College Physics (II) Qingxu Li Tel: 62471347, Email: [email protected] Room 306, College of Mathematics and Physics.

College Physics (II)

Qingxu Li

Tel: 62471347, Email: [email protected] 306, College of Mathematics and Ph

ysics

“The most incomprehensible thing about the universe is that it is comprehensible.”

—-Albert Einstein

About the Course

• College Physics (II)

• Textbook: General Physics, Bin Liang, et al.

• Contents: Mechanics, Oscillation and Wave, Optics; El

ectromagnetism, Relativity, Quantum Physics, etc.

• Course grade: Final Exam (70%) + Performance (30%)

• Exercises and Exam are to be finished in English

(Chapter 2-7, 10)

a. Principle of Physics, 3rd edition, Serway and Jewett

b. Feynman’ Lectures on Physics (Volume I), by R. P. Feynman

c. 物理学，马文蔚，高等教育出版社，第五版

……

Reference books

A Brief Summary of Chapter 1

Units, Dimension, Significant Figures, Order of Magnitude, Vector

a. Units are indispensable for physical quantities. b. Vectors are to be distinguished from scalars.

c. Properties of Vectors:magnitude, direction, components,equality, addition, dot product, cross product, etc.

2 2 2 , , ; ; x y z A x y z A

AA A A A Ae A A A A e

A

cos ;

; sin

AB x x y y z z

ABA B

A B AB A B A B A B

A B A B e A B AB

（单位，量纲，有效数字，数量级，矢量）

Position and Displacement Vectors

'r r r

（位置矢量和位移矢量）

path 路程 , 路线 locus 轨迹 distance 距离

Average Velocity and Instantaneous Velocity

( ) ( ) ( ) ( )r r t x t i y t j z t k

the average velocity : r

vt

0the instantaneous velocity : lim

t

r drv

t dt

f ir r r

（平均速度和瞬时速度）

Fig 1.1 A particle moving in the xy plane

Alternative Expressions

d( )

d

rv v t

t

( , , )x y z vv v v v ve

, ,x y z

dx dy dzv v v

dt dt dt

( )v v t x y z ， ， ，

（其他形式）

Acceleration

The average acceleration of a particle over a time interval is defined as:

f i

f i

v vva

t t t

And the instantaneous acceleration is defined as:

0 lim

t

v dva

t dt

（加速度）

Alternative Expressions

( )( )

( , , )

, ( ); , ,

x y z a

dv ta a t

dta a a a ae

dva a a t x y z

dt

v

Fig 1.2 The Velocity-Time diagram. The magnitude of acceleration vector is the slope of the curve v—t.

Problems Related to Kinematics

position displacement velocity acceleration

( ) f ir r t r r r

d

d

df

i

t

t

rv

t

r v t

0

2

2

0

d d

d d

( ) ( ) dt

t

v ra

t t

v t v t a t

Mechanics

Kinematics

Dynamics

The Laws of Motion

（运动定律）

Part II Dynamics

Nature and nature’s laws lay hid in night.

God said: Let Newton be! and all was light.

--Alexander Pope

The Concept of Force（力的概念）

The force is a vector quantity.

The unit of force is newton, which is defined as the force that,when acting on a 1-kg mass, produces an acceleration of 1m/s2.

21 1 kg m/sN The dimension of force is:

2ML / TF

Newton’s First Law （牛顿第一定律）

Newton’s first law of motion:

In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with aconstant velocity (that is, with a constant speed in a straightline)

In simpler terms, when no force acts on a body, its accelerationis zero.

（在没有外力的情况下，静止的物体会保持静止，而运动的物体则保持运动速度不变，也就是说运动物体做匀速直线运动。）

（简单地讲，如果没有外力作用，物体的加速度为零）

Comments on the First Law

1. The first law tell us that an object has a tendency to maintain itsoriginal state of motion in the absence of the force. This tendency iscalled inertia, and the first law sometimes called the law of inertia.

2. Newton’s first law defines a special set of reference frames calledinertial frames. An inertial frame of reference is one in which thefirst law is valid.

3. Inertial mass is the measure of an objects resistance to change inmotion in response to an external force. Inertial mass is different in definition from gravitational mass, but they have the same value, sowe call them both simply mass.

（牛顿第一定律告诉我们物体在不受外力的情况下有一个保持原来的运动状态的趋势） -称为惯性，因而第一定律有时又被称为惯性定律）

（利用牛顿第一定律可以定义一类特殊的参照系 -惯性系：在惯性系中，第一定律成立）

（惯性质量是物体阻止运动状态发生改变能力即惯性大小的量度。惯性质量和引力质量在定义上不同，但它们具有相同的数值，统称为质量）

Mass and Weight（质量和重量）

Mass and weight are two different quantities, and should not beconfused with each other.

The magnitude of an object is equal to the magnitude of

the gravitational force exerted by the planet on which the objects

resides. While the mass of an object is the same everywhere. A

given object exhibits a fixed amount of resistance to changes in

motion regardless of its location.

E.g. A person of mass 60 kg on Earth also has a mass of 60 kg

on the moon. The same person weighs 588 Newton on Earth, but

weighs 98 Newton on the moon.

Newton’s Second Law （牛顿第二定律）

Newton’s second law of motion:

The acceleration of an object is directly proportional to thenet force acting on it and inversely proportional to its mass.

（物体的加速度和它受到的合外力成正比和它的质量成反比）

Fa

m

net force（合力）

Net Force

① the resultant force

② the sum of the force

③ the total force

④ the unbalanced force

The net force is also known as:

tF F

The Mathematical Form of the Second Law

tF F

am m

, , ,F F

a x y zm m

（第二定律的数学形式）

Comments on the Second Law

1. The Newton’s second law is the central rule of classical mechanics,which bridges dynamics and kinematics and tells that force is thecause of the change of motion (not motion!).

2. The second law has an alternative expressions:

t

dv dpF ma m

dt dt

In special relativity, the mass of an object will vary with its velocityand thus vary with time. The previous form is invalidated in this casebut the new form still holds. Of course, both form are equivalent fornon-relativistic cases.

3. The second law can also be expressed as:2

2t

d rF m

dt

Newton’s Third Law （牛顿第三定律）

Newton’s third law of motion:

If two objects interact, the force exerted by object 1 on object2 is equal in magnitude but opposite in direction to the forceexerted by object 2 on object 1.

（如果两个物体之间（存在）相互作用），则物体 1 作用到物体 2 上的力和物体 2 作用到物体 1 上的力大小相等，方向相反）

Forces always occurs in pairs, i.e., that a single isolated force cannot exist.

（力总是成对出现，也就是说，单个孤立的力是不能存在的）

Comments on the Third Law

The force that object 1 exerts on object 2 may be called action

force and the force of object 2 on object 1 the reaction force. The

action force is equal in magnitude to the reaction force and

opposite in direction. In all cases, the action and the reaction forces

act on different objects and must be of the same type.

（物体 1 作用在物体 2 上的力可以称作作用力，相应地我们称物体 2作用在物体 1上的力为反作用力。作用力和反作用力大小相等方向相反。作用力和反作用力类型相同，并且作用在不同的物体上）

Applications of Newton’s Law

（牛顿定律的应用）

The Particle in Equilibrium

Objects that are either at rest or moving with constant velocity aresaid to be in equilibrium. From Newton’s second law, this conditionof equilibrium can be expressed as:

0tF F

or:

0; , ,tF x y z

（处于平衡状态的质点）

（我们称静止或作匀速直线运动的物体处于平衡状态。根据牛顿第二定律，物体处于平衡状态的条件可以表达为：）

The Accelerating Particle（加速质点）

When a nonzero net force is acting on a particle, the particle isaccelerating, and the second law tell us:

FF ma a

m

In practice, the above equation is broken into components, sothat two or three equations can be handled independently.

（如果质点受到一个非零的合外力，则质点加速运动，由第二定律可知）

（上述方程在应用的时候通常分解为分量形式，这样就可以单独处理两个或三个方程）

E.g. 1.1 When two objects with unequal masses are hung vertically over a light, frictionless pulley as in the figure, the arrangement iscalled an Atwood machine. The device is sometimes used in the labto measure the free-fall acceleration. Calculate the magnitude of theacceleration of the two objects and the tension in the string.

The Atwood Machine（阿特伍德机）

Fig 1.10 The Atwood machine.

1 2Solution : Assuming the masses of the two objects are m , m respectively,

and it is easily to find that both objects must have the same magnitude of

acceleration. We choose the upward direction is alon

1

1 1

2

2

g the positive direction

of y axis.

So, when Newton's second law is applied to m , we find

(1)

Similarly, for m we find

' '

ty y

ty y

F F T m g m a

F F T m g

2

2 1 2 1

2 1

2 1

(2)

From equation (1) and (2), we can obtain

(3)

Solving the above equation, we get the acceleration

.

m a

m m g m m a

m ma g

m m

1 2 1 2

1

E.g. 1.2 Two blocks of masses and , wiht .

are placed in contact with each other on a frictionless,

horizontal surface, as in the figure. A constant horizontal

force is applied to m as show

m m m m

F

n.

(a) Find the magnitudeof the acceleration of the system of

two blocks.

(b) Determine the magnitude of the contact force between

the two blocks.

Solution (a) Both blocks must experience the same acceleration

because they are in contact with each other and keep so. Thus we

model the system of both blocks as a particle under a net force.

For the fo

1 2

1 2

2

rce is horizontal, we have

( ) (1)

So the magnitude of the acceleration reads .( )

(b) It is obviously the contact force exerted on is in the positive

di

xF F m m a

Fa

m m

m

x

2

2 2

22 2 2

1 2

rection, and the net force acting on the is

(2)

From Newton's second law, we get

.( )

x c

x c

m

F F

m FF F m a

m m

Forces of Friction

When an object moves either on a surface or through a viscous

medium such as air or water, there is resistance to the motion. We

call such resistance a force of friction.

（摩擦力）

Force of friction

force of static friction

force of kinetic friction

（静摩擦力）

（动摩擦力）

Simplified model for force of friction

1. The magnitude of the force of static friction between any two surfaces in contact can have the values

s sf n

2. The magnitude of the force of kinetic friction acting between two surfaces is

k kf n

3. The values ofμk andμs depend on the nature of the surfaces, but the former is generally less than the latter.

4. The direction of the friction force on an object is opposite to the actual motion or the impending motion of the object relative to the surface with which it is in contact.

μs : the coefficient of static frictionn: the magnitude of normal force

μk : the coefficient of kinetic friction

Fig 1.5 A graph of the magnitude of the friction force versus thatof the applied force.

The Gravitational Force: Newton’s Law of Universal Gravitation

Newton’s Law of Universal Gravitation

Every particle in the Universe attracts every other particle witha force that is directly proportional to the product of the masses ofthe particles and inversely proportional to the square of the distancebetween them.

1 22g

m mF G

r

11 2 2 6.67 10 / ,

is calle the universal gravitational constant.

G N m kg

引力：牛顿万有引力定律

The electrostatic force

Coulomb’s Law

The magnitude of the electrostatic force between two chargedparticle separated by a distance r is:

1 22e e

q qF k

r

9 2 28.99 10 N m Cek

The Coulomb constant

The Fundamental Forces of Nature

① The Gravitational Force

② The Electromagnetic Force

③ The Strong Force (The Nuclear force)

④ The Weak Force

（自然界中的力）

（引力）

（电磁力）

（强力）

（弱力）

electromagnetic force force (1967,1984)

weak forceelectroweak

Newton’s Second Law

Applied to a Particle in Uniform Circular Motion

A particle moving in a circular path with uniform speed experiencesa centripetal acceleration of magnitude:

2

c

va

r

The acceleration vector is directed toward the centre of the circle andis always perpendicular to its velocity.

Apply Newton’s second law to the particle along the radial direction:

2

t c

mvF ma

r

centripetal force（向心力）

Non-uniform Circular Motion

For non-uniform circular motion, there is, in addition to the radialcomponent of acceleration, a tangential component, that is:

tr t

Fa a a

m

The total force exerted on the particle is

_ _t t r t tF F F

The first term in the RHS is directed toward the center of the circle and is responsible for the centripetal acceleration; and the second termis tangent to the circle and responsible for the tangential acceleration,which causes the speed of the particle to change with time.

Energy of a System（物理体系的能量）

Energykinetic energy

potential energy

（动能）

（势能）

Work （功）

The work done by a force on a system is defined as:

cosdW F dr Fdr

For a finite displacement,

cosf f

i i

r r

r rW F dr Fdr

（作用在一个体系上的力对体系作功定义为）

（对于一个有限位移）

Work Done by a Constant Force（恒力作功）

For a constant force, the work reads:

cos

cos

f f

i i

r r

r rW F dr F dr

F r F r

If the applied force is parallel to the direction of the displacement,

W F r And if the force is perpendicular to the displacement, then

0W

dW F dr F dr

f f

i i

r r

r rW F dr F dr W

From the definition of dot product, we get:

Work done by a Spring

Hooke’s Law

F kx

is (spring constant or

stiffness), and is the distance from

the equili

force c

brium p

onsta

osit .

n

n

t

io

k

x

2 21 1

2 2

f

i

x

s s i fxW F dx kx kx

The work done by the restoring force on a block connected witha spring reads:

restoring force 回复力

Kinetic Energy（动能）

The work done on a system in motion:

2 2 2

1 1 1

2 2 2|

f f

i i

f f

i i

f

i

r r

tr r

r r

r r

f i

vv

W F dr ma dr

dv drm dr m dv

dt dt

mv mv mv

Define the kinetic energy of a particle is:

21

2K mv

From the above definition, we get:

net f iW K K K

Work-kinetic energy theorem

（功能定理）

When work is done on a system and the only change in the

system is in its speed, the work done by the net force equals

the change in kinetic energy of the system.

Work-kinetic energy theorem:

（当外界对体系作的功给体系带来的只是体系运动速率的变化时，合外力所做的功等于体系动能的增量）

E.g. 1.8 A 6.00 kg block initially at rest is pulled to the right alonga horizontal friction less surface by a constant, horizontal force of12.0 N, as shown in the figure. Find the speed of the block after ithas moved 3.00m.

Solution : The work done by the horizontal force is

12.0 3.00 J=36.0 J.W F x

2

Using the work-kinetic theorem and noting that the initial

kinetic energy is zero, we find

1

2f i f fW K K K mv

23.46 m / sf

Wv

m

Potential Energy

Kinetic Energy: related to the motion of an object

Internal Energy: related to the temperature of a system

Potential Energy: related to the configuration of a system

configuration: 构型，结构

Example: gravitational potential energy,

gU mgy

（势能）

Conservative Forces

The work done by a conservative force does not depend on the

path followed by the members of the system, and depends on the

initial and final configurations of the system.

In other words, the work done by a conservative on an object

does not depend on the path of the object, but depends on its

initial and final position.

From above definition, it immediately follows that the work

done by a conservative force when an object is moved through

a closed path is equal to zero.

（保守力）

Conservative Forces and Potential Energy

Conservative force ：( , ) ( ) ( )

0

f

i

r

f i f irF dr f r r U r U r

F dr

'

'

'

0

B A

p pA B

B B

p pA A

B B

p p B AA A

F dr F dr F dr

F dr F dr

F dr F dr U U

Define potential energy function as:

f

i

r

f irU F dr U

From above definition, we can get:

dU F dr

and: ( , , )U U U

F Ux y z

or U

Fr

f

i

r

f i rU U U F dr W

Gravitational Force

Consider a particle of mass m above the Earth’s surface:

The gravitational force on the particle due to the Earth reads:

2g r

GMmF e

r

The work done by the gravitational force

1 1( )

f

i

r

f i irf i

U F dr U GMm Ur r

Assuming: lim 0i

ir

U

And we get: .r

GMmU

r

In summary, the gravitational potential energy for any pair of

particles varies as 1/r. Furthermore, the potential energy is negative

because the force is attractive and we have chosen the potential

energy to be zero when the particle separation is infinity.

( , , )F U Ux z

G

r

y

MmU

3 2 ( , , , , )g r

GMm GMm GMmF x y z e

x y z r r r

E.g. A particle of mass m is displaced through a small vertical distanceΔy near the Earth’s surface. Show that expression for the change ingravitational potential energy reduces to the familiar relationship:ΔUg=mg Δy.

Solution : We can express the potential energy change as:

1 1 ( )

If both initial and final positions of the particle are close to the earth's

surface, then and

f i

f i f i

f i

r rU GMm GMm

r r r r

r r y

2

2 2

. The change in potential energy

therefore, becomes: , where g .

f i E

E E

r r R

y GMU GMm mg y

R R

Electrostatic Force（静电力）

The electrostatic force between two charged particles reads:

1 22e e r

q qF k e

r

In a way similar to gravitational force case, we can get electricpotential energy function （电势能） :

1 2e e

q qU k

r

1 2 1 2 1 23 2

( , , ) , , e e ee r

k q q k q q k q qF x y z e

x y z r r r

The Force of a Spring

According to Hook’s law, a block connected to a spring

experiences a force:

sF kxTherefore, the potential energy stored in a block-spring system is:

2 21( )

2

f

i

x

f i f i ixU kxdx U k x x U

If the initial position of the block is xi=0, Ui is always chosen as

zero, Then,21

2fU kx elastic potential energy

（弹性势能） s

dF U kx

dx

势能函数 保守力

( , , )U U U

F Ux y z

f

i

r

f irU F dr U

Mechanical Energy（机械能）

The sum of kinetic and potential energy is defined as mechanical

energy:

mechE K U If in an isolated system there are only conservative forces which

do work, the mechanical energy will keep unchanged, as is called

conservation of mechanical energy.

（体系的动能和势能之和定义为体系的机械能）

（如果孤立体系中只有保守力做功，则系统的机械能保持不变，称体系的机械能守恒）

The conservation of mechanical energy in a system can be

expressed as:

Constmech i i f fE K U K U

The conservation of energy in an isolated system can be

expressed as:

int int Constt i i i f f fE K U E K U E

Stability of Equilibrium（平衡的稳定性）

Energy diagram: An energy diagram shows the potential energy ofthe system as a function of the position of one of members of thesystem.

Stable equilibrium: When the system locates such a position thatany movement away from this position results in a force directed backtoward the position. (this type of force is called restoring force.)

In general, positions of stable equilibrium correspond to thosepositions for which the potential energy function has a relative minimum value on an energy diagram. And positions of unstableequilibrium correspond to those positions for which the potentialenergy has a relative maximum value.

Energy Transfer

isolated systems Vs non-isolated systems

system and its environment

Work is one means of energy transfer between the system and

its environment. If positive work is done on the system, energy is

transferred from the system to the environment, whereas negative

work indicates that energy is transferred from the system to the

environment.

（孤立系统 Vs 非孤立系统）

（系统和环境）

（能量转移）

Heat and Thermal Conduction（热和热传递）

Except for work, energy can also be transferred through thermal

conduction.

The work done on a system may also increase its internal energy,

in addition to change its kinetic energy. The internal energy of an

object is associated with its temperature. And it’s well known that

heat can be transferred from a warmer object to another object. The

energy transfer caused by a temperature difference between two

regions in space is called thermal conduction.

internal energy 内能 heat 热，热量thermal conduction 热传递

Mechanical Wave

Mechanical wave are a means of transferring energy by

allowing a disturbance to propagate through into air or

another medium. This is the method by which energy

leaves a radio through the loudspeaker-sound-and by which

energy enters your ears to stimulate the hearing process.

（机械波）

disturbance 扰动 propagate 传播medium 介质，媒介

Principle of Conservation of Energy

We can neither create nor destroy energy – energy is conserved.

If the amount of energy in a system changes, it can only be due to

the fact that energy has crossed the boundary by a transfer

mechanism.

（能量既不能创生，也不能消灭——能量是守恒的）

（如果体系的能量发生了变化，只可能是体系的一部分能量通过一种能量转移机制穿出了体系的边界）

（能量守恒定律）

isolated and non-isolated systems

For isolated systems, energy is always conserved, so:

int 0E K U E

While for non-isolated systems, we have:

E H Work

Heat

matter transfer

Power （功率）

Power: the time rate of energy transfer.

Average power: If the work done by a force is W in the time

interval Δt , then the average power during this time interval

is defined as: Wp

t

The instantaneous power:

0lim

t

W dWp

t dt

（功率：能量转化的快慢）

For work done by a varying force:

dW drp F F v

dt dt

In general, power is defined for any type of energy transfer and

the most general expression for power is, therefore

dEp

dt

Unit of Power（功率的单位）

Unit of Power: Watt

2 31 1 / 1 /Watt J s kg m s

Other mostly used units of power:

1hp 746W61kWh 3.60 10 J

Solution : As shown in the figure, the car is the system and the only

force we must consider in the calculation is the horizontal force of

kinetic friction (denoted by ). Thus we can get in the two caskF

2

es:

Case 1. The initial speed is (assuming the mass of the car is m) :

1 0

2Case2. Given the distance the car slides is , then:

k

k

v

F d mv

x

F x

210 2

2From above two equations, it can be easily obtained:

4 4 .

m v

xx d

d

Solution : As shown in the figure, the car is the system and the only

force we must consider in the calculation is the horizontal force of

kinetic friction (denoted by ). Thus we can get in the two caskF

2

es:

Case 1. The initial speed is (assuming the mass of the car is m) :

1 0

2Case2. Given the distance the car slides is , then:

k

k

v

F d mv

x

F x

210 2

2From above two equations, it can be easily obtained:

4 4 .

m v

xx d

d

Momentum and Impulse（动量和冲量）

motion

kinematics

mechanics (force)

energy

momentum

Momentum（动量）

The (linear) momentum of an object of mass m moving with avelocity v is defined to be the product of the mass and velocity:

p mv

Momentum is a vector quantity and its direction is the same asthat for velocity; And it has dimension ML/T. In SI system, themomentum has the units kg·m/s.

, , ,p mv x y z

Momentum and Force

As pointed out before, the Newton’s second law can be rewritten

as: dp

Fdt

From above equation, we see that if the net force on an object is

zero, the time derivative of the momentum is zero, and therefore

the momentum of the object must be constant. Of course, if the

particle is isolated, then no forces act on it and the momentum

remains unchanged——this is Newton’s first law.

Momentum and Isolated Systems

The total momentum of an isolated system remains constant.

0tF

The total momentum t ii

p p

for an isolated system

Thus, we have 0 =Consttt i

i

dpp p

dt

Const, , , .tp x y z The law of conservation of linear momentum!

（孤立体系的动量是一个常数）

（动量守恒定律）

Impulse and Momentum（冲量和动量）

Assuming a net varying force acts on a particle, then we get:

t t

dpF dp F dt

dt

thus the change in the momentum of the particle during the timeinterval Δt = tf − ti reads:

f

i

t

f i ttp p p F dt

The Impulse of a force is defined as:

t

f

i

t

tI F dt p

impulse-momentum theorem（冲量 -动量定理）

Also valid for a system of particles （对于质点系也成立）

Impulse is an interaction between the system and its environment.

As a result of this interaction, the momentum of the system changes.

（冲量是体系和环境之间的一种相互作用，它带来体系动量的变化）

The impulse approximation: We assume that one of the forces

exerted on a particle acts for a short time but is much greater than

other force present. this simplification model allows us to ignore

the effects of other forces, because these effects are be small during

the short time during which the large force acts.

（冲量近似：如果有一个力短时间作用于一个质点，并且作用过程中这个力比该质点所受到的其它力要大很多，这时可以忽略其它力带来的效应。）

Collisions

When two objects collide, it is a good approximate in many cases to assumethat the forces due to the collision are much larger than any external forcespresent, so we can use the simplification model: the impulse approximation.

Collisions

（碰撞）

Elastic collision （弹性碰撞）Inelastic collision （非弹性碰撞） Perfectly inelastic collision

Momentum is conserved in all cases, but kinetic energy isconserved only in elastic collisions.

（动量在所有的碰撞过程中守恒，而动能仅在弹性碰撞中守恒）

1 1 1 1 2 2

1 1 2 2

component: 0 cos cos

component: 0 0 sin sin

i f f

f f

x m v m v m v

y m v m v

For the collision is elastic, we get the third equation forconservation of kinetic energy:

2 2 21 1 1 1 2 2

1 1 1

2 2 2i f fm v m v m v

Solve the set of equations composed of above three equations, we obtain:

1

1 1

22 2 2 21

1 1 1 1 1 1 1 12

sintan

cos

1 12 cos

2 2 2

f

f i

i f i f i f

v

v v

mm v m v v v v v

m

The Centre of Mass（质心）

The center of mass of a system of particles is defined as:

i ii

CM CM CM CM

i i i i i ii i i

m rr x i y j z k

M

i m x j m y k m z

M

For an extended object reads:1

CMr rdmM

The centre of mass of a homogeneous, symmetric body must

lie on an axis of symmetry.

The centre of mass of a system is different from its centre of

gravity. Each portion of a system is acted on by the gravitational

force. The net effect of all of these forces is equivalent to the

effect of a single force Mg acting at a special point called the

center of gravity. The centre of gravity is the average position of

the gravitational force on all parts of the object. If g is uniform

over the system, the centre of gravity coincides with the centre

of mass. In most cases, for objects or systems of reasonable size,

the two points can be considered to be coincident.

E.g. 0.1 A system consists of three particles located at the corners of a right triangle as in the figure. Find thecentre of mass of the system.

E.g. 0.2 A rod of length 30.0 cm has a linear density:

250.0 g/m 20.0 g/mx where x is the distance from one end, measured in meters.

(a)What is the mass of the rod?

(b)How far from the x = 0 end is its center of mass?

Motion of a System of Particles

i ii

CM

m rr

M

d

d dd

ii

CM iCM

rm

r tv

t M

iCM i t

i

drMv m p

dt

tCM i i t

i

dpMa m a F

dt

质心运动定律

E.g. 0.2 A rod of length 30.0 cm has a linear density:

250.0 g/m 20.0 g/mx where x is the distance from one end, measured in meters.

(a)What is the mass of the rod?

(b)How far from the x = 0 end is its center of mass?

Motion of a System of Particles

i ii

CM

m rr

M

d

d dd

ii

CM iCM

rm

r tv

t M

iCM i t

i

drMv m p

dt

tCM i i t

i

dpMa m a F

dt

质心运动定律

Outline of Rotational Motion

Rigid body model: A rigid body is any system of particles in whichthe particles remain fixed in position with respect to one another.

Rotation about a fixed axis: Every particles on a rigid body hasthe same angular speed and the same angular acceleration.

lim

lim

t

t

d

t dtd

t dt

rigid body 刚体 rotation about a fixed axis 定轴转动

Rotational kinematics

The rigid body under constant angular acceleration（常角加速度转动的刚体）

lim

lim

t

t

d

t dtd

t dt

（转动学）

Relations Between Rotational and Translational Quantities（转动量和平移量之间的关系）

translational motion 平动

Rotational Kinetic Energy

the moment of inertial2

i ii

I m r

rotational kinetic energy: 21

2RK I

For an extended system:

2 2 2

0lim

ii im

i

I r m r dm r dV

（转动动能）

The Rigid Body under a Net Torque（力矩作用下的刚体）

The net torque acting on the rigid body is proportional to

its angular acceleration.

t I

( 转动定律 )

The torque vector r F

The Rigid Body in Equilibrium（处于平衡状态的刚体）

Two conditions for complete equilibrium of an object:

0; 0t tF

translational equilibrium rotational equilibrium

（力矩）

Work and Energy in Rotational Motion （转动中的作功与能量）

dW d I d

dW

dt

Work done by a torque

The power of a torque

2 21 1

2 2

f f

i if iW d I d I I

Angular Momentum

The angular momentum of the particle relative to the origin is defined as:

L r p

dL dpr

dt dt

（角动量）

Conservation of Angular Momentum（角动量守恒）

The total angular momentum of a system remains constantif the net external torque acting on the system is zero.

一个刚体的运动，可以视为二种运动所组成，即质量

中心，受到所有外力作用所引起的运动，加上物体在外力作用

下，绕质心的转动。一个刚体的动能，系质心的平移运动的动

能，加上绕质心运动的转动动能。

引自《古典动力学》，吴大猷

刚体的运动总可以分解为质心的平动刚体的转动的合成。

A Brief Summary of Part I

① Kinematics of a Particle

② Newton’s Laws of Motion

③ Work and Energy

④ Momentum and Impulse

⑤ Motion of a System of Particles

⑥ Rotations of a Rigid Body about a Fixed Axis

The End of Part I