大学物理双语版奥本汉姆课件Chap12-Oscillation
大二物理上-课件-第12章-波动
x 2π x
u
λ
y(x,t) y(x,t T ) （波具有时间的周期性）
12– 1
第十二章 平面简谐波
波线上各点的简谐运动图
12– 1
第十二章 平面简谐波
y
A c os [ (t
x) u
0]
A c os [2 π
(t T
x
)
0
]
2） 当 t 一定时，波函数表示该时刻波线上各
u
x
点 O 振动方程 yO Acos(t 0 )
波 函 数
y
A c os [ (t
-
x) u
0]
u 沿x 轴正向
y
A c os [ (t
x) u
0]
u 沿x 轴负向
12– 1
第十二章 平面简谐波
二 波函数的物理意义
y
A c os [ (t
x) u
0]
A c os [2 π
(t T
x
)
0
]
1） 当 x 固定时， 波函数表示该点的简谐运
一 机械波的形成
机械波：机械振动在弹性介质中的传播.
产生条件：1）波源；2）弹性介质.
注意
媒质中任一质元在平衡位置附近振动，不“随 波
逐流”（如河中足球） 即：波传播的是振动，位相，能量，而非质元
波是运动状态的传播，介质的质点并不随波传播.
12– 1
第十二章 平面简谐波
二 横波与纵波
横波：质点振动方向与波的传播方向相垂直的波. （仅在固体中传播 ）
(t
-
6) 20
- π ]cm 3
0.10cos[4π t - 23π ]cm
大连海事大学大学物理(双语)课程教学大纲
7 Magnetic medium 磁介质 8 Magnetic fields acting on currents 磁场对电流
environment • develop a positive attitude toward science
１
• cultivate students' appreciation of the scientific endeavour and their potential to contribute to it
Classroom practice and teaching strategies promotes positive attitudes toward science in all students, including those who are typically under-represented in enrolments and careers in science. All students have equal opportunities to acquire the positive attitudes that foster the development of scientific literacy.
4. COURSE SYLLABUS
No.
Content
1 Kinematics of a point 质点运动学 2 Dynamics of a point 质点动力学 3 Rotation of rigid bodies 刚体转动 4 Electro-static field 静电场
5 Conductors and dielectrics in Electro-static field 静电场中的导体 电介质
大学物理双语课衍射课件
a=2f
λ
a
λ
x0
415 ×109 = 2 × 2.50 × = 2.25 ×105 m = 0.023mm 9.2 ×102
Example
1 a′ = a 2
焦平面上原来3 级暗纹处， 焦平面上原来 级暗纹处，现在明暗情况如何
a sin θ = 3λ
1 3 a sin θ = λ = (2m′ + 1) 1 λ 2 2 2
A
a
R
B
slit
a sin θ = 2
A
A 1
θ
C
L
P
Q
λ
2
o
P
B
λ/2
Hale Waihona Puke AbBR
A
A1
A2
θ L
C
λ/2
Q
a sin θ = 3
λ
2
o
B
2 .The positions on which all rays travel with angle θ
a sin θ = N
Dark fringe
a sin θ = ±2m
S
S
Large obstacle
Small obstacle
2. 惠更斯——菲涅耳原理 惠更斯—— ——菲涅耳原理
惠更斯原理
在波的传播过程中， 在波的传播过程中，波前上的每一点都可看作是发射 子波（次波）的波源，在其后的任一时刻， 子波（次波）的波源，在其后的任一时刻，这些子波的包 迹就成为新的波阵面。 迹就成为新的波阵面。
（sin φ＋1 a ）
φ
a
A
D
C
θ
Principles of Physics--14oscillatory 2
§3. The superposition of simple harmonic motion 1. The superposition of two simple harmonic motions along a straight line with same frequency.
x1 = A cos(ωt +ϕ10 ) 1 x2 = A2 cos(ωt +ϕ20 )
x 2 y 2 2 xy + 2− cos(ϕ 2 − ϕ1 ) = sin 2 (ϕ2 − ϕ1 ) A12 A2 A1 A2
∆ϕ =ϕ2-ϕ1 ∆ϕ = 0 y x ∆ϕ = 5π/4 ∆ϕ = 3π/2 ∆ϕ = 7π/4 π π π ∆ϕ = π/4
P
·
∆ϕ = π/2 ∆ϕ = 3π/4 π
x = Acos(ωt +ϕ0 )
◆ phase:
t
•
ω
ωt +ϕ0
x
ωt
O
( At = 0)
ϕ0
X
= the angle between A and the positive direction of x axis at the time t ◆ displacement: x(t) = the projection of
(2) ∆ϕ = π : straight-line equation
A2
y
x y =− A1 A2
(3) ∆ϕ = π/2 : elliptic equation
− A1
− A2
A1
x
x2 y 2 + 2 =1 2 A1 A2
(4) general condition: elliptic equation
Chapter 15 oscillations(2)
m I / R2 T 2π k
2 T ？
RE
g
R π T 2 ？ g
地球隧道中物体运动的周期
§15.1 Simple Harmonic Motion (SHM) 1. Simple Harmonic Motion (SHM):
An motion in which the net force is directly proportional to the negative of the displacement. Such system is called simple harmonic oscillator (SHO)
Ø
超前或落后
若 = 2- 1>0, 则 x2比x1较早达到正最大, 称x2比x1超前 (或x1比x2落后)。 超前和落后以 < 的相位角来判断 x
x1
o
x2
t
思考： 简谐振动的位移、速度、加速度谁超前？
位移
x(t)=Acos( t+)
速度 v A sin(t ) A cos( t ) 2 加速度 a 2 A cos(t )
Displacement, velocity, acceleration function
A o -A Aω o -Aω Aω2 o -Aω2 a v x
= /2
T t
x ( t ) A cos t
v v A m sin t
= T t T
2
x (t), v (t), a (t) and also F (t) are all periodic function of time
F kA cos t
大学物理教案英文版简谐振动Simple harmonic oscillation
+A +A
+ ωA − ωA
+ω2 A +ω2 A
φ=0
d 2 x( t ) a( t ) = dt 2 = − Aω 2 cos(ωt + φ )
x = + xm v = 0 a = −am x = 0 v = −v m a = 0 x = − xm v = 0 a = +am
§7.2 Simple harmonic oscillation
r A
ϕ
ω
r A1
ω
x
x 1 = A1 cos( ω t + ϕ 1 )
x 2 = A2 cos( ω t + ϕ 2 )
ϕ2
ϕ1
x = x1 + x 2
x2
x1
x
r r r A = A1 + A 2
= A cos( ω t + ϕ )
A=
2 A12 + A2 + 2 A1 A2 cos(ϕ 2 − ϕ 1 ) A1 sinϕ1 + A2 sinϕ 2 ϕ = arctg A1 cosϕ1 + A2 cosϕ 2
§7.4 simple harmonic motion and the uniform circular motion
ωA
A
x ( t ) = A cos(ωt + φ )
v ( t ) = − Aω sin(ωt + φ )
9
§7.4 simple harmonic motion and the uniform circular motion
1
§7.2 Simple harmonic oscillation
《物理双语教学课件》Chapter-9-Oscillations-振动
Chapter 9 OscillationsWe are surrounded by oscillations─motions that repeat themselves. (1). There are swinging chandeliers, boats bobbing at anchor, and the surging pistons in the engines of cars. (2). There are oscillating guitar strings, drums, bells, diaphragms in telephones and speaker systems, and quartz crystals in wristwatches. (3). Less evident are the oscillations of the air molecules that transmit the sensation of sound, the oscillations of the atoms in a solid that convey the sensation of temperature, and the oscillations of the electrons in the antennas of radio and TV transmitters.Oscillations are not confined to material objects such as violin strings and electrons. Light, radio waves, x-rays, and gamma rays are also oscillatory phenomena. You will study such oscillations in later chapters and will be helped greatly there by analogy with the mechanical oscillations that are about to study here.Oscillations in the real world are usually damped; that is, the motion dies out gradually, transferring mechanical energy to thermal energy by the action of frictional force. Although we cannot totally eliminate such loss of mechanical energy, we can replenish the energy from some source.9.1 Simple Harmonic Motion1. The figure shows a sequenceof “snapshots” of a simpleoscillating system, a particlemoving repeatedly back andforth about the origin of the xaxis.2. Frequency: (1). One importantproperty of oscillatory motionis its frequency , or number ofoscillations that arecompleted each second . (2). The symbol for frequency is f, and (3) its SI unit is hertz (abbreviated Hz), where 1 hertz = 1 Hz = 1 oscillation per second = 1 s -1.3. Period: Related to the frequency is the period T of the motion, which is the time for one complete oscillation (or cycle). That is f T 1=.4. Any motion that repeats itself at regular intervals is called period motion or harmonic motion . We are interested here in motion that repeats itself in a particular way. It turns out that for such motion the displacement x of the particle from the origin is given as a function of time by )cos()(φω+=t x t x m , inwhich φωand x m ,, are constant. The motion is called simple harmonic motion (SHM), the term that means that the periodic motion is a sinusoidal of time .5. The quantity m x , a positive constant whose value depends onhow the motion was started, is called the amplitude of the motion; the subscript m stands for maximum displacement of the particle in either direction.6. The time-varying quantity )(φω+t is called the phase of the motion, and the constant φ is called the phase constant (or phase angle ). The value of φ depends on the displacement and velocity of the particle at t=0.7. It remains to interpret the constant ω. The displacement )(t x must return to its initial value after one period T of the motion. That is, )(t x must equal)(T t x + for all t. To simplify our analysis, we put 0=φ. So we then have)](cos[cos T t x t x m m +=ωω. The cosine function first repeats itself when its argument (the phase) has increased by π2 rad, so that we must haveπωπωω22)(=+=+T or t T t . It means f T ππω22==. The quantity ω is called the angular frequency of the motion; its SI unit is the radian per second.8. The velocity of SHM: (1). Take derivative of the displacement with time, we can find an expression for thevelocity of the particle moving with simple harmonic motion. That is, )2/cos()sin()()(πφωφωω++=+-==t v t x dtt dx t v m m . (2). The positive quantity m m x v ω= inabove equation is called thevelocity amplitude .9. The acceleration of SHM:Knowing the velocity for simpleharmonic motion, we can find anexpression for the acceleration ofthe oscillation particle by differentiating once more. Thus we have)cos()2/sin()()(πφωπφωω++=++-==t a t v dtt dv t a m m The positive quantity m m m x v a 2ωω== is called the accelerationamplitude . We can also to get )()(2t x t a ω-=, which is thehallmark of simple harmonic motion: the acceleration is proportional to the displacement but opposite in sign, and the two quantities are related by the square of the angular frequency .9.2 The Force Law For SHM1. Once we know how the acceleration of a particle varies with time, we can use Newton’s second law to learn what forc emust act on the particle to give it that acceleration. For simple harmonic motion, we have kxω. This result-a=-(2=)xmmaF-=force proportional to the displacement but opposite in sign-is something like Hook’s law for a spring, the spring constant here being 2ωmk=.2.We can in fact take above equation as an alternative definition of simple harmonic motion. It says: Simple harmonic motion is the motion executed by a particle of mass m subject to a force that is proportional to the displacement of the particle but opposite in sign.3.The block-spring system forms a linear simple harmonic oscillator(linearoscillator for short),where linear indicatesthat F is proportional to x rather than to some other power of x. (1). The angular frequencyωof the simple harmonic motion of the block is mω. (2). The period of the linear=k/oscillator is k=.2πmT/9.3Energy in Simple Harmonic Motion1.The potential energy of a linear oscillator depends on howmuch the spring is stretched or compressed, that is, on )(t x.We have )(cos 2121)(222φω+==t kx kx t U m . 2. The kinetic energy of the system depends on haw fast the block is moving, that is on)(t v . We have )(sin 21)(sin )(21)(sin 2121)(22222222φωφωφωω+=+=+==t kx t x m k m t x m mv t K m m m 3. The mechanical energy is2222221)(sin 21)(cos 21m m m kx t kx t kx K U E =+++=+=φωφω The mechanical energy of a linear oscillator is indeed a constant, independent of time.9.4 An Angular simple Harmonic Oscillator1. The figure shows an angular version of a simple harmonic oscillator; the element of springinessor elasticity is associated with thetwisting of a suspension wire ratherthan the extension and compressionof a spring as we previously had. Thedevice is called a torsion pendulum ,with torsion referring to the twisting.2. If we rotate the disk in the figure from its rest position and release it, it will oscillate about that position in angular simple harmonic motion. Rotating the disk through an angleθ in either direction introduce a restoring torque given byθκτ-=. Here κ (Greek kappa) is a constant, called the torsion constant , that depends on the length, diameter, and material of the suspension wire.3. From the parallelism between angular quantities and linear quantities (give a little more explanation), we have κπIT 2=for the period of the angular simple harmonic oscillator, or torsion pendulum.9.5 PendulumWe turn now to a class of simple harmonic oscillators in which the springiness is associated with the gravitational force rather than with the elastic properties of a twisted wire or a compressed or stretched spring.1. The Simple Pendulum(1). We consider a simplependulum, which consists ofa particle of mass m (calledthe bob of the pendulum)suspended from an un-stretchable, massless string of length L , as in the figure. The bob is free to swing back and forth in the plane of the page, to the left and right of a vertical linethrough the point at which the upper end of the string is fixed.(2). The forces acting the particle, shown in figure (b), are its weight and the tension in the string. The restoring force is the tangent component of the weight θsin mg , which is always acts opposite the displacement of the particle so as to bring the particle back toward its central location, the equilibrium (0=θ). We write the restoring force as θsin mg F -=, where the minus sign indicates that F acts opposite the displacement.(3). If we assume that the angle is small , the θsin is very nearly equal to θ in radians, and the displacement s of the particle measured along its arc is equal to θL . Thus, we have s L mg L s mg mg F )(-=-=-≈θ. Thus if a simple pendulum swings through a small angle, it is a linear oscillator like the block-spring oscillator.(4). Now the amplitude of the motion is measure as the angular amplitude m θ, the maximum angle of swing. Theperiod of a simple pendulum is g L L mg m k m T /2)//(2/2πππ===. Thisresult hods only if the angularamplitude m θ is small .2. The Physical Pendulum(1). The figure shows a generalizedphysical pendulum, as we shall call realistic pendulum , with its weight g m ρ acting at the center of mass C.(2). When the pendulum is displaced through an angle θ in either direction from its equilibrium position, a restoring torque appears. This torque acts about an axis through the suspension point O in the figure and has the magnitude ))(sin (h mg θτ-=. The minus sign indicates that the torque is a restoring torque, which always acts to reduce the angle θ to zero.(3). We once more decide to limit our interest to small amplitude , so thatθθ≈sin . Then the torque becomes θτ)(mgh -=.(4). Thus the period of a physical pendulum ismgh I T /2π=, when m θ is small. Here I is the rotational inertia of thependulum.(5). Corresponding to any physical pendulum that oscillates about a given suspension point O with period T is a simple pendulum of length L 0 with the same period T . The point along the physical pendulum at distance L 0 from point O is called the center of oscillation of the physical pendulum for the given suspension point.3. Measuring g: We can use a physical pendulum to measurethe free-fall acceleration g through measuring the period of the pendulum.9.6 Simple Harmonic Motion and Uniform circular Motion 1.Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs.2.The figure (a) gives anexample. It shows areference particle P’moving in uniformcircular motion withangular speed ωin areference circle. Theradiusx of the circle ismthe magnitude of theparticle’s position vector.At any time t, theangular position of theparticle is φω+t.3.The projection of particle P’ onto the x axis is a point P. The projection of the position vector of particle P’ onto the x axisgives the location )(t x of P . Thus we find )cos()(φω+=t x t x m . Thus if reference particle P’ moves in uniform circular motion, its projection particle P moves in simple harmonic motion.4. The figure (b) shows the velocity of the reference particle. The magnitude of the velocity ism x ω, and its projection on the x axis is )sin()(φωω+-=t x t v m . The minus sign appears because the velocity component of P points to the left, in the direction of decreasing x .5. The figure (c) shows the acceleration of the reference particle. The magnitude of the acceleration vector ism x 2ω and itsprojection on the x axis is )cos()(2φωω+-=t x t a m . 6. Thus whether we look at the displacement, the velocity, or the acceleration, the projection of uniform circular motion is indeed simple harmonic motion .9.7 Damped Simple Harmonic MotionA pendulum will swing hardly at allunder water, because the water exerts adrag force on the pendulum that quicklyeliminates the motion. A pendulum swinging in air does better, but still themotion dies out because the air exerts a drag force on the pendulum, transferring energy from the pendulum’s motion.1. When the motion of an oscillator is reduced by an external force, the oscillator and its motion are said to be damped . An idealized example of a damped oscillator is shown in the figure: a block with mass m oscillates on a spring with spring constant k. From the mass, a rod extends to a vane (both assumed massless) that is submerged in a liquid. As the vane moves up and down, the liquid exerts an inhibiting drag force on it and thus on the entire oscillating system. With time, the mechanical energy of the block-spring system decreases, as energy is transferred to thermal energy of the liquid and vane.2. Let us assume that the liquid exerts a damped force d F ρthat isproportional in magnitude to the velocity v ρ of the vane and block. Then bv F d -=, where b is a damped constant that depends on the characteristics of both the vane and the liquid and has the SI unit of kilogram per second. The minus sign indicates that d F ρ opposes the motion.3. The total force acting on the block is ∑=-=--=dt dx b kx bv kx F . So we have equation 022=++kx dt dx b dt x d m , whose solution is )'cos()(2/φω+=-t e x t x m bt m , where 'ω, the angular frequency of thedamped oscillator, is given by 224'm b m k -=ω.4. We can regard the displacement of the damped oscillator as a cosine function whose amplitude, which ism bt m e x 2/-, gradually decreases with time.5. The mechanical energy of a damped oscillator is not constant but decreases with time. If the damping is small , we can find )(t E by replacing m x with m bt m e x 2/-, the amplitude of thedamped oscillation. Doing so, we findm bt m e kx t E /221)(-≈, which tells us that the mechanical energy decreases exponentially with time .9.8 Forced Oscillations and Resonance1. A person swing passivelyin a swing is an example offree oscillation. If a kindfriend pulls or pushes theswing periodically, as in thefigure, we have forced, ordriven, oscillations . There are now two angular frequencies with which to deal with: (1) the natural angular frequency ω of the system, which is the angular frequency at which it would oscillate if it were suddenly disturbed and then left tooscillate freely, and (2) the angular frequencyd ω of theexternal driving force.2. We can use the right figure torepresent an idealized forcedsimple harmonic oscillator if weallow the structure marked “rigidsupport” to move up and down ata variable angular frequency d ω. A forced oscillator oscillates atthe angular frequencyd ω of driving force, and its displacement is given by )cos()(φω+=t x t x d m , where m xis the amplitude of the oscillations. How large the displacement amplitudem x is depends on a complicated function of d ω and ω.3. The velocity amplitude m v of the oscillations is easier todescribe: it is greatest whenωω=d , a condition calledresonance . Above equationis also approximately thecondition at which thedisplacement amplitude m xof oscillations is greatest.The figure shows how the displacement amplitude of an oscillator depends on the angular frequencyof thed driving force, for three values of the damped coefficient b. 4.All mechanical structures have one or more naturalfrequencies, and if a structure is subjected to a strong external driving force that matches one of these frequencies, the resulting oscillations of structure may rupture it. Thus, for example, aircraft designers must make sure that none of the natural frequencies at which a wing can vibrate matches the angular frequency of the engines at cruising speed.。
大学双语物理ch12-ch13 Ex.
The Mechanical Energy of SHM
1 2 E U K kxm 2
Damped SHM
x(t ) x m e
t
cos( ' t )
Summary
Superposition of SHM x = x1+ x2 = Acos( t+ )
Solution:
v0 0
58
v0 0
0
o
v0 0
o
A
0
0
x0 0.43
x
0 58x
58
A
v0 0
Ch12-23 (p.317) A mass m is connected to two springs, with spring constants k1 and k2, in two different ways as shown in Fig.12-31a and b. Show that the period for the configuration shown in part (a) is given by T 2 m k1 k1 m And for that in part (b) is given by T 2 k k
58P(H.p397) A wheel is free to rotate about its axel. A spring is attached to one of its spokes a distance r from the axel, as shown in the figure. Assuming that the wheel is a hoop of mass m ,radius R, and inertia (转动惯量) I=mR2,find the frequency of a small oscillations of this system, in terms of m, R, r and the spring constant k. Solution: when the wheel rotates an angle q from the equilibrium position, the spring force is
大学物理英文版
教 学基本 要 求
1. 理解描述质点运动物理量的定义及其矢量性、相 对性和瞬时性； 2. 掌握运动方程的物理意义，会用微积分方法求解 运动学两类问题； 3. 掌握平面抛体运动和圆周运动的规律； 4. 理解运动描述的相对性，会用速度合成定理和加 速度合成定理解题。
重要历史人物
伽利略Galileo Galilei: 1564~1642意大利物理学 家、数学家、天文学家， 近代实验科学的创始人。
大学物理（英文版） 多媒体课件
Volume 1 Introduction Chapter 1 Kinematics Chapter 2 Newton’s Laws of Motion Chapter 3 Work and Energy
Chapter 4 Momentum Chapter 5 Rotation of a rigid body Chapter 6 The Kinetic Theory of Gases Chapter 7 Fundamentals of Thermodynamics
Introduction
2001.9.11 Catastrophe（大灾难）
宇宙：约1250亿个星系， 每个星系由数千亿个恒星 组成。
银河系
太阳系：地球，星星 看得见的：你我他它 分子
物 质
原子
世
原子核
界
基本粒子
相对论 天体物理 经典物理:力学，热等 量子力学 核物理 量子场论
银河系
量子天体 物理学
When we discuss the position and the velocity（速度） of an object ,we must answer the questions:
“position with respect to（相对于） what?” and
大学物理高斯定理课件英文版
E S dS
q
0
写成 E q
0 S dS
e.高斯面部分的通量为0。
E dS, cos 0
Chapter 20 Gauss’s Law
第八章静电场
spherical symmetry
1.Spherical shell of uniform charge
计算步骤: a.场对称性分析。
b.选取高斯面。
c.确定面内电荷代数和 q。
d.应用定理列方程求解。
S EdS
cos
q
0
第八章静电场
Chapter 20 Gauss’s Law
高斯面选取的原则
第八章静电场
a.要求电场具有高度对称性。 b.高斯面要经过所研究的场点。 c.高斯面应选取规则形状。 d.面上各点的场强大小相等，方向与高斯面法线方向一致。
algebraic sum of charges enclosed within the
surface.
Chapter 20 Gauss’s Law
1 n
Φe
E dS
S
0
qi
i 1
Discussion
第八章静电场
1) The relation of the source of electric filed and the field (反映场和源 的关系).
第八章静电场
Karl Friedrich Gauss (1777-1835), German mathematician and physicist. He made a lot of contributions in the fields of experimental physics, theoretical physics and mathematics. He made major contributions to the theory of electromagnetism.
《物理双语教学课件》Chapte...
《物理双语教学课件》Chapte...Chapter 11 The First Law of Thermodynamics11.1Thermodynamics1.With this chapter, we begin a new subject-thermodynamics,which deals with an internal energy of systems, thermal energy, and is governed by a new set of laws.2.The central concept of thermodynamics is temperature. Thisword is so familiar that most of us, because of our built-in sense of hot and cold, tend to be overconfident in our understanding of it.3.Our “temperature sense” is in fact not always reliable. On acolder winter day, for example, an iron railing seems much colder to the touch than a wooden fence post, yet both are at the same temperature. This difference in our sense perception comes about because iron removes energy from our fingers more quickly than wood does.4.Temperature is one of the seven SI base quantities. Physicistsmeasure temperature on the Kelvin scale, which is marked in units called kelvins. Although the temperature of a body apparently has no limit, it does have a lower limit; this limiting low temperature is taken as the zero of the Kelvin temperature scale. Room temperature is about 290 kelvins, or290 K as we write it, abovethis absolute zero . The rightfigure shows the wide rangeover which temperatures aredetermined.11.2 Temperature1. The zeroth law ofthermodynamics : If bodies A and B are each in thermal equilibrium with a third body T, then they are in thermal equilibrium with each other, and have the same temperature .2. By international agreement, the triple point of water , in which liquid water, solid ice, and water vapor (gaseous water) coexist, has been assigned a value of 273.16 K as the standard fixed-point temperature for the calibration of thermometers.3. Celsius temperatures are measured in degrees, and the Celsius degree has the same size as the kelvin. However, the zero of the Celsius scale is shifted to a more convenient value than absolute zero. IfC T represents a Celsius temperature, then 015.273-=T T C .In expressing temperature on the Celsius scale, the degree symbol is commonly used. Thus we write C 000.20 for a Celsius reading about 293.15 K for a Kelvinreading.4. The Fahrenheit scale , used in the United States, employsa smaller degree than the Celsius scale and a different zero of temperature. The relation between the Celsius and Fahrenheit scales is,32590+=C F T T where F T is Fahrenheit temperature.11.3 The Absorption of Heat by Solids and Liquids1. Heat capacity C of an object is the proportionality constantbetween an amount of heat and the change in temperature that this heat produces in the object. Thus)(i f T T C Q -=, in which i T and f T are the initial and final temperature of theobject. Heat capacity C has the unit of energy per degree or energy per kelvin .2. Two objects made of the same material will have heatcapacities proportional to their masses. It is therefore convenient to define a “heat capacity per unit mass” or specific heat c that refers not to an object but to a unit mass of the material of which the object is made. The above equation then becomes )(i f T T cm Q -=.3. Molar specific heat : In many instances the most convenientunit for specifying the amount of a substance is the mole (mol),where units y elementarr mol 231002.61?= of anysubstance. When quantities are expressed in moles, the specific heat must also involve moles; it is then called a molarspecific heat.11.4 A Closer Look at Heat and Work1. Let us take as our system a gasconfined to a cylinder with amovable piston, as shown inthe figure. The system startsfrom an initial state i ,described by a pressurei p , a volume i V , and a temperaturei T . You want to change the system to a final state f , describedby a pressure f p , a volume f V , and a temperature f T . We assume that all changes occur slowly, with the result that the system is always in (approximate) thermal equilibrium .2. Suppose that you remove a few lead shot from the piston of the Fig., The differential work done by the gas during the displacement is pdV ds pA s d F dW ==?=))(( , in which dV is the differential change in the volume of the gas owing to the movement of the piston.3. If the gas change its volume fromi V to f V , the total workdone by the gas is ??==fi V V pdV dW W .4. There are actually manyways to take the gasfrom state I to state f, asin the figure. So asystem can be takenfrom a given initial stateto a given final state byan infinite number ofprocesses. Heat may ormay not involved, and ingeneral, the workW and heat Q will havedifferent value for different process . So we say that heat andwork are path-dependent quantities .5. The figure (f) shows a thermodynamics cycle in which the system is taken from some initial state i to some other state f ，and then back to i.11.5 The first law of thermodynamics1. You have just seen that when a system changes from a given initial state to a given final state, both work W done and theheat Q exchanged depend on the nature of the process. Experimentally, however, we find a surprising thing. The quantity )(W Q - is the same for all process. It depends only on the initial and final states and does not depend at all on how the system gets from one to the other.2. The quantity W Q - must represent a change in some intrinsic property of the system. We call this property the internal energy int E and we write W Q E E E i f -=-=?int,int,int . This is the first law of thermodynamics . If the thermodynamics system undergoes only a differential change, we can write the first law as dW dQ dE -=int .3. Adiabatic process : An adiabatic process is the process that occurs so rapidly or occurs in a system that is so well insulated that no transfer of heat occurs between the system and its environment. Putting0=Q in the first law then lead to W E -=?int . This tells us that if the work done by the system, the internal energy of the system decreases by the amount of work. Conversely, if work is done on the system, the internal energy of the system increases by that amount .4. Constant-volume process : If the volume of a system is held constant, that system can do no work. Putting0=W in the first law yieldQ E =?int . Thus if heat is added to a system, theinternal energy of the system increases. Conversely, if heat removed during the process, the internal energy of the system must decrease .5. Cyclical process . There are processes in which, after certain interchanges of heat and work, the system is restores to its initial state. In that case, no intrinsic property of the system can possibly change. Putting0int =?E in the first law yield W Q =. Thus the net work done during the process must exactly equal the net amount of heat transferred .6. Free expansion . See the following figure. These are adiabatic processes in which nowork is done on or by thesystem. Thus 0==W Qand the first law requiresthat 0int =?E . A freeexpansion differs from all other processes we have considered because it cannot be done slowly and in a controlled way. As a result, at any given instant during the sudden expansion, the gas is not in thermal equilibrium and its pressure is not the same everywhere. So although we can plot the initial and final states on a p-V diagram, we cannot plot the expansion itself.。
大学物理(双语)
y
Multiplying a Vector by a Vector
r v θ
(1) The Scalar (Dot) Product:
vy
x
o
vx
r r r v is resolved to v x and v y , v = v i + v r j
v x = vcosθ and v y = vsinθ
r A
r B
r C'
r B
r C' B r
r
A
The difference between two vectors is the vector that extends from the head of the 2nd vector to the head of the 1st vector.
r r r r A = Ax i + A y j + Az k
r r r i , j , k —— unitary magnitude, only indicate directions
In the coordinate system, For a fixed Cartesian system, Unit vectors:
r r r r r C ′ = A B = A + ( B)
Chapter 1. Tooling up
1-1 Measuring Things (P1-)
In 1971, the 14 General Conference on Weights and Measures, The SI (International System) was adopted. It is based on meter, second and kilogram. It is also called metric system or mks system. Many SI derived units are defined in terms of these base units, such as: 1 watt = 1 W = 1 kg m2 / s3
- 1、下载文档前请自行甄别文档内容的完整性，平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益，请联系客服反馈,我们会尽快为您处理(人工客服工作时间：9:00-18:30)。
or oscillation2 is behind of oscillation1
The concept of phase：
(1) Describing the states of motion; (2) Comparing the difference in steps of two oscillations (比较两个振动在步调上的差异).
P'
the position vector of point
P’ moves in SHM along a
xm
diameter of circle, at t is :
P
0
作匀x速转x动m c矢os量(xrt
0 ).
m,其端点P′
在x轴上的投影点P 的运动是
SHM(规定为逆时针)
Relation between the two ways of expressions:
Hey! Different phase corresponding to different state.
iv. Phase difference (相位差)
Phase difference of two SHM with same frequency:
x1 A1 cos(t 1)
x
Given expression A、T、j Given A、T、j expression
2. Curve method
x A
o -A
j = p /2
Given curve A、T、j T t Given A、T、j curve
3. Uniform circular motion (旋转矢量法) (P306)
12-2 Concepts of Simple Harmonic Motion (SHM)
The simple harmonic
合振动
motion is the most simple x and basic vibration.
分振动1
Every complicated vibration can be
between the limits ±vm=± xm. v(t) is ahead of x(t) for p/2.
5. The acceleration of SHM
a(t) dv(t )
dt
a(t ) 2 xm cos(t ).
The acceleration of the oscillating particle is also
Let t = 0, = 0, then
xmax A x Acos t vmax A v Asin t amax 2 A a 2 Acos t
12-3 Expression Methods of SHM
1. Analytical method
from x=Acos( t+j )
Curve method; t+j x
A
x
Uniform Circular
Motion (Rotating
o
Vector) method.
Hey!
You may familiar to the typical values for the above special cases.
Example:
Initial State 1:
o
v=0
xA
0 0
Initial State 2:
v
Hale Waihona Puke x=00p 2
Initial State 3:
x
V=0
0 ?
x A
O
x
Initial State 4:
v
F=0，x=0
0
3p 2
或 p 2
Example:
A particle is in SHM along x axis, A=0.12m, T=2s. When t = 0, xo= 0.06m, and v > 0 (moves along positive x direction). Try to find out: (1) The expression of this SHM; (2) t = T/4, x=? v=? and a=? (3) At what time will the particle pass the “O” first time (何时物体第一次通过平衡位置）?
1. Features of SHM
Periodic motion is regular, and repeats with a characteristic (特征）time.
As the “snapshots” (快照）of a SHS, a particle moving repeatedly back and forth about the origin of an x axis.
Where frequency (f ), or number of oscillations that are completed each second. Its SI unit is the hertz (Hz)
1 hertz = 1 Hz = 1 oscillation per second = 1 s-1
k m
T
2p
2p
m k
T and f is also called intrinsic period or frequency (固有周期和固有频率) ensure by system property
iii. Phase or phase angle & initial ~(相位和初相)
constants. They
m
are basic characteristic quantities of a SHM (描述简谐振动 的特征量) .
i. Amplitude (振幅) xm or A (A > 0):
t 0,
x0 Acos0 0 Asin0
—– Initial condition of oscillation
Simple harmonic motion is the projection of uniform circular motion on a diameter of the circle in which the latter motion occurs.
The projection in x axis of
SHM varies between the limits ±am = ±2 xm.
v is ahead of x p/2 and a is ahead of x p . The curve of v(t) is shifted (to the left) from the curve of x(t) by T/4 period. The curve of a(t) is always opposite to the direction of x(t) (反位相). Phase difference is p.
2. Motion function of SHM (P299)
The small-amplitude vibration (小幅振动) for the spring,
Object: block-spring system; Origin: equilibrium position
From Hooke’s Law: F kx
o
T
T
2
considered as composing
of several SHM (简谐振动是一
种最简单最基本的振动,一切复杂的 振动都可以看作是若干简谐振动的 合成的结果).
分振动2
3T
2T
t
2
Any motion that repeats itself at regular intervals is called periodic motion or harmonic motion.
For a certain oscillatory object, A and are
fixed. At any time, its motion state ( x & v) will
be determined by (t +o). So (t +o) is called phase. (o is the phase at t = 0 —– initial phase)
12-1 Oscillations (P297)
Oscillations (振动) —–– motions that repeat themselves. ——— Fundamental physical phenomena and basic constituent of wave motion.
3T
2T
t
2
= (2k+1)p (k=0,1, 2,…),
they have opposite (out of)
phase.
x1
对两同频率的简谐振动，
x2
相位差等于初相差.
0 indicates:
t 2 1
Oscillation2 is ahead of oscillation1
Dynamics: f kx or a 2 x
Kinematics: Motion function is sine or cosine form.
3. Basic quantities of SHM (P301)
For the motion function of SHM: