第二章牛顿力学的基本定律 - fudan university · 牛顿第一定律 - 惯性定律...

第二章牛顿力学的基本定律

Prepared by Jiang Xiao

物体运动与力的关系Prepared by Jiang Xiao

物体运动与力的关系

力是物体运动的原因，如果没有力物体运动将会停止。

力是改变物体运动速度的原因，维持物体运动不需要力。

任何物体都保持其静止或匀速直线运动状态，除非有力作用于其上迫使它改变这种状态。

Aristotle384BC - 322BC

Galileo1564 - 1642

Newton1642 - 1727


牛顿第一定律 - 惯性定律

‣牛顿第一定律：任何物体都保持其静止或匀速直线运动状态，除非有力作用于其上迫使它改变这种状态；

• 满足条件：只在（没有任何加速度）惯性参照系中满足，在加速参照系中不满足。

• 地球自转加速度（0.034 m/s2）、地球公转加速度（0.006 m/s2）、银河系转动加速度等。

• 无法证明、但是和现有所有实验相符。


牛顿第二定律

‣牛顿第二定律：物体所获得的加速度的大小与合外力的大小成正比，与物体的质量成反比，加速度的方向与合外力的方向相同，

• 满足条件：惯性参照系、物体运动速度远小于光速。

F = m a

[force] = [M ]

[L]

[T ]2N = kg m/s2


牛顿第三定律

‣牛顿第三定律：两物体间的相互作用力总是大小相等、方向相反并沿着同一直线，作用力 = - 反作用力。

Hero’s engine:http://techtv.mit.edu/videos/9142-heros-engine


http://techtv.mit.edu/videos/9142-heros-engine

http://techtv.mit.edu/videos/9142-heros-engine

牛顿第三定律

‣牛顿第三定律：两物体间的相互作用力总是大小相等、方向相反并沿着同一直线，作用力 = - 反作用力。

地球M⊕ = 6×1024kg

100

m

mapple = 0.5 kg

F

�F

F = mg = 5 N

1

2gt2 = h ) t = 4.5 s

�F = M�a�

) a� = � 5 N

6⇥ 1024kg= �8⇥ 10�25m/s2

1

2a�t

2 = �8⇥ 10�24 m


http://en.wikipedia.org/wiki/Kilogram

http://en.wikipedia.org/wiki/Kilogram

四种基本相互作用

相互作用相对强度作用力程

引力 10-40 长程 (~1/r2)

弱相互作用 10-14 ~ 10-15 m

电磁力 10-2 长程 (~1/r2)

强相互作用 1 ~ 10-18 m

�� : n ! p+ e� + ⌫̄e

�+ : p ! n+ e+ + ⌫e

mp = 1.67⇥ 10�27 kg

qp = �qe = 1.6⇥ 10�27 C

Fgravity = Gmpmp

r2⇠ 10�64 N

FEM =1

4⇡"0

qpqpr2

⇠ 10�28 N


开普勒第三定律

1 10距离太阳 (AU)

0.1

1

10

100

向心加速度

(AU

/yea

r^2)

海王星

水星

金星地球

火星

木星

土星

天王星

a = r!2 = r

✓2⇡

T

◆2

a / r�2

T 2 / r3

f = ma = Km

r2


万有引力定律

f = Gm1m2

r2with G = 6.67⇥ 10�11 m3/kg · s2

rather than increasing as As the body enters the interior of the earth (or other spherical body), some of the earth’s mass is on the side of the bodyopposite from the center and pulls in the opposite direction. Exactly at the center,the earth’s gravitational force on the body is zero.

Spherically symmetric bodies are an important case because moons, planets,and stars all tend to be spherical. Since all particles in a body gravitationallyattract each other, the particles tend to move to minimize the distance betweenthem. As a result, the body naturally tends to assume a spherical shape, just as alump of clay forms into a sphere if you squeeze it with equal forces on all sides.This effect is greatly reduced in celestial bodies of low mass, since the gravita-tional attraction is less, and these bodies tend not to be spherical (Fig. 13.3).

Determining the Value of GTo determine the value of the gravitational constant G, we have to measure thegravitational force between two bodies of known masses and at a knowndistance r. The force is extremely small for bodies that are small enough to bebrought into the laboratory, but it can be measured with an instrument called atorsion balance, which Sir Henry Cavendish used in 1798 to determine G.

Figure 13.4 shows a modern version of the Cavendish torsion balance. A light,rigid rod shaped like an inverted T is supported by a very thin, vertical quartzfiber. Two small spheres, each of mass are mounted at the ends of the hori-zontal arms of the T. When we bring two large spheres, each of mass to thepositions shown, the attractive gravitational forces twist the T through a smallangle. To measure this angle, we shine a beam of light on a mirror fastened to theT. The reflected beam strikes a scale, and as the T twists, the reflected beammoves along the scale.

After calibrating the Cavendish balance, we can measure gravitational forcesand thus determine G. The presently accepted value is

To three significant figures, Because the units of G can also be expressed as

Gravitational forces combine vectorially. If each of two masses exerts a forceon a third, the total force on the third mass is the vector sum of the individualforces of the first two. Example 13.3 makes use of this property, which is oftencalled superposition of forces.

m3>1kg # s22.1 kg # m>s2,1 N =G = 6.67 * 10-11 N # m2>kg2.

G = 6.674281672 * 10-11 N # m2>kg2

m2,m1,

m2m1

1>r 2.

404 CHAPTER 13 Gravitation

Gravitation pulls the small masses toward the largemasses, causing the vertical quartz fiber to twist.

The small balls reach a new equilibrium positionwhen the elastic force exerted by the twistedquartz fiber balances the gravitational forcebetween the masses.

Large mass 1m2 2Small mass 1m1 2

MirrorLaser beam

Quartzfiber

m1 Fg

m2

Scale

Laser

Fg

1

The deflection of the laser beam indicates how farthe fiber has twisted. Once the instrument iscalibrated, this result gives a value for G.

2

13.4 The principle of the Cavendish balance, used for determining the value of G. The angle of deflection has been exaggerated herefor clarity.

100 km100,000 km

Amalthea, one of Jupiter’s small moons, has arelatively tiny mass (7.17 3 1018 kg, only about3.8 3 1029 the mass of Jupiter) and weak mutualgravitation, so it has an irregular shape.

Jupiter’s mass is very large (1.90 3 1027 kg), sothe mutual gravitational attraction of its parts haspulled it into a nearly spherical shape.

13.3 Spherical and nonspherical bodies:the planet Jupiter and one of Jupiter’ssmall moons, Amalthea.


惯性质量和引力质量

牛顿第二定律：

万有引力定律： f = GMmgravity

r2= mgravityg

f = minertiaa

a = g ) mintertia = mgravity


万有引力定律

f = Gm1m2

r2with G = 6.67⇥ 10�11 m3/kg · s2


万有引力定律

f = Gm1m2

r2with G = 6.67⇥ 10�11 m3/kg · s2

木卫五

100 kmPrepared by Jiang Xiao

弹性力

F =a

r13� b

r7中性原子之间相互作用力：

形变弹性力：F = �kx

N

G

Pf


Mona LisaLouvre, Paris, France

摩擦力Leonardo da Vinci

1452 - 1519

(1)摩擦力与接触面积无关

(2)质量加倍、摩擦力加倍


摩擦力N

G

Pf

(1)动摩擦力与正压力成正比，与接触面积无关

(2)速度不大时，动摩擦力与速度无关

(3)静摩擦力可在零与一最大值之间变化，视相对滑动趋势而定。Charles Coulomb

1736 - 1806

动摩擦力： fk = µkN 0 fs µsN静摩擦力：

动摩擦系数静摩擦系数<


摩擦力148 CHAPTER 5 Applying Newton’s Laws

to the box (Fig. 5.19b) and gradually increase the tension T in the rope. At firstthe box remains at rest because the force of static friction also increases andstays equal in magnitude to T.

At some point T becomes greater than the maximum static friction force thesurface can exert. Then the box “breaks loose” (the tension T is able to break thebonds between molecules in the surfaces of the box and floor) and starts to slide.Figure 5.19c shows the forces when T is at this critical value. If T exceeds thisvalue, the box is no longer in equilibrium. For a given pair of surfaces the maxi-mum value of depends on the normal force. Experiment shows that in manycases this maximum value, called is approximately proportional to n; wecall the proportionality factor the coefficient of static friction. Table 5.1 listssome representative values of In a particular situation, the actual force ofstatic friction can have any magnitude between zero (when there is no other forceparallel to the surface) and a maximum value given by In symbols,

(magnitude of static friction force) (5.6)

Like Eq. (5.5), this is a relationship between magnitudes, not a vector relation-ship. The equality sign holds only when the applied force T has reached the criti-cal value at which motion is about to start (Fig. 5.19c). When T is less than thisvalue (Fig. 5.19b), the inequality sign holds. In that case we have to use the equi-librium conditions to find If there is no applied force asin Fig. 5.19a, then there is no static friction force either

As soon as the box starts to slide (Fig. 5.19d), the friction force usuallydecreases (Fig. 5.19e); it’s easier to keep the box moving than to start it moving.Hence the coefficient of kinetic friction is usually less than the coefficient ofstatic friction for any given pair of surfaces, as Table 5.1 shows.

1ƒs = 02. 1T = 02ƒs.1gFS

! 02

ƒs … msn

msn.

ms.ms

1ƒs2max,ƒs

ƒs

ƒs

Box moving; kinetic frictionis essentially constant.

Box at rest; static frictionequals applied force.

1 fs 2max

fs

fk

f

O T

n

w

(e)

No applied force,box at rest.No friction:

fs 5 0

n

w

T

Weak applied force,box remains at rest.

Static friction:fs , msn

fs

n

w

T

Stronger applied force,box just about to slide.

Static friction:fs 5 msn

fk

n

w

T

Box sliding atconstant speed.Kinetic friction:

fk 5 mkn

(a) (b) (c) (d)

5.19 (a), (b), (c) When there is no relative motion, the magnitude of the static friction force is less than or equal to (d) When there is relative motion, the magnitude of the kinetic friction force equals (e) A graph of the friction force magnitude as a function of the magnitude T of the applied force. The kinetic friction force varies somewhat as intermolecular bonds form and break.

ƒmkn.ƒk

msn.ƒs

Application Static Friction and Windshield WipersThe squeak of windshield wipers on dry glassis a stick-slip phenomenon. The moving wiperblade sticks to the glass momentarily, thenslides when the force applied to the blade bythe wiper motor overcomes the maximumforce of static friction. When the glass is wet from rain or windshield cleaning solution,friction is reduced and the wiper blade doesn’t stick.

动摩擦力：fk = µkN

0 fs µsN

静摩擦力：


摩擦力

mg

fN

a

↵

fs = µsmg cos↵

mg sin↵ > fs

) tan↵ > µs

) ↵ > arctanµs

多陡的坡木块开始滑动？

一旦开始滑动，其加速度是多少？

F = ma

mg sin↵� µkmg cos↵ = ma

) a = g sin↵(1� µk cot↵) = g sin↵(1� µk/µs)


力学相对性原理和伽利略变换

y

xO

r

v

S

S’

F0 = m0a0F = ma

F = F0

a = a0m = m0

r = r0 + ut

t = t0a = a0

伽利略变换：

S : (r, t) and S0 : (r0, t0)v0y’

x’O’R

r’ u =dR

dt

�r = �r0

�t = �t0

绝对时空观：


惯性系和非惯性系

牛顿第一定律：任何物体都保持其静止或匀速直线运动状态，除非有力作用于其上迫使它改变这种状态；

• 满足条件：只在（没有任何加速度）惯性参照系中满足，在加速参照系中不满足。

• 地球自转加速度（0.034 m/s2）、地球公转加速度（0.006 m/s2）、银河系转动加速度等。

惯性系：牛顿第一定律（惯性定律）成立的参照系。

非惯性系：牛顿第一定律（惯性定律）不成立的参照系。

a

a


平动非惯性系中的惯性力y

xO

a

a0 = a� a0

S y’

x’O’a0

S0

惯性系S中：

非惯性系S’中：F = ma = ma0 +ma0

F 6= ma0

F0 = F+ (�ma0) = ma0

惯性力 f*

a

a

f⇤ = �ma0


转动非惯性系中的惯性力!

rv

v0 = v + ! ⇥ r, a0 =dv0

dt

静止参照系：

dv

dt= a+ ! ⇥ v

dr̂

dt= !0✓̂

d✓̂

dt= �!0r̂

!0 = ! +v✓r

转动参照系：v = vr r̂+ v✓✓̂, a 6= dv

dt

dv

dt= v̇r r̂+ v̇✓✓̂ + vr

dr̂

dt+ v✓

d✓̂

dt

=

✓v̇r r̂+ v̇✓✓̂ +

vrv✓r

✓̂ � v2✓rr̂

◆+ !(vr✓̂ � v✓ r̂)

= a+ ! ⇥ v

转动参照系中的加速度，与无关!



rv

dr̂

dt= !✓̂

d✓̂

dt= �!r̂

v = vr r̂+ v✓✓̂, a = v̇r r̂+ v̇✓✓̂ 6= dv

dt

v0 = v + ! ⇥ r, a0 =dv0

dt

转动参照系：

静止参照系：

dv

dt= a+ ! ⇥ v

a0 =dv0

dt=

dv

dt+

d!

dt⇥ r+ ! ⇥ dr

dt= a+ ! ⇥ v + ! ⇥ v0

= a+ ! ⇥ v + ! ⇥ (v + ! ⇥ r)

= a+ 2! ⇥ v + ! ⇥ (! ⇥ r)

= a� 2v ⇥ ! � !2r



rv

dr̂

dt= !✓̂

d✓̂

dt= �!r̂

v = vr r̂+ v✓✓̂, a = v̇r r̂+ v̇✓✓̂ 6= dv

dt

v0 = v + ! ⇥ r, a0 =dv0

dt

转动参照系：

静止参照系：

dv

dt= a+ ! ⇥ v

a0 = a� 2v ⇥ ! � !2r

ma 6= F = F0 = ma0

F+ f⇤ = ma

f⇤ = m!2r+ 2mv ⇥ !

离心力科里奥利力


惯性离心力


f⇤ = m!2r+ 2mv ⇥ !



科里奥利力

!

rv

f⇤C

f⇤C

v


f⇤ = m!2r+ 2mv ⇥ !



地球上的科里奥利力的现象

北半球往右偏南半球往左偏

北半球：南北向河流河床右岸冲刷较左岸严重

竖直上抛：落到西边自由落体：落到东边


f⇤ = m!2r+ 2mv ⇥ !



地球上的科里奥利力的现象

北半球往右偏南半球往左偏

北半球：南北向河流河床右岸冲刷较左岸严重

竖直上抛：落到西边自由落体：落到东边


f⇤ = m!2r+ 2mv ⇥ !


西东

!

vz

赤道上上抛


科里奥利力的现象 - 巴黎大炮Prepared by Jiang Xiao

科里奥利力的现象 - 北半球台风和飓风Prepared by Jiang Xiao

科里奥利力的现象 - 南半球气旋Prepared by Jiang Xiao

科里奥利力的现象 - 洋流Prepared by Jiang Xiao

科里奥利力的现象 - 信风Prepared by Jiang Xiao

罗斯贝数 (Rossby Number)

f⇤ = m!2r+ 2mv ⇥ !


特征速度

特征尺度地球转速

R0 =U

2L⌦ sin�

纬度

Rossby Number:

U2

LU ⇥ ⌦

离心力：

科里奥利力：


科里奥利力的现象 - 昆虫定向Prepared by Jiang Xiao

第二章作业

习题：2.2, 2.9, 2.16, 2.20, 2.23

3月21日习题课上交


第二章 牛顿力学的基本定律 - fudan university · 牛顿第一定律 - 惯性定律...

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第二章牛顿力学的基本定律 - fudan university · 牛顿第一定律 - 惯性定律...