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MINITRY OF EDUCATION AND TRANNING

PHYSICS12

2000


2/169Translated by VNNTU Dec. 2001 Page 2

TABLE 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................................................................................82. 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...............................................................................................151. 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



2. 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.................................................................3313. - 14. Alternating current in a circuit containing only resistance, inductance or capacitance.......34

1. Relation between current and voltage............................................................................................34

2. Ohms law for an AC circuit containing only resistance...............................................................34

1. Effect of capacitors to the alternating current................................................................................34


3. Ohms law for an AC circuit containing only capacitance............................................................35

1. Effects of an inductor to the alternating current ............................................................................36


3. Ohms Law for an AC circuit with inductors ................................................................................36

15. Alternating current in an RLC circuit ............................................................................................37Electric current and voltage in an RLC circuit ..................................................................................37

Relation between current and voltage in an RLC circuit...................................................................38

3. Ohms 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

Half-cycle rectifying method.............................................................................................................49



3. 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......541. 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 .............................................................................................583. 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 objects 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



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 .................................................................................................................................784. 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.........................................................................................834. Lens power ....................................................................................................................................84

37. Image of an object through lenses. Lenses equations....................................................................85

1. Observing an objects image through a lens..................................................................................85

2. Method to draw an objects 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. Eyes 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

43. Light interference phenomenon ...................................................................................................105



1. 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............................................................................................1081. 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 ...........................................................................................................................1133. 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. Hertzs 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



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 ...........................................................................................................13356. 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. Einsteins relation between mass and energy..............................................................................138

1. Einsteins axioms.........................................................................................................................138

59. THE LOSS OF MASS. NUCLEAR ENERGY...........................................................................1401. 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 pendulums 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



Part I. OSCILLATIONS AND WAVES

Chapter I MECHANICAL OSCILLATIONS

1.PERIODIC AND SIMPLE HARMONIC MOTIONS. OSCILLATION 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 equilibriumposition. The movement likes that is called the oscillation.An oscillation, or vibration, is a limitedmotion 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 guitars 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 periodicoscillation. Periodic oscillation is the oscillation whose state is repeated as it was after a constant periodof 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

1showing 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 means

that the pendulum carries out 2 oscillations in a second.

The vibration of the guitars strings do not permanently maintained. It is damped then ended. But if it isobserved 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 ofnegligible mass, put in the horizontal plane as shown (figure 1.1a). There is a small hole through the ballso 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 Ois 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



the point O, passes through O. This translation is repeated many times, i.e. the ball oscillates around the

equilibrium position O.

This phenomenon is analyzed as following: when the ball is pulled to an ordinate x, forces exert into itconsist of the pulling force F, the elastic force Fof the spring, the gravity force and the reacting force ofthe 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 horizontal

translation 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 Fis always proportional with the displacement

x of the ball from the equilibrium position (is also the deflection of the spring), and directs toward thepoint O. Since Fis along the coordinate axis, it can be written as:

F = - kx (1-1)

Here k is the spring constant(stiffness) of the spring, and the minus sign indicates that the force F isacting in opposite direction compared with the deflection x of the ball.

According to Newtons second law, it can be written as F = ma, or ma = - kx. Thus a = xm

k .

It is known that a velocity and acceleration are defined by v =t

x

and a =t

v

. If the motion is

investigated in a very small period of time t, thentx

becomes a derivative of x with respect to time

variable, v = x! ; similarly,t

v

becomes a derivative of x respecting to time t, a = v! , i.e. a second order

derivative of x with respect to time variable: a = x!! .

Therefore we have x!! = xm

k (1-2)

Let =m

kthen x!! + 2x = 0 (1-2a)

It can be proved having a solution of x = Asin(t+) (1-3)

where A and are constants and =m

k.

Really , taking derivative of the displacement x (1-3) with respect to the time variable we have the velocity of the

ball : v = x! = Acos(t + ) (1-4)

Taking derivative of the velocity v (1-4) with respect to the time, we get the acceleration of the ball:

a = x!! = -2Asin(t + ) (1-5)

Replacing the value of x into (1-5) we get: x!! = - 2x (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 theoscillated ball is x = Asin(t +).

Since sine function is a periodic function, it is said that the oscillation of the ball (i.e. the oscillation ofthe 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(t+) = Asin(t++/2)

Therefore, it can be defined that a SHM is an oscillation that can be described by a sinusoidal (or

cosinusoidal) function, where A, , are constants.

In the equation (1-3), x is the displacementof the oscillation, showings precisely the deflection of theball from the equilibrium position. A is the amplitudeof the oscillation. It is the maximum value of



displacement, occurred when sin(t+) has the maximum value of 1. The meanings of , and t+will be clarified at 2.

It is known that sine function is a periodic function with the period of 2. Thus, it can be written as

x = Asin(t +) = Asin(t ++2), or x = Asin[(t +2

) + ]

It means that the displacement of the ball at time (t +2

) have the same value at time t. The period of

time T =2

is called the cycleof the SHM. The reciprocal of T, f =T

1=

2

is called the frequencyof

the SHM.

Particularly, for the mass-spring system, we have

T =2

= 2k

m(1-4)

Now the system is taken out from the rod and hung up vertically (figure 1.1c). If the ball is pulled down

then released, it will oscillate in the vertical direction. That is also a mass-spring system. Everything havebeen said about a horizontally oscillated spring system can be applied to a vertically oscillated spring

system as well. In this case, the equilibrium position is no longer the point O that corresponds with thetime the spring was not deflected, but is a point O that corresponds to the time the spring was deflecteddue to the gravity of the ball.

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 ofharmonic oscillation?

4. Give more example about oscillation and harmonic oscillation?

2.EXPLORING 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 (measured inrad/s). A point C in the circle is chosen to be an origin. At the initialtime t = 0, the position of the moving point is M0, is specified by an

angle of . At an arbitrary time, the position of the moving point is Mtspecified by an angle of (t + ).

We project the path of point M onto an axis xx pass through point O

and perpendicular to OC. At time t, the projection of point M onto xxaxis is point P which has the ordinate of x = OP. Since OP is the

projection of OMtonto the xx axis so we have:

x = OMtsin(t +)

x = Asin(t +) (1.8)

(1.8) has the same format as (1.3) so we can conclude that the motion of point P on the xx axis is aSHM. 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.



2.Angular phase and angular frequency of a simple harmonic motion

From figure 1.2, the angular (t +) 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 specifies the position of P at the initialtime t = 0, and is called the initial phase (or initial angular phase) of the oscillation. The angular velocity

allows us to determine f =

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

is called the angular frequency(circular frequency) of the oscillation. Here , and (t +) arespecified angles and can be measured directly.

In equation (1-3) for the mass-spring system, the quantities , and (t+) 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

Lets 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(t +) satisfied, we must have sin(t +) = 1, and since t =0 so = /2.

Therefore, the oscillation equation of the ball is x = Asin(t +/2) (1-9)

So we have determined the amplitude, initial phase and the cycle of the oscillation. The amplitude and

initial 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 springconstant, not on other factors. If the initial conditions are changed then the amplitude A and initial phase

will be changed as well but , T are constant.

An oscillation the period of which depends only on the systems characteristics (here is a mass-spring

system), and not on other stimulating factors, is called afree oscillation. A system that can implement afree oscillation by itself is called a self-oscillation system. After being stimulated, a self-oscillationsystem 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

As we know from 1,

v = x! = Acos(t +/2) = Asin(t +) (1-10)

a = v! = x!! = - 2Asin(t +/2) = 2Asin(t -/2) (1-11)

In figure 1.3, there are curves presenting functions (1-9), (1-10) and (1-11). It is observed that after each

cycle T =2

, the values of displacement, velocity and acceleration are in same values as before, and the

behaviors of curves are also unchanged. The phase of the oscillation (t +) not only determines theposition of the oscillating thing but also allow to specify the value and behavior of the velocity and

acceleration. The phase of the oscillation determine the state of the oscillation. Similarly, the initialphase specifies the initial state of the oscillation.

When the ball is in the harmonic oscillation, its velocity and acceleration fluctuate following a sinusoidalor cosinusoidal function, i.e. theyfluctuate harmonicallywith the same frequency of the ball.



5.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 its

size 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 P

corresponding to a deflection angle of . We only investigate the case in which is small enough to havethe arc "OP coincide with the chord OP and sincan be approximated as (in radian). If 10othen theerror is not greater than 6/1000.

Thus we have: sin=l

s(1-12)

Now, the ball is released and it swings itself. The force acting on the ball include of the gravity Ft= mg,the tend force Tof the string. The force Ftis resolved into 2 components: F in the direction of the stringand Fis perpendicular to the string. The component F balances the tend force T, hence the ball does not

move in the string direction. The direction of component Fis tangential to the arc

OP , but since isvery small it can be considered lying along the chord OP and direct to point O.

From Newtons 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 aand Fare in OPaxis and the direction of Fis opposite with the ordinate s = OP so

ma = F = -Ft sin= - mg= mg l

s

, or

a = -l

gs; and s!!= -

l

gs (1-14)

Equation (1.14) has the same format as (1.2). Herel

gplays the role of

m

k, and s plays the role of x.

Therefore (1.14) has the same meaning with (1.2). We can applied the analysis process in 1 and 2 as

well then it can be concluded that the motion of a single pendulum is a harmonic oscillation with angular

frequency =l

g.



The time constant of the pendulum is:

T =2

= 2l

g (1-15)

For small oscillation, i.e. with 10o, the cycle of a simple pendulum is not dependent on the oscillationamplitude. 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 Thas 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 Tis 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 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 = 4cos4t. 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/s

2. 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.ENERGY 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 hasa maximum magnitude.

When the force is not exerted, the spring compressed, theelastic force directs the ball toward point O. Its velocity isincreasing, 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, its

velocity 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 is

decreasing.

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 kinetic

energy is increasing while the potential energy is decreasing.



During 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

2mv

2.

Replacing v by its expression from (1-10): v = Acos(t +/2), we have :

Ed =1

2m2A2cos2(t +

2

) (1-16)

It was proved that the potential energy of a ball is equal to the work done of the elastic force in order to

bring it from the position x to the equilibrium: Et =1

2kx

2

Replacing x by its expression from (1-9): x = Asin(t +/2) and replace k by m2, we get

Et =1

2

m2A2sin2(t +

2

) (1-17)

(1-16) and (1-17) are the expressions of kinetic and potential energy of the ball at an arbitrary time t, andthe total energy of the ball at this time is

E = Et+ Ed

E =1

2m2A2[sin2(t +

2

) + cos

2(t +

2

)]

E =1

2m2A2= constant (1-18)

The total energy of oscillation is conserved. During the oscillation process, the total energy is unchanged

and proportional with the square of amplitude.

There is a transformation between potential and kinetic energy. Based on (1-18), we rewrite (1-16) and

(1-17) in new forms:

Ed= Ecos2(t +

2

) (1-19a)

Et= Esin2(t +

2

) (1-19b)

Note that the kinetic energy of a single pendulum is dependent on the initial excitation. If it is excited bya powerful force to make a bigger displacement then the amplitude is bigger hence the total energy is also

bigger. Certainly, we just can increase the amplitude to a limited value within elastic limitation of the

spring.

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 while

the amplitude is reduced 2 times?



4.- 5. THE 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 isa combination of many different oscillations. When the hammock is hung on a ship, it will swing with its

own frequency. However, the ship is also oscillated due to wave. Finally, the oscillation of the hammockis 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, two

identical mass-spring systems are hung next to each other, they have the same angular frequency of .The balls are pulled to displacements x1= A1and x2= A2respectively. At the time t = 0, ball 1 is releasedto 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 that

the 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

x1= A1sin(t + 1)

x2= A2sin(t + 2)

As we have known (from 2), the oscillation equation of ball 1 is:

x1= A1sin(t + /2) (1-20)

For ball 2, at the time t =4

Tthen x2= A2, thus:

2 2 2sin( )4

TA A

= + ; 2sin( ) 1

4+ =

T ; 2

4 2+ =

T ; 2

2. 0

2 4 2 4= = =

T T

T

The oscillation equation of ball 2 is

x2= A2sint (1-21)

In general, the phase difference between two oscillations that have the same frequency is:

(t + 1) - (t + 2) = 1- 2

The phase difference is a constant quantity and equal to the difference between the initial phases. It is

called the phase differenceand these two oscillation are called phase-different oscillations. When= 1- 2> 0, it is said that oscillation 1 is a lead in phase to oscillation 2 or oscillation 2 is a lag in

phase to oscillation1. When = 1- 2< 0, it is said in a contrast way.

In this example, it is said that ball 1 leads ball2 by an angle of2

or ball 2 lags ball 1 by an angle of

2

(note that these angles only appear in calculations but are not real angles that can measured by angular

ruler).

The phase difference is a characteristic quantity for the discrepancy between two oscillations that have

the same frequency. If the phase difference is zero, or generally is 2nthen they are in phase. If it is or(2n+1)then they are out of phase (n is an arbitrary integer, n = 0; 1; 2; 3; ...)



3.Vector-diagram method

In order to combine two harmonic oscillations with the same directions, frequencies but differentamplitudes and phases, it is usually used a very convenient method called Fresnels vector diagram. Thismethod is based on a property having been discussed in 2: a simple harmonic oscillation can be treatedas 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(t+) need to berepresented. A horizontal axis () and a vertical axis xx thatintersects () at point O are built (figure 1.6). A vector A whoseorigin is at point O, magnitude is proportional with amplitude A and

makes with axis () an angle of initial phase . At the time t = 0,vector A(its head is M0) is rotated in positive direction

(conventionally is counter-clockwise) with angular velocity of .When the head M of vector Ais projected on to xx axis then themotion of the projection P on xx is a harmonic oscillation. At anytime t, the head of Ais M, its projection on xx is P and we have:

x = OP = Asin(t + )

That is the simple harmonic oscillation necessary to express. It is said that a simple harmonic oscillation

x = Asin(t + ) 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 in

two oscillations of the same directions and the same frequency , but they have different amplitudes A1,A2 as well as initial phases 1 , 2

x1= A1sin(t + 1) (1-22)

x2= A2sin(t + 2) (1-23)

The resultant motion is the combination of two components (1-22) and (1-23). The Fresnels vectordiagram method will be applied to find the equation of resultant motion.

Two axes () and xx are drawn as in figure 1.7. Draw a vector,namely A1, whose magnitude is proportional with amplitude A1makes an angle of 1with () (figure 1.7). Similarly, drawvector A2whose magnitude is proportional with amplitude A2makes an angle of 2with (). Draw vector Awhich is theresultant vector of A1and A2, this vector makes an angle of with ().

In figure 1.7, the angle between A1and A2is (2- 1) (the phasedifference of two components x1and x2). Since 1 and 2areconstant then (2- 1)is also constant.

Now rotate A1and A2around point O in positive direction withthe same frequency of . Then a trapezoid OM1MM2is notdeformed since both sides OM1, OM2and the angle

"2 1M OM are unchanged. Therefore Ahas a constant

magnitude and rotates around O in positive direction with angular velocity of A1and 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 xx is the combination of P1(projection ofM1) and P2(projection of M2) on xx 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 (figure 1.7).

Similarly, if it is necessary to combine various oscillations x 1, x2, x3 it is recommended to draw theresultant vector Aof A1, A2, A3...



Figure 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(t +) (1-24)

where A is proportional with the magnitude of amplitude vector A.

It is necessary to evaluate A and in (1-24). For triangle OMM2in figure 1.7, we have:

"2 2 2

2 2 2 2 2

OM OM M M 2OM .M M cos OM M= + , or

A2= A1

2+ A1

2 2A1A2cos" 2OM M

Since " 2OM M and"

2M OM are supplementary angles thus:

cos" 2OM M = - cos"

2M OM = - cos(2 - 1) (1-25)

From figure1.7 we can deduce that

tg=MP '

OP '=

OP

OP '= 1 1 2 2

1 1 2 2

A sin A sin

A cos A cos

+ +

(1-26)

Finally, the total oscillation is a harmonic oscillation described by (1-24), has the same frequency withconstituent oscillation, has amplitude specified by (1-25) and initial phase determined by (1-26)

From (1-25), the amplitude of total oscillation is dependent on the phase difference ( 2 - 1) ofconstituent oscillations.

If the constituencies are in phase (2 - 1 = 2n) then cos(2 - 1) = 1, the resultant amplitude hasmaximum value of A = A1 + A2.

If the constituencies are out of phase (2 - 1 = (2n+1)) then cos(2 - 1) = -1, the resultant amplitudehas minimum value of A = |A1 - A2|.

If the phase difference is arbitrary then the resultant amplitude satisfies |A1- A2| < A < A1+A2.

Questions

1.What is the phase difference?

2.How are in phase oscillations, out of phase oscillation, leading phase oscillation, lagging phaseoscillation?

3.From figure 1.3 in 2, compared with the oscillation of a mass-spring system, are the velocity and

acceleration of ball lagging or leading and how much are they?

4.Briefly state the Fresnels vector-diagram method?

5.Two harmonic oscillations have the same direction and the same frequency f = 50Hz, and have the

amplitudes A1 = 2a, A2= a and the initial phases 1 =3

, 2= .

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) = /2.



6.- 7. UNDERDAMPED AND FORCED OSCILLATIONS

1.Underdamped oscillation

In research of harmonic oscillation of mass-spring system, simple pendulum and other things, it isobserved that their frequencies and amplitudes are time-independent quantities. It means that they will

repeats 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, thereforewhen 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 thisoscillation is examined in a short time, the damping is negligible and it can be seenas a harmonic oscillation.

Let a pendulum fluctuates in a container of water (figure 1.8). The friction of wateris strong enough so it will be damped fairly quickly and it will stop at theequilibrium 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) thenreturns and stops there. (figure 1.9b)

If the friction of lubricating oil is much stronger, theball even can not passes through the equilibriumposition and stops immediately (figure 1.9c)

In the real life and technology, in some cases the

underdamping is harmful and it is required toovercome this phenomenon (i.e. for clock pendulums).

In contrast, in some cases it is useful and needed sopeople make use of that. For example, we all knowthat 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 compressedor stretched. After that, the frame continues vibratinglike a spring, makes travelers tired and uncomfortable.

In order to make it damped faster, vehicles areequipped with a special equipment. It consist of a

piston that can travel in a vertical cylinder containlubricating oil, piston is assembled to the frame andthe cylinder is mounted to a shaft of wheel. When the

frame vibrates on buffer springs, the piston is alsofluctuated inside the cylinder. The lubricating oil

make the vibration damped faster and so the vibrationof the frames does.

2.Forced oscillation

In order to make an oscillation not be damped, the simplest way is exerting on it an externally periodicforce. This force supplies energy to the whole system to compensate the frictional losses.

It is known that a mass-spring system and simple pendulum are free oscillations. If there is no friction,they will oscillate ever and ever with their own frequencies. However, it is ideal. In fact, externalenvironment exert on the ball a strong or light frictional force Fms , make the vibration damped (see the

diagram in figure 1.9).

An externally periodic force is applied to the ball called the forced force:



Fn= Hsin(t +)

where H is the amplitude and is the frequency of the forced force. Generally, the frequency

of the external excitation f =2

is different from that of the free oscillation f0of 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, freevibrations are ended, the ball oscillates due to the external excitation. Its frequency is the frequency of the

external force and the amplitude is dependent on a relationship between the externally excited frequencyf and free frequency f0. That is why a vibration after along time is a forced oscillation. If the excitation is

maintained 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 duringt.

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 alight 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 onto a thin calibrated metal rod. The frequency f is determined correspondingto 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 isallowed to swing in the plane that is perpendicular to the plane of paper. Thefrequency f of pendulum B is transferred to pendulum A as a forced

excitation by the spring. This force makes pendulum A vibrating, and after atime, 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 isobserved that the vibration of pendulum A reaches the maximum value when

f f0, but when f is smaller or greater than f0then the amplitude of pendulumA 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 issmaller 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 increase

significantly. 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 for

people.

A child can use a small force to swing an adults 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 hammocks frequency, a child can swing it to higher level, i.e. hammocksamplitude 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 andthey have their free frequencies. They can be a bridge, frame, shaftDue to some reasons, they vibrate



resonantly with the other (e.g. a big electric generator), and they will vibrate dramatically and can be

broken, 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 coincidedaccidental 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.

5.Self-oscillation

There is another way to maintain oscillation, keep it not be damped and there is no external excitation. Asimple example is the pendulum clock.

The pendulum of the clock swings freely with its specified cycle T. Due to the friction with the air and atthe shaft, its fluctuation will be damped if it is not compensated the losses.

When the clock is wound up, it is accumulated a certain amount of potential energy. The spring is relatedwith the pendulum by a system of cog-wheel and proper mechanism. When the pendulum reaches to themaximum displacement, after half of cycle, the spring is stretched a little bit and a part of this potentialenergy is transferred to the pendulum through agent mechanism. This amount of energy is sufficient tocompensate the frictional losses. Therefore, the pendulum can swing for a long time with the same

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, suchas 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 unchanged

from 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 1fT

= is the

number 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(t+) or x = Acos(t+). The displacement x is the deflection from an equilibrium position.The amplitude is the maximum displacement. The angular frequency is a quantity to specify the

frequency2

f

= and the cycle

2T

= of the harmonic oscillation. The phase (t+) specifies the

state of oscillation at any time of time t. The initial phase specifies the initial state, i.e. at the time t = 0.



A harmonic oscillation can be regarded as the projection of a uniform circle motion in the projectile

plane. 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 = 2k

mdepending only on the characteristics of the

pendulum. It is called free oscillation and its cycle is called free cycle. The simple pendulum has a cycle

of T = 2l

gdepending on the gravity acceleration. When the simple pendulum is placed in a specific

location (g is constant), it can be seen its fluctuation as free oscillation.

When the pendulum oscillates harmonically, the velocity and acceleration of the ball behave follow asine or cosine law, i.e. they behave with the same frequency of the ball.

During the oscillating process, there is a transformation from kinetic energy to potential energy and vice

versa, but the mechanical energy is constant and proportional to the square of amplitude of the ball.

3. The phase difference of two oscillation is the difference of initial phase and is call the phase difference

= 1- 2. Two oscillations are in phase if = 2n, are out of phase if = (2n+1).

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 of

A=| 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 f0is not damped, it is

applied 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.



Chapter II MECHANICAL WAVE. ACOUSTICS

8.WAVE IN MECHANICS

1.Natural mechanical waves

When we throw a rock into a still water surface, we can observe a number of circular water waves

spreading out to every direction from the place where the stone is thrown. If we drop a cork or a leaf

down 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 the

nearby 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 direction

will 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 thatspread out with time in a material medium.

Note that in mechanical waves only the oscillation states, i.e. the phases of the oscillation, is spreading

away, while the mediums 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 gas

form. For example, if dropping some grains of sandinto 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 atransverse waves. Thereexists 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 in

details in this chapter.

2.Oscillation phase transmission. Wavelength.

A stone thrown into a water surface can create only a fewsmall waves, the oscillation will soon die out. To make

better research in mechanical waves, a small equipment iscreated 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 slightlytouches the surface of a large water tray (figure 2.1). Thenwe just need to flip on the right end of the metal piece to

make the ball harmonically vibrate with period T. Then allwater molecules contacting with the ball will also vibrate

with period T in a relatively long time. On the watersurface, a number of circular waves will start to spread inall directions.

We can simplified by imagining viewing the water tray through a vertical projection through P. The cuttrace will have the form displayed in figure 2.2. It is the truly form of the water waves in different instant

of time.

Suppose at t = 0 the waves have the form displayed in figure 2.2a. We can see that points A, E and I

vibrate in phase: they all go through their equilibrium positions and move downwards. Points C and Gare out of phase to points A, E and I: they also go through their equilibrium positions but moving



upwards. 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 betweentwo successive in-phase points along the direction of wave transmission is called the wavelength,denoted by !(the Greek letter lambda). In general, those points the distance between which is a multipleof 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 wavelengthwill 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 sameperiod, which equals to the period T of the wave source. This period is called the periodof wave. Thereciprocal of the period, f = 1/T, is called the frequencyof wave.

In the above example, we can see that after each period T the oscillation phase travels through a distanceequal to the wavelength. Thus, we can also say that: the wavelength is the distance the wave travels in aperiod 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 , we have the following relation:

= v T (2-1)

or =v

f(2-2)

4.Amplitude and energy of waves

Once the wave reached a point, it makes the medium elements at that point oscillate with a particularamplitude. This amplitude is called the wave amplitudeat 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

transmission is also the process of transferring energy.



For waves that originating from one point and spreading out in a surface, the wave energy is also

stretched 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. For

waves that originating from one point and spreading out in a space, the wave energy reduces proportionalto 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 reduced

along 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. SOUND 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

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