Chapter 2 Dynamics( 动力学） (Newton’s Laws of Motion)

§2-1 Newton’s Laws( 牛顿定律 )

§2-2 General Properties of Forces in Mechanics 力的基本性质

§2-3 Applying Newton’s Law of Motion 牛顿定律应用

The Fundamental Forces of Nature 四种基本力

--momentum--momentumvmp

Second lawSecond law

dt

pdF

dt

vmd

First lawFirst law

whenwhen 0Fv

Constant vector (including Constant vector (including zerozero))

§2-1 Newton’s laws§2-1 Newton’s laws

In inertial reference frame,In inertial reference frame,

baab FF

Third lawThird law

For m =const.dt

vdmF

am

Action-reaction lawAction-reaction law

iFFF

21 iFF

iamamam 21

am iam

NotesNotes

The principle of superposition of forces or principle of independence of forces.

　　 The components of second lawThe components of second law

Newton’s law are used only for Newton’s law are used only for inertial reinertial re

ference frameference frame

Cartesian

Coor. Sys.

zz

yy

xx

maF

maF

maF

NatureCoor. Sys.

tt

nn

maF

maF

dt

dvmF

vmF

t

n

2

§§2-2 2-2 General Properties of Forces

11..Weight ( (Earth gravity)

gmP

gmP

m

rr

mMGF ˆ

2

rr

MGg ˆ

2

r

Gravitational Force

The Fundamental Forces of Nature

2. Molecular force and elastic force2. Molecular force and elastic force

--The force between atoms--The force between atoms

0 r

F

d

ts rrF

ts

,,,,ss,,tt-- -- constant depend constant depend on experimenton experiment

kxF

-- -- Hooke’s law

xm0

0 xm

0F

F

F --- --- Restoring force

xx--- --- deformationdeformation

3. Frictional force between two solid surfaces 3. Frictional force between two solid surfaces

F

sf

0v

When 0v

Static frictional force

Ff s

Maximum Maximum static frictional force

Nf ssm

s ---- coefficient of static friction

WhenWhen 0v

Kinetic frictional force

Nf ----Coefficient of kinetic friction.Coefficient of kinetic friction.

vF

f

s

What is the reason of friction What is the reason of friction ？？？？

4. Fluid friction4. Fluid friction

2vcF (at high speed)(at high speed)

(at low speed)vF

(Drag force)

Experienced formula

5. The Fundamental Forces of Nature

Electromagnetic Force

Gravitational Force Action at a distance force

Strong Force (~104N, acting on nuclear particles, hold the atomic nucleus together)

Distance of force <10Distance of force <10-15-15mm

Short distance forceShort distance force

Weak Force (~10-2N acting on most elementary particle, only manifest itself in certain kinds of radioactive decay reactions )

Distance of force <10<10-17-17mm

Short distance forceShort distance force

Solving procedureSolving procedure

Select a body to which Newton’s laws Select a body to which Newton’s laws

will be applied. will be applied.

§§2-3 2-3 Applications of Newton’s Law

Draw a free-body diagram. Draw all thDraw a free-body diagram. Draw all th

e forces acting on the chosen body. Anae forces acting on the chosen body. Ana

lyze the accelerations of the chosen bodlyze the accelerations of the chosen bod

y.y.

Set up a coordinate system. Set up a coordinate system.

Write the component equations of NewWrite the component equations of New

ton’s Second Law for each body. Solvton’s Second Law for each body. Solv

e the equations to find unknown quane the equations to find unknown quan

tities. tities.

Analyze the results if necessary.Analyze the results if necessary.

a

[[Example1Example1] A incline of mass ] A incline of mass M M is placed on is placed on a table and a block of mass a table and a block of mass mm is put on is put on MM. S. Suppose all contact surface are frictionless anuppose all contact surface are frictionless and angle d angle is known. Find the Acce. of is known. Find the Acce. of MM , an , and the Acce.of d the Acce.of mm with respect to with respect to M.M.

M

mm

Ma

gM

'N

MNN

gm

Ma

Ma

gM

'N

MN

a mN

gm

Ma

For For MM:: MMaN sin'

ForFor m m::

)cos(sin MaamN

sincos amNmg

[[SolutionSolution]]

'NN andand

---- MM with respect to the earth with respect to the earth

---- m m with respect to with respect to MM

Ma

a

x

y

We getWe get

2sin

cossin

mM

mgaM

2sin

sin)(

mM

gMma

[Example 2] A ball of mass m is sinking in the water. Suppose the drag force exerted on the ball is , the floating force is F. The initial condition is that the ball is at rest on the surface of water at the beginning.

vkf

Find (1) the speed of the ball, (2) the sinking

distance of the ball at any time.

Solution

(1)(1) Draw all the forces acting on the ball. Draw all the forces acting on the ball. Analyze its accelerationAnalyze its acceleration

X

o

)(tv

mg

f

F

a

Apply New.’s Law to the ball

dt

dvmFkvmg

Separate the variables

m

dt

Fkvmg

dv

Integrate in both side of “=”

ttvdt

mFkvmg

dv0

)(

0

1

Discuss

When t ， v(t) ，When t ， Tv

k

Fmgtv

)(

vT -- terminal speed

We can get)(

)()(

tm

k

ek

Fmgtv

1

(2) From dt

dxv

We have vdtdx dtek

Fmg tm

k

)1()(

Make a integration, we can get

)1()()(

2

tm

k

ek

Fmgmt

k

Fmgx

[Example 3] A small block of mass m slides on a horizontal frictionless surface as it travels around the inside of radius R. Suppose its initial speed is v0 and the between it and the wall is known. Find its speed at any time.

R

A

v0

Solution

Set up nature coordinate system

n̂

t̂

Draw forces Draw forces Analyze accelerationsAnalyze accelerations

f

v

t̂

n̂

na

taN

From

We can get the dynamics equations

R

vmN

2

Normal direction

Tangential directiondt

dvmN

R

vmmaF

dt

dvmmaF

nn

tt

2

Solve the equations, we have

tR

vv

tv0

0

1)(

v

[Example 4 ] One end of a weightless rope of length l is fixed on a nail. Its another end connects with a ball of mass m. Pull the ball and make the rope horizontally first. Then let the ball fall down. Calculate the speed of the ball and the tension of the rope at any angle.

Solution

Set up nature coordinate system

Draw forces Draw forces Analyze accelerationsAnalyze accelerations

vT

mg

n l

vmmgT

2

sin

Normal direction

Tangential direction

dt

dvmmg cos

…..

…..

Rewrite Eq.

dt

dvg cos

dt

d

d

dv

d

dv

l

v

Separate the variables dglvdv cos

After integration, we have

sin2glv

Substitute this solution to Eq.

we have sin3mgT

[ 例 5] 均质软绳单位长质量为，开始时盘绕在桌面。若以恒定加速度 a 竖直向上提绳，求当提起高度为 y 时，作用在绳上的力？若以恒定速度 v 竖直向上提绳，情况又如何？（设 t=0 时， y=0,v=0 ）

gayF 3

ygvF 2

[ 例 6] 直九型直升机的每片旋翼长 L ，质量m ，若视其为均匀薄片，求旋翼以角速度 旋转时根部所受拉力？

Lm 2

2

1

or dr

F+dF FdmadF n

drr 2

dm 对 o 轴的拉力 :

整个旋翼对 o 的拉力 :

dFF

v

0a

　　§§2-4 2-4 非惯性系 惯性力非惯性系 惯性力

gm

T 对地对地

对车对车

gm

T

对地对地

对车对车

0Tgm

0amTgm

0Tgm -------- 牛律不成立牛律不成立0a

牛律适用的参照系牛律适用的参照系 ---- 惯性系惯性系 ,,

反之反之 ---- 非惯性系非惯性系

一一 .. 惯性系和非惯性系惯性系和非惯性系

说明 : 一参照系是否是惯性系，要靠实验判断

在惯性系内进行任在惯性系内进行任何力学实验均不能何力学实验均不能确定该系作匀速直确定该系作匀速直线运动的速度线运动的速度

-------- 力学相对性原理力学相对性原理

相对于惯性系作相对于惯性系作匀速直线运动匀速直线运动的参照系都的参照系都是惯性系是惯性系，，作作变速运动变速运动的参照系为非惯性的参照系为非惯性系系

船走吗 ?

'x

'y

'0

'K0a 二二 . . 直线加速参照系直线加速参照系

x

y

0

K 'a

KK ：：惯性系惯性系KK ：：非惯性系非惯性系 (( 车车 ))

：： ------KK 对对 KK0a

：： ------ 球对球对 KK 'a

则球对则球对 KK ：： 0' aaa

在在 KK 系系 amF

)'( 0aam

'amFF

惯

有有 ')( 0 amamF

定义：定义： -------- 惯性力惯性力0amF

惯

-------- 非惯性系中的牛二律牛二律

惯性力惯性力与真实力有区别与真实力有区别说明：

在非惯性系中研究物体的运动时，才在非惯性系中研究物体的运动时，才考虑惯性力考虑惯性力

'amF

[[ 例例 55]] 用惯性力的方法解用惯性力的方法解 [[ 例例 3]3]

M

m

am

N

gm

惯F

Ma

gM

'N

MN

解：解： 以劈为参照系以劈为参照系劈和木块的受力如图劈和木块的受力如图

对对 mm

水平方向水平方向 cossin ammaN M 垂直方向垂直方向 sincos amNmg

对对 MM 0sin MMaN 即可解得即可解得

MamF

惯

三三 . . 转动参照系转动参照系1. 匀速转动参考系中的惯性离心力

一人站在一转盘边缘随盘以 转动，一人站在一转盘边缘随盘以 转动， 从地面观察，人作匀速圆周运动，从地面观察，人作匀速圆周运动，

nRanˆ2

向心力由摩擦力提供，向心力由摩擦力提供，

ns amf

nRm ˆ2R

sf

R

在转盘上观察，人静在转盘上观察，人静止，即止，即 0'a

namF

惯

惯Fff s

则则

惯F

惯F

sf

nam nRm ˆ2

-------- 惯性离心力惯性离心力

引入：

0 am