
phenomenon of wave, namely wave interference, which
will be studied in this section
Using experimental equipment similar to that is described
in §8 (figure 2.1) but here a light bar is used instead of the
ball In each end of the bar a small ball is attached
contacting the water surface (figure 2.5) When the bar P is
stimulated to oscillate, two balls at points A and B will
produce on the water surface two systems of circular waves
originated from two corresponding points Two waves
spread out, and mix up with each other
After a while, when the features of the waves are stabilized, we can observed on the water surface thatthere is a group of curves of which the oscillation amplitude is maximum, coming between them is
another group of curves in which the water surface does not vibrate These curves keep standing, unlikewaves under observation previously which spread out
2 Theory of interference
Suppose that A and B are two sources of oscillation with the same
frequency and phase, and their waves both travel to a point M on
the same plane with A and B following two corresponding paths
d1 and d2 (figure 2.6) Two sources of wave having the same
frequency and the same phase, or having the same frequency and
a constant difference in phase are called constructive sources,
and their corresponding waves are called constructive waves.
In the example described above, the two sources do not vibrate
independently They always oscillate with the same frequency and
phase as that of the bar P, and thus they are exactly two
constructive sources
Suppose that the equation of oscillations at both A and B is u = asinωt If the distance l between A and B
is negligible in comparison with lengths of paths d1 and d2, then the amplitudes of waves traveling topoint M can be safely considered to be equals
Let’s denote v as the traveling speed of the wave, then (d1/v) is the time needed for the oscillation totravel from A to M As such, the oscillation at M at the instant t is in phase with the oscillation at thesource A at the instant (t - d1/v) Thus the equation describing the oscillation at point M originating fromsource A is:
) d v t sin(
a ) v
d t ( sin a
M A
ω
− ω
=
− ω
Similarly, the equation describing the oscillation at point M originating from source B is:
) d v t sin(
a ) v
d t ( sin a
M B
ω
− ω
=
− ω
Trang 29same frequency but different phase The phase deviation equals:
thus we have the conclusion: at the points that the difference in the wave’s paths equals an integer
multiple of the wavelength, d = nλ (n = 0, 1, 2, ), the difference in phase will equals 2nπ – which meansthe two oscillating components are in phase with each other – will making the amplitude of the combinedoscillation is twice as that of each component The oscillation at these points are maximum Using
mathematical tools, it can be proved that the locus of these points is a family of hyperbolic curves thatreceive points A and B as two focuses, and also included the equidistant line of AB (figure 2.7,
continuous lines)
Similarly, at those points that the difference in the waves’ paths d
equals an odd number time of half of the wavelength,
d=(2n+1)λ/2, then the phase difference of the two wave
components is (2n+1)π – which means the two oscillating
components are in out of phase to each other - thus making the
oscillation amplitude at these points equals zero In other words,
the water elements at this points do not fluctuate at all The locus
of these points is also a family of hyperbolic curves having two
points A, B as its own focuses (figure 2.7, slash lines)
At all other points, the amplitude of the oscillation stays
somewhere between those two above mentioned numbers
The above observation is called the wave interference The wave
interference is the combination of two or more coordinate waves
in space, in which there are fixed places where the oscillation
amplitudes are either strengthened or dismissed.
The combination of three or more waves will create a complex
picture of interference that goes beyond the scope of this book
3 Standing wave
Prepare a string of length 2m, one end (M) was tied to
the wall, the other end (P) is kept in the our hand (figure
2.8) Stretch the string and swing our hands back and
forth, making point P oscillate Adjust slightly the
frequency of oscillation of point P until we can see a
stabilized oscillation of the string, in which some points
in the string are fluctuating very heavily while there are
other points where the string does not seems to have in
fluctuation
The observation can be explained as follows: the oscillation of point P travels along the string from P to
M Since M is tied, there will be a reflected waves traveling from M to P These incident and reflected
waves satisfy the condition of coordinate waves Here, point M does not oscillate, i.e at this point two
waves are out of phase to each other The result is that, in the string there is a combination of two
coordinate waves with the same frequency but opposite phases at point M (two points P and M can beconsidered as two sources of coordinate waves)
To better investigate the experiment, let’s consider a string with two tied end points A and B, over which
Trang 30case here is very similar to the experiment we have just described above, but both two end points are tieddown and not oscillate at all.
Take t = 0 at the instant when two waves are in opposite phase to each other at a certain point M in thestring The string AB will have a form as described in figure 2.9a The first wave going from left to rightwhile the second wave going from right to left The combined oscillation amplitude at every points at thisinstant is 0 At t = T/4, each wave has traveled a distance of λ/4, and the combined oscillation of twowaves on the string AB has the form as described in figure 2.9b Similarly, at t = T/2 and t = 3T/4 thecombined wave has the form as depicted in figures 2.9c and 2.9d respectively
Observing the string AB over time, we can see that point M and other points that distancing it an integer
multiple of half of the wavelength always remain stationary They are called nodes Point N and other
points distancing it an integer multiple of half of the wavelength are the points that fluctuate the most.They are called antinodes (or bells) of the wave The locations of nodes and antinodes do not changeover time The distance between two successive nodes or antinodes equals half of the wavelength (λ/2)
The wave of which all nodes and antinodes are fixed in the space is called the standing wave A standing
wave does not travel in the space It should be noted that even though two component waves still travelfollowing two different directions, the combined wave still “stand”
In the case of longitudinal waves, although the picture of the stand waves is a little bit different, it stillhave nodes and antinodes, and their distance is still λ/2 In the string of the string musical instrument, thestanding wave is a transverse one, while in the flute or clarinet, the standing wave is a longitudinal one.The phenomenon of standing waves allows us to observe a wavelength by normal eyes, and accuratelymeasure a wavelength as well For sound waves and other types of waves, measuring the wave frequency
is rather simple Since the frequency f, the wave speed v and the wavelength λ are related through thecorrelation v = λf, based on the standing wave phenomenon it is possible to calculate the wave speed bymeasuring λ and f
Questions
1 What is the coordinate source?
2 What is the wave interference? Show how to create the interference of water waves
3 Show how to create a standing wave in a stretched elastic string Which points are nodes, and whichpoints are antinodes?
4 How to determine the wave speed on a string with a standing wave?
5 One musical string of length 60cm produces a sound note of frequency 100Hz Observing the string, it
is recorded that there are 4 nodes (including two ends) and 3 antinodes Calculate the wave speed on thisstring
Hints: 5) 40m/s
Trang 31SUMMARY OF CHAPTER II
1 Mechanical waves are mechanical oscillations that spread out with time in a material medium Themechanical wave can travel in either solid, liquid, or gas mediums Wave transmission means the
transmission of oscillation phase, in which the medium particles do not travel but vibrate around
equilibrium positions The longitudinal wave has its direction of oscillation coincided to the travelingdirection of wave, while the transverse wave has its direction of oscillation perpendicular to the travelingdirection of wave
The period of a wave is a common period of oscillation for all substance elements through which thewave travels, and is equal to the oscillation period of the wave source The reciprocal of wave period isthe frequency of wave The wave speed is the speed of phase transmission The wavelength is the
distance between two successive in-phase points along the direction of wave transmission It is also thedistance the wave travels in a period
Between the wave speed v, the wavelength λ and the frequency of wave f (or the period of wave T) existthe correlation λ = vT or λ = v
f .The amplitude of oscillation of medium elements at the point the wave travels though is called the waveamplitude at this point The transmission of wave is also the transmission of energy The wave energy at
a point is proportional to the square of the wave amplitude at this point Generally, the farther the wavetravels from the wave source, the smaller the wave amplitude and the wave energy are
2 Sound waves are those longitudinal waves traveling in a material medium, having frequencies in arange of 16Hz – 20,000Hz, and causing acoustic senses inside human ears Sound waves have bothphysical and biophysical characteristics From the physical point of view, both sound waves, ultrasonicwaves and infrasonic waves are similar to other types of mechanical wave The biophysical
characteristics of sound waves depend on the structure of human ear
Musical sounds are those which have specific frequencies Noises are those which have no specificfrequencies The sound altitude is a biophysical characteristic of sound, characterized by the frequency ofsound Sounds created by human or musical instruments are combinations of the fundamental sound andother harmonics, creating the timbre which is a biophysical characteristic of sound
The loudness of sound is a biophysical characteristic of sound, depending on the intensity of sound Eachsound frequency corresponds to a threshold of hearing, therefore two sounds with the same intensity butdifferent frequencies will have different loudness The intensity level is measured in decibel (dB) Anaudible sounds has an intensity level in a range of 0dB to 130dB
There are two main sources of music: vibrating strings in stringed instruments and air columns in windinstruments
3 The wave interference is the combination of two or more coordinate waves in space, in which there arefixed places where the oscillation amplitudes are either strengthened or dismissed Coordinate waves arethose produced by coordinate sources of oscillation, i.e sources having the same frequency and the samephase, or having the same frequency and a constant phase difference In a plane, two coordinate sourceswith the same frequency and the same phase produce an interferential image in which the points ofmaximum oscillation and the points of non-oscillation lie on two alternate family of hyperbolic curves.The points of maximum oscillation are those the difference in traveling distances from two sources towhich equals an integer multiple of the wavelength The points of non oscillation are those the difference
in traveling distances from two sources to which equals an odd integer multiple of the wavelength
When an incident wave and its reflected wave travel in the same path, they will interfere with each otherand produce a standing wave with nodes and antinodes The distance between two successive nodes orantinodes equals a half of the wavelength Based on the phenomenon of standing wave, it is possible toeasily measure the wavelength λ and to determine the speed of wave v when the wavelength λ and thefrequency f are all known
Trang 32Chapter III – ELECTRIC OSCILLATIO N, ALTERNATING CURRENT
§12 H ARMONIC OSCILLATION VOLTAGE A LTERNATING CURRENT
1 Harmonic oscillation voltage
Let a metal loop of area S and turn N uniformly rotate around its
symmetrical axis xx’ in a uniform magnetic field B whose direction
is perpendicular to xx’ (figure 3.1) The angular velocity of the loop
is ω
Suppose that at time t = 0 the normal On of the frame is parallel to
the direction of the magnetic field The magnetic flux according to
one turn is BS
At time t > 0, the normal On makes an angle ωt with vector B, the
magnetic flux according to one turn is:
Φ = BScosωtThe magnetic flux through each turn varies with time, inducing an
induced electromotive force (emf) with magnitude:
where ω, N, B, S are constant with respect to time It can be concluded that the emf exists in the loop is avariable harmonic one If two points A, B are connected to a external circuit, an electric current willappear in a closed loop The loop acts as an electric generator, an the induced emf is the emf of the
electric source
As the emf varies harmonically with an angular frequency of ω, the voltage of the external circuit alsovaries harmonically with the same angular frequency ω With suitable initial conditions, the equation ofvoltage has a simple form
Trang 33The electric circuits are usually composed of bulb, electric engine, electric cooker, etc there are alsoresistors, capacitors and inductors They are damped oscillation circuits, and when being connected to theplugs, the potential difference produces a forced oscillation electric current with angular frequency ω:
The phase difference φ between i and u depends on the properties of the circuit As the velocity of
electric field flowing in wires is so high, nearly equal to velocity of light, at a certain time t, the electricfields at all points of the series circuit are the same Hence, the currents at any point are the same
The current described in (3-5) is a harmonic fluctuating one It is called the alternating current In fact,
there are currents with direction changed but they are not harmonic When saying about alternatingcurrent, it should be understood that we are talking about harmonic oscillation current
3 Root mean square (rms) value of intensity and voltage
The alternating current we used is of frequency 50Hz (60Hz in some countries), the intensity variesrapidly with respect to time When using the alternating current, we are not interested in the effect of theelectric current at a certain time, but in the effect in a long time
Let an alternating current i = I0sinωt go through a resistor R for quite a long time t, and then measure thethermal energy the resistor R emitted There is a relation between Q, I0, R and t as follows
Q = R
2 0I
2 tNow, if there is a direct current I going through this resistor R in the same time t, emitting the samethermal energy Q, then the relation between I, R, t, Q is:
Q = RI2t
Comparing these two equations we got I =
2 0I
2 ; or I=
0I
It means that the alternating current i = I0sinωt is equivalent to a direct current I = I0
2 on the heatemission effects Theoretical calculations also give the same result The intensity I determined by (3-6) iscalled the rms (root mean square) value of the alternating current (sometimes called the effective value)
The rms value of an alternating current is equal to the intensity of a direct current which would produce the same thermal energy when they go through the same resistance R in the same time interval The rms
current is equal to the peak current divided by 2
Similarly, the rms value of an emf of an alternating electric source is E = E0
Questions:
1 What is the alternating current?
2 Give the definitions of rms intensity of an alternating current
3 What are the rms value of current, emf and voltage?
Trang 344 Write the oscillation equations of alternating voltage in the case that the rms voltage and the frequencyare: a) 220 V, 50 Hz; b) 127 V, 60 Hz.
Hints: 4 a) u = 311sin100πt (V); b) u = 180sin120πt (V)
§13 - §14 A LTERNATING CURRENT IN A CIRCUIT CONTAINING ONLY RESISTANCE , INDUCTANCE OR CAPACITANCE
Normally, an AC circuit in a household equipment contains both resistance, inductance and capacitance.However, here we will study those circuits containing only resistance, inductance or capacitance, beforecontinue with a common case
B - A LTERNATING C URRENT I N A C IRCUIT CONTAINING ONLY R ESISTANCE
1 Relation between current and voltage
Consider a circuit containing a resistor R (figure 3.2), and applied to two ends
of this circuit is an alternating voltage
For a very small period of time ∆t, the current is considered unchanged
According to Ohm’s law:
i = u
R = 0U
Because U0 and R are not changed in time, let I0 = U0
R and rewrite (3-10) to
Comparing (3-9) and (3-11), we can see that the voltage applied
to two end of a circuit containing only resistance fluctuate
harmonically and in phase with the current Figure 3.3 shows the
vector diagram demonstrating the relationship between voltage u
and current i Axis Ox is called the axis of current, since the
direction of vector I0 is coincident with that of axis Ox In this
case vector U0 lies on the axis of current
2 Ohm’s law for an AC circuit containing only resistance
In the equation I0 =U0
R , if the two sides are divided by 2 : I =
U
where I and U are rms values of current and voltage respectively Formula (3-12) shows Ohm’s law for
an AC circuit containing only resistance, in the same form as for a DC circuit Note that (3-10) shows therelationship between i and u which is valueless in the real life, while (3-12) shows the relationship
between the rms values I and U that we interest in when using the alternating current
C - A LTERNATING C URRENT I N A C IRCUIT CONTAINING ONLY C APACITANCE
1 Effect of capacitors to the alternating current
Consider a circuit as shown in figure 3.4 There is an alternating
voltage between A and B When K is switched to M, the bulb D is
lightened Then switch K to N, the bulb is also lightened, but not as
bright as when K is switched to M If we replace the AC voltage by a
DC voltage when K is switched to N, the bulb D can not be lightened
It shows that the capacitor prevents the direct current from going
through it, but allows the alternating current to go through The
Trang 35capacitor also has a preventive effect to the alternating current, i.e it has a resistance, which is called the
capacitive impedance to distinguish with the normal resistance The issue of an alternating current going
through a capacitor will be discussed farther in §25
2 Relation between current and voltage
Connect two terminals A, B of a capacitor C (figure 3.5) with an alternating voltage u = U0sinωt (3-13)The amount of charge q of the capacitor at time t is q = Cu = CU0sinωt
The amount of charge of capacitor varies harmonically with an angular
frequency ω, i.e there are always electrons flowing from one terminal to a
conducting plate of the capacitor, or vice-versa In other words, there is an
alternating current in AB Considering a very small time interval ∆t, the
current i will become the derivative of q with respect to time t:
i = q! = ωCU0cosωt = ωCU0sin(ωt +
Comparing (3-13) and (3-14), it can be seen that the current also varies
harmonically with the angular frequency ω, but leads the voltage
2
π in
phase In other words, the voltage between two ends of a circuit
containing only conductance varies harmonically with a lag of π/2 in
phase compared with the current.
By changing the time origin we can rewrite (3-13) and (3-14) to
u = U0sin(ωt –
2
Figure 3.6 shows the relationship between the voltage and the current In this case vector U0 is
perpendicular to the axis of current, and has a direction downward
3 Ohm’s law for an AC circuit containing only capacitance
When two sides of the expression Io = ωCU0 are divided by 2 , it leads to a new expression
ω The capacitiveimpedance depends on the angular frequency ω of the current, and plays the role of resistance in Ohm’slaw for direct currents The expression (3-15) can be rewritten as:
I =C
Trang 36D - A LTERNATING C URRENT I N A C IRCUIT WITH I NDUCTORS
1 Effects of an inductor to the alternating current
Consider a circuit as shown in figure 3.7 There is an alternating
voltage between two terminals A, B The resistance of the inductor is
negligible
When K is switched to M, the bulb D is lightened Then k is switched
to N, the bulb is also lightened, but not as bright as before
It means that the inductor has a preventive effect to the alternating
current, i.e it has a resistance, which is called the inductive
impedance to distinguish with the normal resistance.
2 Relation between current and voltage
Apply an alternating voltage to two terminals of an inductor of inductance L and negligible resistance(figure 3.8) The alternating voltage produces an alternating current:
Suppose that at time t, the current through L is increasing L then
plays the role of a consumer with:
Complete calculations show that (3-17) is satisfied at any arbitrary time
Comparing (3-16) and (3-17), it can be seen that the voltage between two
ends of a circuit containing only inductance varies harmonically with a
lead of π/2 in phase compared with the current.
Figure 3.9 shows the relationship between the voltage and the current In
this case vector U0 is perpendicular to the axis of current, and has a
direction upward
3 Ohm’s Law for an AC circuit with inductors
From the equation ωLI0 = U0, we deduce: I0 = U0
LωDividing two sides by 2 , we get: I = U
L
Trang 37It is the expression of Ohm’s law for circuits containing only inductance I and U are rms values ofcurrent and voltage respectively, ωL is the inductive impedance, ZL = ωL The inductive impedancedepends on the angular frequency ω, and plays the role of resistance in Ohm’s law for the direct current.The expression of Ohm’s law, (3-18), can be rewritten as:
I =L
U
An inductor does not prevent the direct current but the alternating current The higher the frequency is,the more the current is prevented
The inductor without resistance is just an ideal definition In practice, every inductor has a resistance
A circuit containing an inductor of inductance L and resistance R should be seen as a circuit with aninductor L without resistance and a resistance R without inductance connected in series, as there is anonly current flowing from one end of the inductor to the other Ohm’s law for this case is discussed in
§15
Questions
1 In a circuit with only resistance, how does the voltage vary with current Write the equation of Ohm’slaw for it
2 How does the voltage vary with current in a circuit containing: a) only inductance; b) only capacitance
3 Write the expression of Ohm’s law for a circuit containing: a) only inductance; b) only capacitance
4 Write the equations demonstrating the magnitudes of capacitive impedance and inductive impedance.What are their effects on alternating currents with different frequencies?
5 A circuit contains an inductor with inductance L = 0.8 H The resistance of the circuit is negligible.There is an alternating voltage 220V, 50Hz between the two terminals Calculate the capacitive
impedance and the current intensity
6 A circuit has a capacitor, capacitance C = 20μF The resistance is negligible There is an alternatingvoltage 127V, 60 Hz between two terminals of the circuit Calculate the capacitive impedance and thecurrent intensity
§15 A LTERNATING CURRENT IN AN RLC CIRCUIT
An electric cooker with both the resistance and the inductive impedance is equivalent to a cooker with aresistor and an inductor connected in series A normal electric fan consists of resistors, inductors andcapacitors connected complicatedly In this section, we concentrate on the case of a simple non-branchedcircuit that contains a resistor R, an inductor L and a capacitor C connected in series, namely the RLCcircuit
1 Electric current and voltage in an RLC circuit
Figure 3.10 shows a RLC series circuit with a voltage between
two terminals and an alternating current From §12, it’s known
that at any time t, the intensities at any point of the series circuit
are the same Suppose that the intensity of the alternating current
at time t is:
i = I0sinωt
We have known the method to determine voltages uR (between A and M), uL (between M and N), and uC(between N and B) These voltages vary sinusoidally with the same angular frequency ω; uR and current iare in phase; uL lead
2
πof phase to current i; and u
C lags 2
πof phase to current i Their descriptiveequations are uR = URosinωt; uL = ULosin(ωt + π ); and u
C = UCosin(ωt – π ) in turn.
Trang 38As this is a series circuit, the voltage between two terminals A and B at time t is:
2 Relation between current and voltage in an RLC circuit
The oscillation of voltage u is the combination of three
oscillations of uR, uL and uC By using Fresnel diagram, the
equation of oscillation could be found
In the same vector diagram, draw three vectors URo, ULo, UCo
(figure 3.11) As ULo and UCo are in opposite direction, the
summation ULo + UCo has a magnitude of │ULo – UCo│ The
direction is upward if ULo > UCo , downward if ULo < UCo and
equals to zero if ULo = UCo Vector U0 is the summation of
URo + ULo + UCo, making an angle φ with the current axis
The equation of the voltage u is as follow:
u = U0sin (ωt + φ)
In the right-angled triangle OSP:
OP = │URo│ = URo, SP = OQ = │ULo + UCo│ = │ULo - UCo│
ω , the
voltage and current are in phase
3 Ohm’s Law for an RLC circuit
Let both sides of (3-21) divide by 2 and let
Trang 39(3-24) is the expression of Ohm’s law for a RLC circuit Here, I and U are rms values of intensity andvoltage respectively, Z is the impedance playing the role of a resistance as in Ohm’s law for direct
Questions
1 Draw a vector diagram for a RLC circuit when ULo = 1
2UCo In this case, the voltage u leads or lags in
phase to the current i?
2 Write the formulae to calculate the impedance of a RLC series circuit In which case the impedancereaches maximum? What will happen then?
3 In what conditions u and i are out of phase and in phase What happen in the RLC circuit when u and iare in phase?
4 An RLC series circuit has R = 140Ω, L = 1H and C = 25μF The alternating current in the circuit has afrequency f = 50 Hz and I = 0.5 A (rms value) Calculate the impedance of the circuit and the voltage
§16 P OWER OF THE ALTERNATING CURRENT
1 Power of the alternating current
Applied to two terminals of a circuit a certain voltage Using a voltmeter and an ammeter to measure therms value of voltage and current, U and I respectively Using a wattmeter to measure the power P
absorbed by the circuit
If the circuit contains resistors only, the relationship between U, I and P is expressed as follow;
k = cosφWhen the circuit has only resistors, I and U are in phase and cosφ = 1, in both cases we can write:
The quantity cosφ is called the power factor of the current As shown in Figure 3.11:
Trang 40cosφ = OP
o
U U
But URo = IoR and Uo = IoZ Hence:
cosφ =R
2 Significance of the power coefficient
According to (3-26), when U and I are constant the power P will increase if cosφ increases Let us
consider specific cases:
a) when cosφ = 1, i.e φ = 0: In this case, the circuit has only resistors, or there is a resonance in thecircuit The power consumed is maximum and equals to UI
b) when cosφ = 0, i.e φ = ±
2
π
: In this case, the circuit does not have resistors, but only has capacitors orinductors or both of them The power consumed is minimum and equals to zero A large power can beprovided, making U and I become large, but no power can be absorbed
c) when 0 < cosφ < 1, i.e –
2
π < φ < 0 or 0 < φ <
2
π :
The power absorbed UIcosφ is less than the power provided UI To enhance the effectiveness of
consuming, cosφ should be enlarged Hence, the circuit can use most of the power provided
In fact, systems with cosφ < 0.85 are normally not be used Usually, electric engines have inductiveimpedance larger than resistance, cosφ therefore is small A capacitor is connected in parallel to increasecosφ For a RLC circuit, when it has only R and L, φ can be quite large (figure 3.12a) The connection of
C to the circuit could make φ decrease, i.e make cosφ increase (figure 3.12b)
Questions:
1 In what cases does the power factor cosφ = 1? Draw the correspond diagram
2 In what cases does the power factor cosφ = 0? Draw the correspond diagram
3 A coil with inductance L = 0.2H and resistance R = 10Ω is effected by alternating voltage 220 V Thefrequency is 50 Hz Calculate the intensity of the current and the thermal energy emitted in 5 seconds