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Chapter 1. Vectors and Scalars

Units（International , dimensions), Scalars，Vectors, Adding vectors, parallelogram of vectors, multiplying vectors, components, equilibrium 重点：Addition，Subtraction and Production of Vector 难点：scalar product,vector product

Chapter

2

Motion and Momentum

law of inertia )

I.Fundamental Laws of Motion(Newton's Law of Motion)

a) Newton's first Law of Motion (the

An object at rest will remain at rest unless acted on by an unbalanced force. An object in motion continues in motion with the same speed and in the same direction unless acted upon by an unbalanced force.

b) Newton's Second Law of Motion

Acceleration is produced when a force acts on a mass. The greater the mass (of the object being accelerated) the greater the amount of force needed (to accelerate the object).

or FORCE = MASS times ACCELERATION

Everyone unconsciously knows the Second Law. Everyone knows that heavier objects require more force to move the same distance as lighter objects.

c) Newton's third Law of Motion

For every action there is an equal and opposite re-action. This means that for every force there is a reaction force that is equal in size, but opposite in direction. That is to say that whenever an object pushes another object it gets pushed back in the opposite direction equally hard. ii) Drawing Free-Body Diagrams Free-body diagrams are diagrams used to show the relative magnitude and direction of all forces acting upon an object in a given situation. The only rule for drawing free-body diagrams is to depict all the forces that exist for that object in the given situation. Thus, to construct free-body diagrams, it is extremely important to know the various types of forces.

iii) Types of Forces A force is a push or pull acting upon an object as a result of its interaction with another object. Contact Forces Frictional Force Tension Force Normal Force Air Resistance Force Applied Force Spring Force Action-at-a-Distance Forces Gravitational Force Electrical Force Magnetic Force

II. Momentum

a) Momentum can be defined as "mass in motion." Momentum = mass ? velocity is a vector quantity

Attention: Linear momentum is dependent on the frame of reference:It is important to note that the an object can have momentum for one frame of reference but the same object if kept in another reference frame can have zero momentum.

b) Conservation of momentum

The total momentum in a closed or isolated system remains constant. It states that the momentum of a system is constant if there are no external forces acting on the system. e.g. a collision Suppose we have two interacting particles 1 and 2, possibly of different masses. The forces between them are equal and opposite. According to Newton's second law, force is the time rate of change of the momentum, so we conclude that the rate of change of momentum P1 of particle 1 is equal to minus the rate of change of momentum P2 of a particle 2, (1) Now, if the rate of change is always equal and opposite, it follows that the total change in the momentum of particle 1 is equal and opposite of the total change in the momentum of particle 2. That means that if we sum the two momenta the result is zero, ( 2 ) But the statement that the rate of change of this sum is zero is equivalent to stating that the quantity P 1?P 2 is a constant. This sum is called the total momentum of a system, and in general it is the sum of all individuals momenta of each particle in the system. Attention: 1) It holds true for any component. 2) An alternative of this is the law of conservation of angular momentum.

3) If no external force acts on a system in a particular direction then the total momentum of the system in the direction remains unchanged.

c)Impulse-momentum theorem

1)The change of momentum is also termed as impulse. Impulse = Change in Momentum 2) When large force acts for a very short time,a force comes in to play. This ? is called Impulse. It is a vector quantity denoted by I .

? ? dI ? Fdt ? t2 ? ∴ I ? ? Fdt

t1

If the force does not vary with time, then

The unit of Impulse are gcms-1

? ? I ? F?t or kgms-1.

3) What is the relationship between impulse and momentum? It is simple - and extremely powerful. The impulse-momentum equation says: The change in momentum of a particle during a time interval equals the impulse of the net force that acts on the particle during that interval.

III. The Work-Energy Theorem a) Mechanical Energy: Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). E.g. A moving car possesses mechanical energy due to its motion (kinetic energy). A moving baseball possesses mechanical energy due to both its high speed (kinetic energy) and its vertical position above the ground (gravitational potential energy).

When net work is done upon an object by an external force, the total mechanical energy (KE + PE) of that object is changed. Attention: It is vital to distinguish between external forces and internal forces. external forces include the applied force, normal force, tension force, friction force, and air resistance force. Because external forces are capable of changing the total mechanical energy of an object, they are sometimes referred to as nonconservative forces. internal forces include the gravity forces, magnetic force, electrical force, and spring force. When only forces doing work are internal forces, energy changes forms - from kinetic to potential (or vice versa); yet the total amount of mechanical is conserved. Because internal forces are capable of changing the form of energy without changing the total amount of mechanical energy, they are sometimes referred to as conservative forces. b) Deriving Kinetic Energy

Kinetic energy is closely linked with the concept of work, which is the scalar product (or dot product) of force and the displacement vector over which the force is applied.

An examination of how the theorem was generated gives us a greater understanding of the concepts underlying the equation. Because a complete derivation requires calculus, we shall derive the theorem in the one-dimensional case with a constant force. Proof: From Newton’s Second Law of motion, we know that F = ma, and because of the definition of acceleration we can say that If we multiply both sides by the same thing, we haven’t changed anything, so we multiply by v: But remember that v = dx/dt:

We rearrange and integrate:

F dx = mv dv Fx = m(?v2) = ?mv2 = Ek

But Fx = Work; therefore Work = Δ Ek. It is powerfully simple, and gives us a direct relation between net work and kinetic energy. Stated verbally, the equations says that net work done by forces on a particle is equal to the change in kinetic energy of the particle. IV. Conservation of Energy Energy can be defined as the capacity for doing work. It may exist in a variety of forms and may be transformed from one type of energy to another. However, these energy transformations are constrained by a fundamental principle, the Conservation of Energy principle. 1) One way to state this principle is "Energy can neither be created nor destroyed". 2) Another approach is to say that the total energy of an isolated system remains constant.

V.Gravity 1. Newton's Law of Gravity Newton's law of gravity defines the attractive force between all objects that possess mass. Understanding the law of gravity, one of the fundamental forces of physics, offers profound insights into the way our universe functions. - Gravitational Forces In the Principia, Newton defined the force of gravity in the following way (translated from the Latin): Every particle of matter in the universe attracts every other particle with a force that is directly proportional to the product of the masses of the particles and inversely proportional to the square of the distance between

them. Mathematically, this translates into the force equation shown to the equation mm Fg ? G 1 2 2 , where the quantities are defined as: r ? Fg= The force of gravity (typically in newtons) ? G= The gravitational constant, which adds the proper level of proportionality to the equation. The value of G is 6.67259 x 10-11 N * m2 / kg2, although the value will change if other units are being used. ? m1& m1 = The masses of the two particles (typically in kilograms) ? r= The straight-line distance between the two particles (typically in meters) 2. Gravitational Fields (P34:textbook) Sir Isaac Newton's law of universal gravitation (i.e. the law of gravity) can be restated into the form of a gravitational field, which can prove to be a useful means of looking at the situation. Instead of calculating the forces between two objects every time, we instead say that an object with mass creates a gravitational field around it. The gravitational field is defined as the force of gravity at a given point divided by the mass of an object at that point, as depicted to the right.

3. Gravitational Potential Energy Gravitational Potential Energy on Earth On the Earth, since we know the quantities involved, the gravitational potential energy U can be reduced to an equation in terms of the mass m of an object, the acceleration of gravity (g = 9.8 m/s), and the distance y above the coordinate origin (generally the ground in a gravity problem). This simplified equations yields a gravitational potential energy of: U = mgy There are some other details of applying gravity on the Earth, but this is the relevant fact with regards to gravitational potential energy. Notice that if r gets bigger (an object goes higher), the gravitational potential energy increases (or becomes less negative). If the object moves lower, it gets closer to the Earth, so the gravitational potential energy decreases (becomes more negative). At an infinite difference, the gravitational potential energy goes to zero. In general, we really only care

about the difference in the potential energy when an object moves in the gravitational field, so this negative value isn't a concern. This formula is applied in energy calculations within a gravitational field. As a form of energy, gravitational potential energy is subject to the law of conservation of energy.

Chapter 4 Dynamics of Rigid Bodies

Part I

Center of the mass

i) concept of center of mass

--point mass --Calculating center of mass

ii) law of center of mass

? ? Fnet ? Macm

Ask students to derive the equation, based on the newton’s second law. iii) application of center of mass A fisherman stands at the back of a perfectly symmetrical boat of length L. The boat is at rest in the middle of a perfectly still and peaceful lake, and the fisherman has a mass 1/4 that of the boat. If the fisherman walks to the front of the boat, by how much is the boat displaced? If you’ve ever tried to walk from one end of a small boat to the ot her, you may have noticed that the boat moves backward as you move forward. That’s because there are no external forces acting on the system, so the system as a whole experiences no net force. If we recall the equation , the center of mass of the system cannot move if there is no net force acting on the system. The fisherman can move, the boat can move, but the system as a whole must maintain the same center of mass. Thus, as the fisherman moves forward, the boat must move backward to compensate for his movement.

Because the boat is symmetrical, we know that the center of mass of the boat is at its geometrical center, at x = L/2. Bearing this in mind, we can calculate the center of mass of the system containing the fisherman and the boat:

Now let’s calculate where the center of mass of the fisherman -boat system is relative to the boat after the fisherman has moved to the front. We know that the center of mass of the fisherman-boat system hasn’t moved relative to the water, so its displacement with respect to the boat represents how much the boat has been displaced with respect to the water. In the figure below, the center of mass of the boat is marked by a dot, while the center of mass of the fisherman-boat system is marked by an x. At the front end of the boat, the fisherman is now at position L, so the center of mass of the fisherman-boat system relative to the boat is

The center of mass of the system is now 3 /5 from the back of the boat. But we know the center of mass hasn’t moved, which means the boat has moved backward a distance of 1/5L, so that the point 3/ 5Lis now located where the point 2 /5 L was before the fisherman began to move. Part II angular momentum& Torque

Review: terms for circle parameter 1)Angular displacement: measure in radius(why?) 2)Angular velocity 3)Period 4) Centripetal acceleration/force

i) angular momentum (vs linear momentum)

-rigid body -centripetal acceleration/force an idealization of a solid body in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. -angular momentum of a material object Example: http://www.lightandmatter.com/html_books/lm/ch15/ch15.html#Section15.1 The angular momentum of a moving particle is L= mv⊥r , (explain why?) where m is its mass, v⊥ is the component of its velocity vector perpendicular to the line joining it to the axis of rotation, and r is its distance from the axis. Positive and negative signs are used to describe opposite directions of rotation. The angular momentum of a finite-sized object or a system of many objects is found by dividing it up into many small parts, applying the equation to each part, and adding to find the total amount of angular momentum. Note that r is not necessarily the radius of a circle. (As implied by the qualifiers, matter isn't the only thing that can have angular momentum. Light can also have angular momentum, and the above equation would not apply to light.) ii) Torque and Work Energy Theorem http://oyc.yale.edu/physics/phys-200/lecture-9 -momentum of inertia

I? 1 N ? mi vi2 2 i ?1

Note that it is dependent on the point of rotation -Torque(Derive from the previous kinetic energy) represented by the Greek letter tau, ? , the rate of transfer of angular momentum iii)Conservation of angular momentum Conservation of angular momentum has been verified over and over again by experiment, and is now believed to be one of the three most fundamental principles of physics, along with conservation of energy and momentum. Example: skater pulls in her arms so that she can execute a spin more rapidly

When a figure skater is twirling, there is very little friction between her and the ice, so she is essentially a closed system, and her angular momentum is conserved. If she pulls her arms in, she is decreasing r for all the atoms in her arms. It would violate conservation of angular momentum if she then continued rotating at the same speed, i.e., taking the same amount of time for each revolution, because her arms' contributions to her angular momentum would have decreased, and no other part of her would have increased its angular momentum. This is impossible because it would violate conservation of angular momentum. If her total angular momentum is to remain constant, the decrease in r for her arms must be compensated for by an overall increase in her rate of rotation. That is, by pulling her arms in, she substantially reduces the time for each rotation.

Chapter 5 Special Relativity

Review: nertial frame/non-inertial frame

I. Galilean transformation

1. Galilean transformations, also called Newtonian transformations, set of equations in classical physics that relate the space and time coordinates of two systems moving at a constant velocity relative to each other. Adequate to describe phenomena at speeds much smaller than the speed of light, Galilean transformations formally express the ideas that space and time are absolute; that length, time, and mass are independent of the relative motion of the observer; and that the speed of light depends upon the relative motion of the observer.

1) Derivation

Suppose there are two reference frames (systems) designated by S (un-primed frame) and S' (primed frame) such that the co-ordinate axes are parallel (as in figure 1). In S, we have the co-ordinates {x, y, z, t} and in S' we have the co-ordinates {x' , y' , z ' , t '} . S' is moving with respect to S with velocity (as measured in S) in the x direction. The clocks in both systems were synchronised at time and they run at the same rate.

We have the intuitive relationships

This set of equations is known as the Galilean Transformation. They enable us to relate a measurement in one inertial reference frame to another. For example, suppose we measure the velocity of a vehicle moving in the -direction in system S, and we want to know what would be the velocity of the vehicle in S'.

This is the result our intuition is familiar with.

(2) Explanation

We have stated the we would like the laws of physics to be the same in all inertial reference frames, as this is indeed our experience of nature. Physically, we should be able to perform the same experiments in different reference frames, and find always the same physical laws. Mathematically, these laws are expressed by equations. So, we should be able to ``transform'' our equations from one inertial reference frame to the other inertial reference frame, and always find the same answer.

Suppose we wanted to check that Newton's Second Law is the same in two different reference frames. (We know from experiment that this is the case.) We put one observer in the un-primed frame, and the other in the primed frame, moving with velocity relative to the un-primed frame. Consider the vehicle of the previous case undergoing a constant acceleration in the -direction,

Indeed, it does not matter which inertial frame we observe from, we recover the same Second Law of Motion each time. In the parlance of physics, we say the Second Law of Motion is invariant under the Galilean Transformation. 2. The Medium of Light It was thought in the 1800's that the frame-independence of the speed of light predicted by Maxwell's equations should be interpreted as the speed with reference to the ether, a mythical medium supporting the propagation of electro-magnetic waves. The ether would have to have the properties that material objects would pass through it with negligible friction. Many experiments were stimulated to measure the velocity of the earth through the ether. Figure 3 demonstrates schematically what one might expect for measuring the speed of light , which would be constant w.r.t. the ether, in an earth-bound laboratory moving with velocity w.r.t. the ether. From earth, the ether would appear to be a ``wind'', so we may use the velocity addition formula we are used to for an aircraft traveling in a wind. What is the minimum accuracy for measuring the speed of light in order to detect the motion of the earth through the ether, assuming the Galilean Transformation.

Michelson-Morley experiment (Detecting the Aether Wind) http://galileoandeinstein.physics.virginia.edu/more_stuff/flashlets/mmexpt6 .htm The most famous experiment designed to detect changes in the speed of light is now known as the Michelson-Morley experiment, performed in 1881. In this experiment, a Michelson Interferometer is used to produce an interference pattern from two beams which recombine at the detector after having been separated and sent on perpendicular paths by a half-silvered mirror. Assuming Galilean Relativity, interference fringes would pass the detector reflecting the changing optical path difference as the device was rotated through 90 . In this way the two perpendicular arms of the interferometer would experience the ether flowing past in different but correlated directions, leading to different optical path lengths in each arm.

To quantify this statement, we will calculate the time difference for the light beams to travel in Arm 1 and Arm 2 of the apparatus, once with Arm 2 parallel to the motion of the earth, and once with Arm 1 parallel to the motion of the earth.

Arm 2 parallel to the motion of the earth The time difference between light traveling in Arm 1 and Arm 2 is To quantify this statement, we will calculate the time difference for the light beams to travel in Arm 1 and Arm 2 of the apparatus, once with Arm 2 parallel to the motion of the earth, and once with Arm 1 parallel to the motion of the earth. Arm 2 parallel to the motion of the earth The time difference between light traveling in Arm 1 and Arm 2 is

Arm 1 parallel to the motion of the earth After rotating the apparatus by 90 , the time difference between light traveling in Arm 1 and Arm 2 is

The change in time difference between the un-rotated and rotated configurations is

This can be simplified using the binomial expansion. If x ? 1 , then Since c / ? ?? 1 , we find .

and

Therefore

sing 500 nm light, an effective arm size in the Michelson interferometer of 11 m, the speed of light equal to m/s and the speed of the earth around the sun km/s, we expect an optical path difference for the two arms of This would be an observable fringe shift of

No such fringe shift was observed. It is now accepted that the ether concept, as well as the Galilean Transformation are wrong.

II. Special theory of relativity

i) The Two Postulates of Relativity In 1905, at the age of only 26, Einstein published his special theory of relativity. Regarding his theory, he wrote : The relativity theory arose from necessity, from serious deep contradictions in the old theory from which there seemed no escape. Einstein based the Special Theory of Relativity on two postulates: ? The Principle of Relativity: The laws of physics must be the same in all inertial reference frames. ? The constancy of the speed of light:The speed of light in vacuum has the same value, 3 ?108 m/s , in all inertial reference frames. ii)Length Contraction and Time Dilation

iii) Deriving the Lorentz Transformation Lorentz transformations, set of equations in relativity physics that relate the space and time coordinates of two systems moving at a constant velocity relative to each other. Required to describe high-speed phenomena approaching the speed of light, Lorentz transformations formally express the relativity concepts that space and time are not absolute; that length, time, and mass depend on the relative motion of the observer; and that the speed of light in a vacuum is

constant and independent of the motion of the observer or the source. The equations were developed by the Dutch physicist Hendrik Antoon Lorentz in 1904. These co-ordinate transformations are known as the Lorentz Transformation. Note that the Lorentz Transformation reduces to the Galilean Transformation when ? ?? 1 so that as required.

Chapter 5 Oscillations and Simple Harmonic Motion

We have already studied the most common types of motion: linear and rotational motion. We have developed the concepts of work, energy, and momentum for these types of motion. In this chapter,we start to examine the the case of oscillations.

Part I Basic Concepts

1. An oscillating system -a system in which a particle or set of particles moves back and forth. (e.g.a ball bouncing on a floor, a pendulum swinging back and forth, or a spring compressing and stretching.) http://www.animations.physics.unsw.edu.au/mechanics/chapter4_simpleharmoni cmotion.html -In an oscillating system, the force always acts in a direction opposite to the displacement of the particle from the equilibrium point.(restoring force,equilibrium point） 2. Variables of Oscillation A simple oscillator generally goes back and forth between two extreme points; the points of maximum displacement from the equilibrium point. We shall denote this point by x m and define it as the amplitude of the oscillation. (E.g. If a pendulum is displaced 1 cm from equilibrium and then allowed to oscillate we can say that the amplitude of oscillation is 1 cm.)

-Amplitude

-Period In simple oscillations, a particle completes a round trip in a certain period of time. This time, T , which denotes the time it takes for an oscillating particle to return to its initial position, is called the period of oscillation. -Frequency, denoted by ν, is defined as the number of cycles per unit time and is related to period as such:

ν = 1/T Period, of course, is measured in seconds, while frequency is measured in Hertz (or Hz), where 1 Hz = 1 cycle/second.

-Angular frequency Angular frequency defines the number of radians per second in an oscillating system, 2? ?? T

Part II simple harmonic motion

Of all the different types of oscillating systems, the simplest, mathematically speaking, is that of harmonic oscillations. For now, however, we simply define simple harmonic motion, and describe the force involved in such oscillation. To develop the idea of a harmonic oscillator we will use the most common example of harmonic oscillation: a mass on a spring. For a given spring with constant k, the spring always puts a force on the mass to return it to the equilibrium position. Recall also that the magnitude of this force is always given by: F(x) = - kx where the equilibrium point is denoted by x = 0 . In other words, the more the spring is stretched or compressed, the harder the spring pushes to return the block to its equilibrium position. This equation is only valid if there are no other forces acting on the block. If there is friction between the block and the ground, or air resistance, the motion is not simple harmonic, and the force on the block cannot be described by the above equation. Though the spring is the most common example of simple harmonic motion, a pendulum can be approximated by simple harmonic motion, and the torsional oscillator obeys simple harmonic motion. 1. Deriving the Equation for Simple Harmonic Motion From our concept of a simple harmonic oscillator we can derive rules for the motion of such a system. We start with our basic force formula, F = - kx . Using Newton's Second Law, we can substitute for force in terms of acceleration: ma= - kx Here we have a direct relation between position and acceleration. For you calculus types, the above equation is a differential equation, and can be solved quite easily. Note: The following derivation is not important for a non-calculus based course, but allows us to fully describe the motion of a simple harmonic oscillator.

Rearranging our equation in terms of derivatives, we see that: d 2x m 2 ? ? kx dt Let us interpret this equation. The second derivative of a function of x plus the function itself (times a constant) is equal to zero. Thus the second derivative of our function must have the same form as the function itself. What readily comes to mind is the sine and cosine function. Let us come up with a trial solution to our differential equation, and see if it works. As a tentative solution, we write: x ? Ac o ? s ?t ? ?0 ? k It is clear that, if ? 2 ? ,then the equation is satisfied. Thus the equation m governing simple harmonic oscillation is: Simple x ? Ac o ? s ?t ? ?0 ? Where ? is the initial phase. 2. Rotating vector diagram http://www.physics.uoguelph.ca/tutorials/shm/phase0.html

Simple harmonic motion can be visualized as a projection of uniform circular motion http://www.animations.physics.unsw.edu.au//jw/SHM.htm

3. The superposition of simple harmonic motions 1)in phase and out phase Let us consider two oscillators with simple harmonic motions vibrating with the same angular frequency ? , and their initial phases are ? 1 and ?2 respectively, x1 ? A1 c o s ? (t ? ?1 ) x2 ? A2 c o s ? (t ? ?2 ) Where A1 and A2 are their amplitudes, the phase difference between these two vibrations is: ?? ? ?2 ? ?1

When , they are “in phase” ?? ? ?2 ? ?1 ? ?2k? , k ? 0,1,2,3... ,they are “out of phase”. ?? ? ?2 ? ?1 ? ?(2k ? 1)? , k ? 0,1,2,3... 2)two vibrations in the same direction with same frequency Say one consider one particle undergoes two simple harmonic motions in the dame direction with the same frequency simultaneously. x1 ? A1 cos(?t ? ?1 ) x2 ? A2 c o s ? (t ? ?2 ) As we know, the resultant displacement obeys the law of superposition: x ? x1 ? x2 Ask students to derive resultant displacement via the rotating vector diagram.

Chapter 6

Part I Basic Concepts

Wave Motion

1) Wave A wave is defined as the transfer of energy from one point to another.

Classification of Waves

(i) There are two large, all encompassing categories of waves: mechanical and non-mechanical. -Mechanical waves are waves that require a medium for the transfer of their energy to occur. -Non-mechanical waves are waves that do not require a medium for the transfer of their energy to occur. Electromagnetic waves are the only type of non-mechanical waves. They can travel through the vacuum of space. Light from distant stars travel hundreds of thousands of millions of years to reach us. Although the electromagnetic radiation spans a large spectrum of wavelengths and frequencies, all electromagnetic radiation travels through a vacuum at 3 x 108 m/sec, or c, the speed of light. (ii)Transverse Waves and Longitudinal Waves

http://dev.physicslab.org/Document.aspx?doctype=3&filename=WavesSound_IntroductionWaves.xml

A wave may be a combination of types. Water waves in deep water are mainly transverse. However, as they approach a shore they interact with the bottom and acquire a longitudinal component. When the longitudinal component becomes very large compared to the transverse component, the wave breaks. 2) Wave Fronts and Rays -A wave front is a line representing all parts of a wave that are in phase and an equal number of wavelengths from the source of the wave. The shape of the

wave front depends upon the nature of the source; a point source will emit waves having circular or spherical wave fronts, while a large, extended source will emit waves whose wave fronts are effectively flat, or plane. -A ray is a line extending outward from the source and representing the direction of propagation of the wave at any point along it. Rays are perpendicular to wave fronts. But, How can we quantitatively describe a vibrating object? What measurements can be made of vibrating objects that would distinguish one vibrating object from another?

Part II Derive Wave function

http://oyc.yale.edu/physics/phys-200/lecture-18#ch7

-Parameters of Waves

amplitude, A; the period, T; wavelength, λ

The maximum displacement of the medium in either direction is the amplitude of the wave. The distance between successive crests or successive troughs (corresponding to maximum displacements in the same direction) is the wavelength of the wave. The frequency of the wave is equal to the number of crests (or troughs) that pass a given fixed point per unit of time. Closely related to the frequency is the period of the wave, which is the time lapse between the passage of successive crests (or troughs). The frequency of a wave is the inverse of the period. One full wavelength of a wave represents one complete cycle, that is, one complete vibration in each direction. The various parts of a cycle are described by the phase of the wave; all waves are referenced to an imaginary synchronous motion in a circle; thus the phase is measured in angular degrees, one complete cycle being 360° . Two waves whose corresponding parts occur at the same time are said to be in phase. If the two waves are at different parts of their cycles, they are out of phase. Waves out of phase by 180° are in phase opposition. The various phase relationships between combining waves determines the type of interference that takes place.

Part III Principle of Superposition for Waves

1) superposition principle

http://www.acs.psu.edu/drussell/demos/superposition/superposition.htm l

Waves on a string and waves in other elastic bodies usually obey a superposition principle:when two or more waves arrive at any given point simultaneously, the resultant instantaneous deformation is the sum of the individual instantaneous deformations.Such a superposition means that the waves do not interact; they have no effect on one another. Each wave propagates as though the other were not present, and the contribution that each makes to the displacement of a particle in the elastic body is as though the other were not present. For instance, if the sound waves from a violin and a flute reach us simultaneously, then each of these waves produces a displacement of the air molecules just as though it were acting alone, and the net displacement of the air molecules is simply the (vector) sum of these individual displacements. For waves of low amplitude on a string and for sound waves of ordinary intensity in air, the superposition principle is very well satisfied. However, for waves of very large amplitude or intensity, the superposition principle fails. When a wave of very large amplitude is propagating on a string, it alters the tension of the string, and therefore affects the behavior of a second wave propagating on the same string. Likewise, a very intense sound wave (a shock wave) produces significant alterations of the temperature and the pressure of the air, and therefore affects the behavior of a second wave propagating through this same region. In this section, we will not worry about such extreme conditions, and we will assume that the superposition principle is a good approximation. 2) Huygens' Principle Huygens (1629-1695), basically states that: Every point of a wave front may be considered the source of secondary wavelets that spread out in all directions with a speed equal to the speed of propagation of the waves. What this means is that when you have a wave, you can view the "edge" of the wave as actually creating a series of circular waves. These waves combine together in most cases to just continue the propagation, but in some cases there are significant observable effects. The wave front can be viewed as the line tangent to all of these circular waves. -Huygens-Fresnel Principle Wave at P is superposition of all wavelets from wavefront at aperture. Although Huygens’ principle was initially stated without any proof, a slightly modified form of it was later (about 1815) derived by Fresnel from the mathematical theory of waves. Note that Huygens was a contemporary of Newton, and that it would probably have been much more difficult to publish his

theory if he had lived in England, where disagreeing with Newton was not an easy or popular position to take.

Part IV Phenomenon

1) Interference (i) Thomas Young’s double-slit experiment

http://video.mit.edu/watch/thomas-youngs-double-slit-experiment-8432/

In 1801 Thomas Young was able to offer some very strong evidence to support the wave model of light. ? He placed a screen that had two slits cut into it in front of a monochromatic (single color) light. ? The results of Young's Double Slit Experiment should be very different if light is a wave or a particle. ? Let’s look at what the results would be in both situations, and then see how this experiment supports the wave model. (ii) Interference Patterns(conditions) -constructive interference -destructive interference

(From http://curricula2.mit.edu/pivot/book/ph1605.html?acode=0x0200) -Beats (iii) Diffraction A special case of interference is known as diffraction and takes place when a wave strikes the barrier of an aperture or edge. At the edge of the obstacle, a wave is cut off, and it creates interference effects with the remaining portion of the wave fronts. Since nearly all optical phenomena involve light passing through an aperture of some kind - be it an eye, a sensor, a telescope, or whatever - diffraction is taking place in almost all of them, although in most cases the effect is negligible. 2) reflection

Throwing back or deflection of waves, such as light or sound waves, when they hit a surface.

Reflection occurs whenever light falls on an object. The law of reflection states that the angle of incidence (the angle between the ray and a perpendicular line drawn to the surface) is equal to the angle of reflection (the angle between the reflected ray and a perpendicular to the surface). Looking at an image on the surface of the water in a lake is an example of light rays reflecting towards the observer. Reflection of light takes place from all materials. Some materials absorb a small amount of light and reflect most of it back; for example, a shiny, silvery surface. Other materials absorb most of the light and reflect only a small amount back; for example, a dark, dull surface. Reflected light gives objects their visible texture and colour. Light reflected from a surface can be either regular (plane), where the surface is flat and smooth and light rays are reflected without any scattering; or scattered, where the surface is irregular (in effect, many different surfaces). The colour of the sky is due to scattering of sunlight by particles in the atmosphere, such as dust and gas particles, water droplets, or ice crystals. On a clear day the sky appears blue due to the scattering of shorter wavelength light. -Total internal reflection When light passes from a dense medium to a less dense medium, such as from water to air, both refraction and reflection can occur. If the angle of incidence is small, the reflection will be relatively weak compared to the refraction. But as the angle of incidence increases the relative degree of reflection will increase. At some critical angle of incidence the angle of refraction is 90° . Since refraction cannot occur above 90° , the light is totally reflected at angles above this critical angle of incidence. This condition is known as total internal reflection. Total internal reflection is used in fibre optics to transmit data over long distances, without the need of amplification. - Phase shift by Reflection (self-study)

3)Refraction

Bending of a wave when it passes from one medium into another. It is the effect of the different speeds of wave propagation in two substances that have different densities. For example, when light passes from air (less dense) into glass (more dense) it slows down (from 300 million to 200 million metres per second) and is refracted. The amount of refraction depends on the densities of the media, the angle at which the wave strikes the surface of the second medium, and the amount of bending and change of velocity corresponding to the wave's frequency (dispersion). Refraction occurs with all types of progressive waves – electromagnetic waves, sound waves, and water waves – and differs from reflection, which involves no change in velocity. Refraction of light The degree of refraction depends in part on the angle at which the light hits the surface of a material. A line perpendicular to that surface is called the normal. The angle between the incoming light ray and the normal to the surface is called the angle of incidence. The angle between the refracted ray and the normal is called the angle of refraction. The angle of refraction cannot exceed 90° . An example of refraction is light hitting a glass pane. When light in air enters the denser medium, it is bent toward the normal. When light passes out of the glass into the air, which is less dense, it is bent away from the normal. The incident light will be parallel to the emerging light because the two faces of the glass are parallel. However, if the two faces are not parallel, as with a prism, the emerging light will not be parallel to the incident light. The angle between the incident ray and the emerging ray is called the angle of deviation. The amount of bending and change in velocity of the refracted wave is due to the amount of dispersion corresponding to the wave's frequency, and the refractive index of the material. When light hits the denser material, its frequency remains constant, but its velocity decreases due to the influence of electrons in the denser medium. Constant frequency means that the same number of light waves must pass by in the same amount of time. If the waves are slowing down, wavelength must also decrease to maintain the constant frequency. The waves become more closely spaced, bending toward the normal as if they are being dragged. The refractive index of a material indicates by how much a wave is bent. It is found by dividing the velocity of the wave in the first medium by the velocity of the wave in the second medium. The absolute refractive index of a material is the velocity of

light in that material relative to the velocity of light in a vacuum. See also apparent depth. 4) Doppler Effect So far we have only discussed cases where the source of waves is at rest. Often, waves are emitted by a source that moves with respect to the medium that carries the waves, like when a speeding cop car blares its siren to alert onlookers to stand aside. The speed of the waves, v, depends only on the properties of the medium, like air temperature in the case of sound waves, and not on the motion of the source: the waves will travel at the speed of sound (343 m/s) no matter how fast the cop drives. However, the frequency and wavelength of the waves willdepend on the motion of the wave’s source. This change in frequency is called a Doppler shift.

Appendix: http://ocw.mit.edu/courses/physics/8-03-physics-iii-vibrations-and-waves-fall-2004/video-lectures

Chapter 7 Thermodynamics Section 1

Part I Basic concepts

1. Temperature （Macroscopic quantity） -Thermometer -Thermal equilibrium Zeroth law of thermodynamics If C is initially in thermal equilibrium with both A and B, then A and B are also in thermal equilibrium with each other. It was recognized only after the first, second, and third laws of thermodynamics had been named. It may seem trivial and obvious, but even so it needs to be verified by experiment. Two systems are in thermal equilibrium if and only if they have the same temperature. -temperature scales (1) Celsius temperature scale (centigrade scale) Freezing temperature of pure water “zero”; boiling temperature “100”. (2) Fahrenheit temperature scale(US) ? Freezing temperature of water is 32 F (thirty-two degrees Fahrenheit) and the boiling ? temperature 212 F , both at standard atmospheric pressure.

9 TF ? Tc ? 32 ? C 5

(3) Kelvin scale (named for the British physicist Lord Kelvin(1824-1907) Ideally, we would like to define a temperature scale that doesn’t depend on the properties of a particular materials .(a truly material-independent scale)

Tk ? Tc ? 273.15

293k: is read “293 kelvins”. 2. Heat -heat Energy transfer that takes place solely because of a temperature difference is called heat flow or heat transfer; and energy transferred in this way is called heat. Ask students：to distinguish temperature and heat. Temperature depends on the physical state of a material and is a quantitative description of its hotness or coldness; while heat always refers to energy in transit from one body or system to another because of a temperature difference, never to the amount of energy contained within a particular system. -conductor

A material which provides good conduction is called a conductor, while materials that provide poor conduction are called insulators.

-convention

convection is the process by which heat transfers through the movement of a fluid (or gas) from one region to another. If the fluid is circulated by a pump or fan, it is called forced convection. In other cases, the flow is by natural means of fluid expansion and interaction, and this is called free convection or natural convection.

3. Phases of matter Gas, liquid, solid

4. The Ideal-gas equation

PV ? nRT

Where R is a proportionality constant(it turns out to be the same value for all gases, at least at sufficiently high temperature and low pressure), called gas constant. (Pression, volume, amount of substance, mass, number of moles) An ideal gas is one for which the above equation holds precisely for all pressures and temperatures.

Part II Thermodynamic processes

1.Thermodynamic system e.g. Popcorn kernels in a pot with a lid 2. Thermodynamic processes A system undergoes a thermodynamic process when there is some sort of energetic change within the system, generally associated with changes in pressure, volume, internal energy (i.e. temperature), or any sort of heat transfer. There are several specific types of thermodynamic processes that have special properties:

? Adiabatic process- a process with no heat transfer into or out of the system. ? Isochoric process- a process with no change in volume, in which case the system does no work. ? Isobaric process- a process with no change in pressure. ? Isothermal process- a process with no change in temperature. --First Law of Thermodynamics (1) The change in a system's internal energy is equal to the difference between heat added to the system from its surroundings and work done by the system on its surroundings. -internal energy Internal energy of a system as the sum of the kinetic energies of all of its constituent particles, plus the um of all the potential energies of interaction among these particles. One of the special properties of an ideal gas is that its internal energy depends only on temperature, not on its pressure or volume. (2) Mathematical Representation of the First Law Physicists typically use uniform conventions for representing the quantities in the first law of thermodynamics. They are: ? U1(or Ui) = initial internal energy at the start of the process ? U2(or Uf) = final internal energy at the end of the process ? delta-U = U2 - U1 = Change in internal energy (used in cases where the specifics of beginning and ending internal energies are irrelevant) ? Q= heat transferred into (Q > 0) or out of (Q < 0) the system ? W= work performed by the system (W > 0) or on the system (W < 0). This yields a mathematical representation of the first law which proves very useful and can be rewritten in a couple of useful ways: U2- U1 = ? U = Q - W Q = ?U + W (3) The First Law & Conservation of Energy The first law of thermodynamics is seen by many as the foundation of the concept of conservation of energy. It basically says that the energy that goes into a system cannot be lost along the way, but has to be used to do something. Taken in this view, the first law of thermodynamics is one of the most far-reaching scientific concepts ever discovered. (4) heat capacities for an ideal gas

-molar heat capacity at constant volume Cv -molar heat capacity at constant pressure C p

Section 2

I. Quasi-static Processes Sometimes a process takes place relatively slowly. Slowly enough so that all the intensive variables can have actually definite values through the entire path taken by the process. Such a process is called a quasi-static process. For example, let us assume that our system is an amount of a gas trapped inside a cylinder, pushing on a piston forming one end of the cylinder. If the piston moves out rapidly, the gas expands rapidly and the pressure will vary from point to point inside the gas. But if the piston moves very very slowly, possibly due to friction or some artificial restriction limiting its speed, the pressure may well have a common value throughout the system at every instant in time. The time scale involved is easy to figure. If the piston moves rapidly, pressure waves will exist inside the cylinder and there will be no one pressure for the system. But if the piston moves slowly, compared to the speed of sound in the gas in the cylinder for example, then the pressure will be essentially uniform all the time. Such a process is a quasi-static one. The advantage of a quasi-static process is that at each point along the path taken by the system in going from the initial state to its final state the system is effectively at equilibrium. So we can define this path very precisely. This can be illustrated by means of a pressure-volume diagram: Figure 1.1: p-V diagram showing an initial and a final state.

The initial and final states are labelled 1 and 2. These are equilibrium states. If a process takes us from State 1 to state 2 by means of a series of non-equilibrium situations there is no way to show the path on a p-V diagram. That's because on the path the system does not have a single uniform pressure. On the other hand, if the path is traversed through a series of quasi-static states, that path can be shown on the diagram: Figure 1.2: p-V showing a path between two states

because each and every point on the line connecting State 1 and State 2 is an equilibrium state of the system. II. Two key thermodynamic processes: (1) Adiabatic process- a process with no heat transfer into or out of the system. Definition: An adiabatic process is a thermodynamic process in which there is no heat transfer (Q) into or out of the system. In other words Q = 0. An adiabatic process is generally obtained by surrounding the entire system with a strongly insulating material or by carrying out the process so quickly that there is no time for a significant heat transfer to take place. Applying the first law of thermodynamics to an adiabatic process, we obtain: delta-U = -W Since delta-U is the change in internal energy and W is the work done by the system, what we see the following possible outcomes: ? A system that expands under adiabatic conditions does positive work, so the internal energy decreases. ? A system that contracts under adiabatic conditions does negative work, so the internal energy increases. (2) Isothermal process- a process with no change in temperature. Definition: An isothermal process is a thermodynamic process in which the temperature of the system remains constant. The heat transfer into or out of the system typically must happen at such a slow rate that the thermal equilibrium is maintained. The internal energy of an ideal gas, depends solely on the temperature, so the change in internal energy during an isothermal process for an ideal gas is also 0. If the pressure and volume change, it is possible for a substance to change its state of matter even while its temperature remains constant, if you're careful about how you apply or remove heat from the system. III. Second Law of Thermodynamics The law of entropy, or the second law of thermodynamics, along with the first law of thermodynamics comprise the most fundamental laws of physics. Entropy (the subject of the second law) and energy (the subject of the first law) and their

relationship are fundamental to an understanding not just of physics, but to life (biology, evolutionary theory, ecology), cognition (psychology). According to the old view, the second law was viewed as a 'law of disorder'. The major revolution in the last decade is the recognition of the "law of maximum entropy production" or "MEP" and with it an expanded view of thermodynamics showing that the spontaneous production of order from disorder is the expected consequence of basic laws.

(i) It is impossible to construct a heat engine with an efficiency of 1.00 . (ii) It is impossible to build a cyclic machine that produces no other effect than to transfer heat continuously from a cold temperature to a hot temperature. The Carnot Engine http://chemwiki.ucdavis.edu/Physical_Chemistry/Thermodynamics/Thermodynam isc_Cycles/Carnot_Cycle (1) A Carnot engine -- or a Carnot cycle -- is a combination of isothermal expansions and compressions and adiabatic expansions and compressions. From an initial stat A, the gas is placed in contact with the hot temperature reservoir (Th) and expands isothermally (keeping T = Th = constant) to some state B. During this isothermal expansion heat Qh flows into the gas from the hot temperature Th. From state B, the gas undergoes an adiabatic expansion to state C. No heat is exchanged during this expansion. Expanding an insulated gas means work is done at the "expense" of the internal energy. That means the gas will have a lower temperature. This is the cold temperature Tc. At state C, we place the gas in contact with the cold temperature heat reservoir (like a large tank of water) and do an isothermal compression to state D. In compressing the gas, work is done on the gas by the outside. But the temperature remains constant -- meaning the internal energy U of the gas remains constant. For this to happen, heat Qc is given out to the cold temperature heat reservoir. From state D we do an adiabatic compression back to state A. Remember, "adiabatic" means insulated so there is no heat exchange.

These processes may be seen in the drawings of the gas and piston or on the PV diagram immediately above. It will be useful to go back and forth between the two diagrams and the words describing them. We already know the efficiency of a heat engine,

It can be shown that the ratio of the heat expended to the heat absorbed, Q c/Qh, is also equal to the ratio of the cold temperature to the hot temperature, Tc/Th. Then the efficiency of a Carnot engine can also be written as The significance of this is that all Carnot engines operating between the same temperatures have the same efficiency. It also shows us that the only way to have an efficiency of 100% is if the cold temperature Tc is absolute zero -- that is, Tc = 0 K. For most practical situations the cold temperature is around room temperature. So increasing efficiency usually means increasing the hot temperature Th. That is why most fuel-efficient automobile engines run hotter than most poorly efficient automobile engines. The efficiency of a real engine will always be less than the efficiency of a Carnot engine running between the same temperatures. (2) The efficiency of a Carnot engine -- or any reversible heat engine -- is the greatest that is possible to achieve. Call the efficiency of the Carnot engine ec. Then suppose the efficiency of some real heat engine, e r, is greater than that of the Carnot engine, er > ec Then we could use the real heat engine to power a Carnot cycle heat pump. If er > ec, then the net result would be the transfer of heat from a cold temperature to a high temperature. But this violates the Second Law.

Qh is the heat absorbed from the high temperature by the real heat engine and Qh' is the heat expelled to the high temperature by the Carnot cycle heat pump. For this analysis, both Qh and Qh' are intrinsically positive. er > ec W/Qh > W/Qh' 1/Qh > 1/Qh' Qh < Qh' Qh' > Qh Qh,net = Qh' - Qh Qh,net > 0 Since we believe the Second Law, that means our assumption that er > ec is wrong.

(3) Entropy

Entropy, S, is another "state variable" -- along with P, V, and T. We define the change in entropy by dS is the change in entropy. dQr is the amount of heat that would flow out of the system along a reversible path between the two states and T is the absolute temperature. Entropy is a description of the disorder of a system. For a reversible process, the total entropy of the system and its surroundings will be unchanged. For an irreversible process, the total entropy of the system and its surroundings will increase. In general, we will need to add up or integrate these small entropy changes,

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