物理Lectures3and4第三,四讲

Lectures

3

and

4(第三,四讲)Newtonian

Mechanics

(

力学)【

or

Newtonian

dynamics

(

动力学)】In

Newtonian

dynamics(动力学)------------Chapters

5-6：the

effect

offorces

acting

on

objects(作用在物体上的力)will

be

the

main

concern.Newtonian

Kinematics(运动学)-------------chapters

2-4：Only

Dealswiththe

relationships

among

position(x),velocity(v),and

acceleration(a)andtime

(t).

Four

variables

(x,

v,

a,

t).Under

what

circumstances

Newton’s

2nd

law

holds:𝐹

=

𝑚𝑎𝐹

≠

𝑚𝑎---------------------inertial

reference

frame (惯性参考系中成立)---------------------Non-inertial

reference

frame(非惯性参考系中不成立)University

Physics

I1Q:

What

is

the

cause

for

the

changeof

the

status

of

motion

of

an

object?A:

It

is

the

force

exerting

on

it.In

Newtonian

dynamics(动力学，Chapters

5-6）：the

effect

of

forcesacting

on

objects(作用在物体上的力)will

be

the

main

concern.University

Physics

I2Lectures

3(第三讲)：If

objects

are

very

small

(at

atomic

10-10

m

or

nano

10-9m

scale),Newtonian

mechanics

must

be

replaced

byquantum

mechanics(量子力学).Chapter

5

Force

and

Motion

(I)

Newtonian

Mechanics

describes

therelationship

between

force

and

motion

basedon

the

laws

discovered

by

British

physicistSir

Isaac

Newton

(1642-1727,明末清初).《自然哲学的数学原理》1687(

)At

very

high

speeds

(an

appreciable

fraction

of

the

speed

oflight

c),

Newtonian

mechanics

must

be

replaced

byEinstein’s

special

theory

of

relativity(

狭义相对论).University

Physics

I3ForceForce

causes

the

acceleration

of

an

object(or

changes its

momentum).Force

can

be

exerted

on

an

object

by

otherobjects,either

physically

touched(如绳子拉力)or

at

a

distance(如电磁力).Force

is

a

vector

tywith

bothmagnitudeand

direction.University

Physics

I4Net

Force(净力)University

Physics

I5Net

force

is

the

sum

of

allforces

on

a

body.Net

force

is

an

overall

effect,

not

a

real

force.In

the

case

of

𝐹net

=0,the

forces

cancel

with

each

other；a

status

called

equilibrium(平衡状态).But

each

individual

force

still

exists.Only

the

net

effect

on

the

body

is

like

no

forces

on

it

atall.Principle

of

superposition

for

force(力的叠加原理)1.Newton’s

Law

(Law

of

Inertia，惯性定律)If

no

net

force

acts

on

a

body

(

𝐹net

=

0

),

then

the

body‘svelocity willnot

change(or

to

say

its

momentum,

p=mv,

will

not

change);

that

is,

the

body

cannot

beaccelerated.Newton’s

Law

(

第一定律--Law

of

Inertia，惯性定律)：Given

all

forces

on

a

body

balance

one

another,if

the

body

is

at

rest

(stationary ),

it

stays

at

rest;if

it

is

moving(运动),it

continues

to

move

at

constant

velocity.In

such

cases,

thebody

is

called

in

equilibrium(平衡).University

Physics

I67Draw

Free-Body

Diagram(分析受力图)In

a

coordinate

system,

we

represent

the

body

as

a

dot,

eachforce

on

the

body

is

a

vector

arrow

with

its

tail

on

the

body.External

force(外力):A

collection

of

two

or

more

bodies

iscalled

a

system(质点系),and

any

force

on

the

bodies

insidethe

system

from

bodies

outside

the

system

is

called

anexternal

force.Internal

force(内力):are

forces

between

two

bodies

inside

thesystem.Internal

forces

cannot

accelerate

the

system.If

the

bodies

are

rigidly

connected,

then

we

can

treat

thesystem

as

one

composite

body,

and

the

net

force

𝐹net

on

it

isthe

vector

sum

of

all

external

forces.University

Physics

I8Sample

ProblemIn

a

two-dimensional

tug-of-war(拉力比赛，拔河),Alex,

Betty,and

Charles

pull

horizontally

on

an

automobile

tire

at

theangles

shown

in

the

overhead

view

of

the

figure.The

tire

remains

stationary( )

in

spite

ofthe

three

pulls.Alex

pulls

with

force

𝑭𝑨

of

magnitude

220

N,and

Charles

pulls

with

force

𝑭𝑪

ofmagnitude

170

N.The

direction

of

𝑭𝑪

is

not

given.What

is

the

magnitude

of

Betty's

force

𝑭𝑩

?Solution:

equilibrium(平衡状态)

FB

FA

FCFA

FB

FC

0A

C0

F

cos

133

F

cos

Force

components

in

x-axis:FBX

FAX

FCX0

(

220

N)(cos133

)

(170

N)

cos 170

N

91

(220

N)(cos133

)

cos

28.04F.B.D.Force

components

in

y-axis:A

CF

sin

133

F

sin

,90

)

F

sin(B

FB

(220

N)(sin

133

)

(170

N)sin

28.04FB

241

NTherefore,FBy

FAy

FCyF.B.D.University

Physics

I10Some

Particular

ForcesGravitational

force(引力or

重力),Normal

force(法向支撑力or正压力),Frictional

force(摩擦力),Tension(

张力).University

Physics

I11The

Gravitational

Force（地球引力，重力）Magnitude:Direction:Fg

mgPointing

towards

the

center

of

EarthThe

gravitational

force

𝐹𝑔

on

a

body

is

the

forceexerted

by

Earth

to

pull

the

body

down

to

the

ground.General

Physics

I12Weight（重量）:The

weight

W

of

a

body

is

equal

to

the

magnitude

of

thegravitational

force

on

the

body.The

Normal

Force

(法向支撑力or正压力)When

a

body

presses

against

asurface(表面),the

surface

(even

a

seemingly

rigid

surface)deforms(形变)and

pushes

on

thebody

with

a

normalforce，which

isperpendicular

to

the

surface.F.B.D.The

surface

deforms这个表面发生形变The

normal

forceis

perpendicular

tothesurfaceUniversity

Physics

I13Frictional

Force(or

friction

摩擦力)If

we

slide

or

attempt

to

slide

a

body

over

a

surface,

the

motion

isresisted

by

a

bonding

between

the

body

and

the

surface.

Theis

considered

to

be

a

single

force

𝑓,

called

thefrictional

force,

or

simply

friction.This

force

is

directed

along

the

surface,

opposite

the

direction

ofthe

intended

motion.In

ideal

cases,to

simplify

situation,friction

is

assumed

to

benegligible(namely

“the

surface

is

frictionless”,称作“光滑表面”).University

Physics

I14Tension

(张力)The

tension

force

pulls

on

a

bodyalong

a

taut

cord(绷紧的绳子).The

cord

is

often

assumed

to

be

massless

and

un-stretchable(不可拉伸).A

string

has

a

single

tension

force

.The

direction

depends

on

thebody

on

which

this

force

acts

upon.We

usually

assume

a

pulley(滑轮)to

be

frictionless

and

massless.The

tension

forces

at

the

two

ends

of

thecord

are

equal

in

magnitude152.

Newton’s

Second

Lawalong

that

same

axis,

and

not

by

forcecomponents

along

any

other

axis.The

netforce

on

abody

is

equal

tothe

product

of

the

body's

mass

and

the

acceleration

of

the

body.Fnet

ma

(Newton's

second

law)

.𝐹net,𝑥

=

𝑚𝑎𝑥,𝐹net,𝑦

=

𝑚𝑎𝑦

,𝐹net,𝑧

=

𝑚𝑎𝑧The

acceleration

component

along

agivenorthogonal

axis

iscaused

onlyby

the

sum

of

the

force

componentsThis

expression

of

Newton’s

2ndlaw

in

the

right

is

what

welearned

in

high

school

physics.More

scientifically,

we

can

say:Net

force

acting

on

an

objectchanges

its

momentum𝑝

=

𝑚𝑣,

or

force

equals

thechanging

rate

of

momentumwith

respect

to

time,

i.e.𝑑𝑝【𝑭

=𝑑𝑡

=𝑑𝑡𝑑(𝑚𝑣)𝑑𝑣=

𝑚

𝑑𝑡

=

𝑚𝒂𝐹

=

𝑑𝑝𝑑𝑡第二定律16University

Physics

I17Unit

of

Force（力的单位）In

SI

unit,the

unit

of

force

is

newton.1N

(1kg)

(1

m/

s2

)

1

kgm/

s2General

Physics

I18MassThe

mass

of

a

body

is

an

intrinsic

characteristicthat

relates

a

force

on

the

body

to

the

resultingacceleration.Mass

is

a

scalar.It

is

a

measurement

of

an

object’s

inertia.The

greater

an

object’s

mass,

the

more

the

objectresists

being

accelerated.University

Physics

I193.

Newton’s

Third

LawWhen

two

bodies

interact,

the

forces

on

the

bodies

from

eachother

are

always

equal

in

magnitude

and

opposite

in

direction.Third-law

force

pair:

FBC

FCB𝐹𝐵𝐶

is

the

force

on

book

from

crate.𝐹𝐶𝐵

is

the

force

on

crate

from

book.Action

(force)equals

reaction

(force)---作用力等于反作用力General

Physics

I20Conceptual

Question:How

dose

a

horsetry

to

use

physicsto

esc

work!Applying

Newton’s

Laws

to

solve

problemsStep

1:

yze

the

forces

exerted

on

the

object

of

the

study.Draw its

free-body-diagram(画受力图)Step

3:

yze

its

status

of

motion(including

its

v,

a)

andfind

out

the

initial

conditions(找出初始条件)Step

2:Construct

a

coordinate

system(建立坐标系)convenient

forposition

to

get

𝐹𝑥

,

𝐹𝑦Step

4:

Apply

Newton’s

2nd

law，to

establish

equationsforx-direction,

and

y-direction

to

get

𝑎𝑥,

𝑎𝑦University

Physics

I21Sample

Problem

5-3The

hanging

block

H

falls

as

the

sliding

block

S

accelerates

to

the

right.University

Physics

I22The

figure

shows

a

block

S

(the

Slidingblock)

with

mass

M

=

3.3

kg.

The

blockis

free

to

move

along

a

horizontalfrictionless

surface

such

as

an

air

table.The

block

on

the

table

is

connected

bya

cord

that

wraps

over

a

frictionlesspulley

to

a

second

block

H

(theHanging

block),

with

mass

m

=

2.1

kg.The

cord

and

pulley

have

negligiblemasses

compared

to

the

blocks

(theyare

“massless”).HS23Find

(a)

the

acceleration

of

the

sliding

block,the

acceleration

of

the

hanging

block,

andthe

tension

in

the

cord.SOLUTION:F.B.D.The

cord

is

taut.

Therefore,

twoblocks

have

the

same

magnitudeof

acceleration

𝒂

along

the

cordat

any

moment.The

Acceleration

of

block

S

pointsto

the

right;

acceleration

of

blockH

points

downward.Tensions

at

two

ends

of

the

cordhave

the

same

magnitude

𝑻.University

Physics

I24T

Maponent

of

block

S:g

13

NM

mM

mT

M

ma

m

g

3

.

8

m

/

s

2If

𝑚

→

0,

𝑇

and

𝑎 →

0.ponent

of

block

H:T

mg

m(a)Two

unknown

variables

，T

and

𝒂，can

be

solved𝑀

→

0,

𝑇

→

0,

and

𝑎

→

𝑔.25Sample

Problem

5-4Like

ndulum(单摆)，a

small

ball

withmass

of

m

connected

to

a

cord

of

length

lis

released

from

horizontal

level(𝜃=0°).𝜃

=

0°𝜃𝑇𝜃𝑚𝑔𝑣𝑚𝑔

cos

𝜃

=

𝑚𝑎𝑡

=

𝑚𝑑𝑣𝑑𝑡-------Eq1Along

tangent(切向):Along

normal(法向)(or

radial径向):𝜃

𝑛𝑇 −

𝑚𝑔

sin

𝜃

=

𝑚𝑎 =

𝑚𝑣2𝑙----Eq2𝜃

=

𝜃1What

will

be

the

speed

of

the

ball

whenthe

cord

sweeps

an

angle

of

𝜃

=

𝜃1?What

is

the

tension

force

of

the

cord

atthis

angle

of

𝜃

=

𝜃1From

Eq.1,

we

can

obtain

how

the

speed

varies

with

the

angle𝜃

=

0°𝜃𝑇𝜃𝑚𝑔𝑣Along

tangent(切向):

𝑚𝑔

cos

𝜃

=

𝑚𝑎𝑡

=

𝑚𝑑𝑣𝑑𝑡𝜃

=

𝜃1So

that

𝑑𝑣

= 𝑔

cos

𝜃

𝑑𝑡𝑑𝑠

=

𝑣𝑑𝑡

=

𝑙𝑑𝜃𝑑𝜃𝑑𝑣

= 𝑔

cos

𝜃

𝑑𝑡

= 𝑔

cos

𝜃𝑑𝑠

=𝑔

cos

𝜃𝑙𝑑𝜃𝑣So

that

𝑣𝑑𝑣

=

𝑔𝑙

cos

𝜃

𝑑𝜃𝑣Integrate

both

sides:𝑣

𝑣𝑑𝑣

=0𝜃

𝑔𝑙

cos

𝜃

𝑑𝜃02We

get:

1𝑣2

=

𝑔𝑙

sin

𝜃,→

𝑣

= 2𝑔𝑙

sin

𝜃1.

What

will

be

the

speed

of

the

ball

when

the

cord

sweeps

an

angle

of

𝜃

=

𝜃1?26Since

𝑑𝑠

=

𝑣𝑑𝑡 →

𝑑𝑡

=

𝑑𝑠/𝑣,also

the

arc

length

𝑑𝑠

=

𝑙𝑑𝜃From

Eq.2,

we

can

obtain

how

the

tension

of

thecord

at

the

angle

of

θ𝜃

=

0°𝜃𝑇𝜃𝑚𝑔𝑣𝜃

=

𝜃1𝑑𝑠

=

𝑣𝑑𝑡

=

𝑙𝑑𝜃𝑑𝜃Replace

v

by

the

previous

result

𝑣

=2𝑔𝑙

sin

𝜃Along

normal(法向),or

radial(径向):𝜃

𝑛𝑇 −

𝑚𝑔

sin

𝜃

=

𝑚𝑎 =

𝑚𝑣2𝑙We

get:

𝑇𝜃

=

3𝑚𝑔

sin

𝜃𝜃𝑇 =

𝑚𝑔

sin

𝜃

+

𝑚𝑣2𝑙2.

What

is

the

tension

force

of

the

cord

at

this

angle

of

𝜃

=

𝜃1When

𝜃

=

0°, 𝑇𝜃

=

0When

θ

=

90°,

𝑇𝜃

=

3𝑚𝑔27General

Physics

I28Sample

Problem

5.06 Force

withinan

elevator

cabIn

the

figure, senger

of

mass

𝑚

=

72.2

kgstands

on

a

platform

scale

in

an

elevator

cab.

Weare

concerned

with

the

scale

readings

when

thecab

is

stationary,

and

when

it

is

moving

up

ordown.F.B.D.Find

a

general

solution

for

the

scale

reading,

whateverthe

vertical

motion

of

the

cab.Fnet

N

mg

maN

m

(g

a)𝒂

=

𝟎，𝑵

=

𝒎𝒈，when

the

cab

is

stationary

or

moving

at

constant

velocity𝒂

>

𝟎，𝑵

>

𝒎𝒈，when

the

cab

is

accelerating

upward.𝒂

<

𝟎，𝑵

<

𝒎𝒈，when

the

cab

is

accelerating

downward.(4)

𝒂

=

−𝒈，𝑵

=

𝟎,which

is

called

weightless(失重).N

m

(g

a)The

scale

reading

isWhat

if

𝒂

𝒊𝒔

𝒍𝒆𝒔𝒔

𝒕𝒉𝒂𝒏

−

𝒈

?

(𝒂

<

−𝒈)University

Physics

I29Setting

up

rectangular

coordinate

systems

and

solving

problemsUniversity

Physics

I30Coordinate

systems

can

be

arbitrarily

(or

intentionally)

set

upto

solve

physical

problems.But

selecting

a

correct

one

should

be

more

convenient

insolving

problems.Physical

phenomena

(results

or

solutions)should

beindependent

on

the

intentionally

selected

coordinate

systems.物理结果与怎样选取坐标系应该是无关的！Sample

ProblemA

cord

holdsstationary

a

blockof

mass

m

=

15

kg,on

a

frictionlessplane

that

isinclined

at

angleθ=

27°F.B.D.We

now

cut

the

cord.As

the

block

then

slides

down

the

inclined

plane(斜面),does

itaccelerate?If

so,what

is

itsacceleration?Think?There

is

no

movement

of

the

sliding

block

perpendicularto

the

inclined

plane.

Therefore,

Acceleration

must

beparallel

to

inclined

plane.31Fx

mg

sin

maxax

gsin

(9.8

m/s2

)

(sin

27

)=−4.4

m/s2The

acceleration

points

down

along

the

inclined

plane.ponent(Method

1)

We

choose

our

coordinate

system

with

axes

parallel

and

perpendicular

to

theinclined

plane.The

arrangement

aligns

mostforces

along

axes,

simplifying

theforceresolving

procedure32Method-2

Alternatively,

We

set

up

our

rectangular

coordinate

system

with

axes

along

horizontal

and

vertical

directions.There

is

no

movement

perpendicular

to

the

inclinedplane.Therefore,acceleration

𝑎

must

be

along

theramp(斜坡),assuming

down

along

ramp.ponent𝐹𝑥

=

𝑁𝑥

=

−𝑁

sin

𝜃𝑎𝑥

=

−𝑎

cos

𝜃ponent𝐹𝑦

=

𝑁𝑦

−

𝑚𝑔

=

𝑁

cos

𝜃

−

𝑚𝑔𝑎𝑦

=

−𝑎

sin

𝜃𝜃xy𝑎𝑥University

Physics

I33𝑎𝑎𝑦We

get

the

same

answer

for

acceleration

as

we

did

in

method-1.ponent𝐹𝑥

=

𝑚𝑎𝑥−𝑁

sin

𝜃

=

−𝑚𝑎

cos

𝜃(1)𝑎

cos2

𝜃

+

sin2

𝜃 =

𝑔

sin

𝜃𝑎

=

𝑔

sin

𝜃(1)∙

cos

𝜃

+(2)∙sin

𝜃𝜃xySolving

in

horizontal-vertical

coordinate

systemUniversity

Physics

I34ponent(2)𝐹𝑦

=

𝑚𝑎𝑦𝑁

cos

𝜃

−

𝐹𝑔

=

𝑚(−𝑎

sin

𝜃)𝑎

𝑥𝑎𝑎𝑦Under

what

circumstances

Newton’s

2nd

law

holds:1.

𝐹

=

𝑚𝑎---------------------inertial

reference

frame

(惯性参考系中成立)2.

𝐹

≠

𝑚𝑎---------------------Non-inertial

reference

frame(非惯性参考系中不成立)University

Physics

I3536Inertial

Reference

Frames(惯性参考系)An

inertial

reference

frame

is

one

in

which

Newton's

laws

hold.(𝐹

=

𝑚𝑎)A

reference

framewith

acceleration【有加速度的参考系，

including

rotatingframe(旋转的参考系,即是有向心加速度)】is

anon-inertial

reference

frame(非惯性参考系)e.g.

an

accelerating

elevator

is

a

non-inertial

frame.A

reference

frame

at

rest(

的)orwith

constant

velocity(或匀速运动的)is

an

inertial

reference

frame(惯性参考系).Ground

is

usually

assumed

as

an

inertial

frame,

neglecting

Earth’sastronomical

motion.Observing

from

a

non-inertial

referenceframe(非惯性参考系),Newton‘s

laws

do

not

hold(

定律不成立)-----to

discuss

it

laterUniversity

Physics

I37As

the

saying

goes:水火不相容？Conclusion：There

is

no

absolute

correctness

or

absolute

truth，except

this

statement(没有绝对真理).Correctness

is

relative(真确性都是相对的)Newton‘s

2nd

law

is

true

(𝐹

=

𝑚𝑎),if

it

is

observed

in

inertial

frame(

定律在惯性参考系中是真理)In

non-inertial

reference

frame，it

is

not

true！(在非惯性参考系中，

定律不是真理)True

gold

does

not

fear

the

fiery

真金不怕火炼？Observing

from

a

non-inertial

reference

frame(非惯性参考系),Newton‘s

laws

do

not

hold(

定律不成立)What

does

“Newton‘s

laws

do

hold”

mean(

定律成立是什么意思)?If

there

is

a

net

force,there

is

an

acceleration----------有力就有加速度！(无力就无加速度)e.g.

Look

into

Newton’s

2nd

lawWhat

does

“Newton‘s

laws

do

not

hold”

mean(

定律不成立是什么意思)?2.

If

there

is

a

net

force,there

is

no

acceleration----------有力却没加速度(有受到净力但不动)！1.If

there

is

no

force,there

is

an

acceleration---------------没力却有加速度(不受力却会动)38Observing

from

a

non-inertial

reference

frame(非惯性参考系),Newton‘s

laws

do

not

hold(

定律不成立)e.g.

Look

into

Newton’s

2nd

law1.

If

there

is

no

force,there

is

an

acceleration---------------没力有加速度！Bus

StopNo

horizontal

𝐹𝑥

Observing

𝑎𝑥Conclusion:without

force,

but

acceleratingi.e.

Newton’s

2nd

law

breaksNo

horizontal

𝐹𝑥

Conclusion:without

force,

but

acceleratingi.e.

Newton’s

2nd

law

breaks39Observing

from

a

non-inertial

reference

frame(非惯性参考系),Newton‘s

laws

do

not

hold(

定律不成立)e.g.

Look

into

Newton’s

2nd

law2.If

there

is

a

net

force,there

is

no

acceleration----------有力没加速度！AlexBettyCharlesAlex

asks

:

what

do

you

feel

at

the

bottom

of

your

feet?Betty

answers:

I

feel

some

force

acting

on

thebottom

of

my

feet?Alex:

Then

why

you

are

not

moving

towards

me?

Newton’s

law

does

not

hold?40General

Physics

I41

𝜽

MGroundWedgeBlockmGiven

the

inclined

surface

ofthe

wedge

and

the

groundsurface

are

both

frictionless,wedge

and

the

blockfind

accelerations

of

thewithrespect

to

the

ground.Tutorial

problem

1（相对运动---chapter4）As

shown

in

figure,

a

block

of

mass

m

is

placed

on

the

inclinedplane

of

a

wedge.

The

wedge

is

then

placed

on

the

ground.The

mass

of

wedge

is

M

and

the

angle

between

its

inclinedsurface

and

ground

is

𝜽.What

about

the

case

that

two

objects

move

with

a

relative

acceleration?University

Physics

I