7&8 Work and Energy
Chapter 7 & 8
Energy and Conservation Law
1,Work and Energy
2,Kinetic Energy （动能） & Work- Energy Principle
3,Conservative（保守） and Nonconservative
Forces
4,Potential Energy（势能）
5,Conservation of Energy
7&8 Work and Energy
New words
Conservative and non-conservative forces（保守力与非保守力）
Potential energy（势能）
Conservation of mechanical energy
（机械能守恒）
Conservation of energy（能量守恒）
7&8 Work and Energy
§ 7-1&7-3 Work and Power (P147-156)
Work W Provides a link between force and energy.
1,Work done by a constant force(P148)
Φ Φ xFrFrFW x )c o s(?
力的 空间累积 效应,
WrF,
Work is product of the
magnitude of the
displacement of the force
parallel to the displacement.
7&8 Work and Energy
if
4321 FFFFF
321
321
WWW
rdFrdFrdFrdFW
The work done by several forces = algebraic
sum of the work done by the individual force.
2 Work done by several forces
合力的功 = 分力的功的代数和
7&8 Work and Energy
3 work done by a general variable force(变力 P152-156)
a
b
1.无限分割路径；
The path is divided into short intervals,
2.以直线段代替曲线段；
The intervals could be treated as straight lines.
3.以恒力的功代替变力的功；
During each small interval,the force is
approximately constant.
4.将各段作功代数求和；
The total work done is the sum
of all terms.
2F
2?
1F
1?
r?
7&8 Work and Energy
111 c o s?rFW
222 c o s rFW?
iii rFW?c o s
nnn rFW?c o s+）
i
n
i
WW
1 ii
n
i
rF c o s
1
7&8 Work and Energy
0?r?Let
ii
n
ir
rFW
c o slim
10
c o sF d rba b
a rdF
BABA rFrFW dc o sd
co sF
Ar Brrd
r
o
When a particle moves from A to B along a
curve path,the total work done by the force
equals the integral from A to B.
7&8 Work and Energy
a n d, kFjFiFF zyx, kdzjdyidxrd
f
i
f
i
f
i
z
z
z
y
y
y
x
x
x
B
A
zyx
B
A
dzFdyFdxF
dzFdyFdxFrdFW )(
4 Special expression in scalar product(p154)
zFyFxFW zyx ddd
zyx WWWW
7&8 Work and Energy
5 Work is a scalar quantity,no direction,
0d,900 W?
sFrFW dc o sdc o sd
0d,1 8 090 W?
rFW dd
0dd90 WrF
F?
r?d?
iF
1dr?
ir
d B*
*
i?
1?
A 1F?
positive work;
negative work.
zero work;
7&8 Work and Energy
6 Geometrical Representation of Work:
The work done by a force
equals the area under the F
versus x curve —— 变力曲线与位移轴在极限 x1,x2 之间所包围的面积,
co sF
Ar Brrd
r
o
Example 7-5 and 7-6
7&8 Work and Energy
质量为 10kg 的质点，在外力作用下做平面曲线运动，该质点的速度为 jit 164 2v
解 24dd ttxxv ttx d4d 2?
16dd tyyv ty 16?
ttmF xx 80dd v 0dd?tmF yy v
J 1 20 0d3 2021 3 tt
在质点从 y = 16m 到 y = 32m 的过程中，外力做的功。求例
,开始时质点位于坐标原点。
时16?y 1?t
时32?y 2?t
yFxFW yx dd
7&8 Work and Energy
F?
L
缓慢拉质量为 m 的小球，
解 0s in θTF
0c o s mgθT
θmgF ta n?
sθθmg dc o st a n
x
y
θ
G?
T?
0c o s1 θm g L
例
=?0 时，求已知用力 F? 保持方向不变F?
作的功。F?
00 c o st a nθ θθθL m g d
sθFrFW dc o sd
7&8 Work and Energy
P164:37
7&8 Work and Energy
c o sFvvFdt rdFdtdWP
Instantaneous power:
The instantaneous power P is defined as the
rate at which work is performed.
The SI unit of power is the watt,
1 W = 1 J/s
1 horsepower=1 hp = 746 W
§ 8-8 Power (功率 ) (P188)
7&8 Work and Energy
§ 7-2 Kinetic Energy and the work-Energy
Priciple (P156-p160)
Kinetic energy(动能 ) K is energy associated with
the state of motion of an object,（ The faster the
object moves,the greater is its kinetic energy.
p156）
When the object is stationary,its kinetic energy is zero.
● SI unit of energy is joule (J)
1 Kinetic Energy (动能 ) P156
2
2
1 vmK?
7&8 Work and Energy
A particle moves from a to b along an arbitrary path,
and may vary from point to point along the path.sd?
F?
2 Work Energy Principle (动能定理 p157-158)）
当外力移动物体从 a到 b过程中，力对物体作功，将外力分解为切向分力和法向分力
a
bFnF
F?
sFrFrFW ddd tt
t
mF
d
d
t
v?
2
1
2
2 2
1
2
1dd
d
d 2
1
2
1
vvvvv v
v
v
v
mmms
t
mW
7&8 Work and Energy
动能（ 状态 函数 ）
m
pmE
22
1 22
k v
k1k2 EEW
动能定理 Work-energy principle
合外力对质点所作的功,等于质点动能的增量,
Net work done on an object equals to the
change in its kinetic energy,P157
7&8 Work and Energy
This principle is not a new independent law.
Rather,it has been derived from the
definitions of work and kinetic energy using
Newton’s second law,p158
7&8 Work and Energy
Descriptions:
(1) Work is the measurements for the change in
the kinetic energy (“K”) of a body,
if
if
KKW
KKWIf
,0
,0
——Body gains the kinetic energy.
—Body loses,K” & does work outside.
(2) Both work and,K” are scalars,They have
same units and dimensions;,K” only depends on
the speed of initial and final states,but work
depends on the real process(动能是状态量,功是过程量 ).
Example 7-10
7&8 Work and Energy
3 Work Energy Principle for a system with more
particles
如果研究的对象为质点系，动能定理又如何表示？以最简单的两个质点组成的质点系为研究对象。
两个质点质量为 m1、
m2,受外力 F1,F2，内力为 f12,f21，初速度为
v10,v20,末速度为 v1,v2,
位移为,21,rr
1?
'1?
2?
'2?21f
12f
1F
2F
1r?
2r?
1m
2m
110 vv?
220 vv?
7&8 Work and Energy
对 m1,m2 应用质点动能定理，
10111 kk EEWW 内外
20222 kk EEWW 内外由于 m1,m2 为一个系统，将上两式相加：
内外 i
n
ii
n
i
WW
11
011 ki
n
iki
n
i
EE
ki
n
i
k EE
1
令 为质点系的动能，
内外 i
n
ii
n
i
WW
11 0kk
EE k
7&8 Work and Energy
1m
2m
im
exiF?
iniF?
内力功外力功
0kk0kk
inex EEEEWW
i
i
i
i
i
i
i
i
对质点系，有
0kk
inex
iiii EEWW
对第 个质点，有i
推广到有 个质点的质点系i
7&8 Work and Energy
质点系 动能定理
0kk
inex EEWW
work-kinetic energy theorem should be,The
sum work done on a system by all external and
internal forces equal to the change in kinetic
energy of the system.
7&8 Work and Energy
EXAMPLE:
A simple pendulum of mass m=1kg
is released from position?o=30o,It
moves in a circular arc of radius
R=1m in a vertical plane,Use the
work-kinetic energy theorem to
determine the speed of the
pendulum bob at the angle?=10o.
Solution:
sTsGsFW n e t ddd
G?,constant T?,variable
sT d sGW n e t d
o?
R?
O
ds
d
G?
T?
7&8 Work and Energy
)c o s( c o s 0 m g R
KKKW ifn e t
2
0 2
1)c o s( c o s
fg mvmg RW
smgRv /53.1)co s( co s2 0
Gn e t smgsgmW?
)(
CBn e t sGsGW dd o?
R?
A
BB’
C’’C’
O
C
G?
T?'' Cm g B?
7&8 Work and Energy
§ 8-1 Conservative and Nonconservative Forces
1,Work Done by and Its Features
gF
Considering a body near the
surface of Earth,from a to b
(P168-170)
0d zmgW
)( AB m g zm g z
kmgP
zmgrPW B
A
z
z
B
A
dd
A
B
Az
Bz
mg
o
x
y
z
7&8 Work and Energy
The Feature of the gravity:
Wg on a particle depends only on the initial
and final position (or height),and does not
depend on the real path taken by the particle.
1.
0 m g hm g hW a e b c a
b c aa e ba e b c a WWW
,)( m g hyymgW fiae b
m g hyymgW fibc a )(
Wg on a particle moving around a closed path=0,2.
x
y
o
mg
yi
f
a
b
c
d
h
e
7&8 Work and Energy
Based on Hooke’s law:
2,Work Done by and Its Features (P172):
sF
have the same features!
gs Fa n dF
ikxF
Ax Bx
F?
x
o
0d xkxW
B
A
B
A
x
x
x
x
xkx
xFW
d
d
)
2
1
2
1( 22
AB kxkxW
7&8 Work and Energy
3,Conservative & Nonconservative Forces
Conservative Forces,The work done by the force
on an object moving from one point to another
depends only on the initial and final positions and
is independent of the particular path taken.
结论：万有引力、重力、弹性力做功都与路径无关。
保守力,力所作的功与路径无关,仅决定于相互作用质点的 始末 相对 位置,
7&8 Work and Energy
A
B
C
D
物体沿 闭合 路径运动 一周时,保守力对它所作的功等于零,
Conservative Forces:
The net work done by the force on
an object moving around any
closed path is zero,P169
另外一种表述：
7&8 Work and Energy
A
B
C
D
0dl rF
BD AAC Bl rFrFrF d d d
A
B
C
D
AD BAC B rFrF d d
AD BAC B rFrF d d
These two definitions are equivalent and powerful
because they allow us to simplify difficult
problems when only conservative forces are
involved.
7&8 Work and Energy
4 Work done by friction 摩擦力的功 p170
1M 2M
v?
F?
mgF
摩擦力方向始终与质点速度方向相反结论,摩擦力的功，不仅与始、末位置有关，而且与质点所行经的路径有关,
摩擦力 在这个过程中所作的功为
2
1
dc o sM
LM
sFW?
m g sW
7&8 Work and Energy
If work done by force is dependent on the path
L 0rdFthe is called nonconservative force.F?
A nonconservative force is also called dissipative
force (耗散力,P182),such as a friction force,
sfE kth
The work done by frictional force equals to the
increase in thermal energy.
In general,
7&8 Work and Energy
§ 8-2 Potential Energy (P170-173)
Potential energy U is energy that can be
associated with the configuration (or
arrangement) of a system of objects that exert
forces on one another.
Gravitational potential energy
Elastic potential energy
Such as:
If the configuration of the system changes,then
the potential energy of the system can also
change.
7&8 Work and Energy
we have discussed the
work done by gravitational
force:
)( if yymgW
1,Gravitational potential energy（重力势能）
mg
rd
y
yi
yt
x0
p
h
= -mgh
m g yU g?
Then,
gif
x
x
s Um g xm g ydxFW
f
i
)(
7&8 Work and Energy
2
2
1 kxU
s?
The work done by conservative force = minus
change of,U” related to the conservative force
(保守力作功等于该过程始末两状态势能增量的负值 ).
2,Elastic potential energy（弹性势能）
UkxkxdxFW if
x
x
s
f
i
)
2
1
2
1
( 22
this work does not depend on the path too.
define
3,potential energy in general（势能）,
UUUrdFW if
f
i
c o n s e r )(
7&8 Work and Energy
Descriptions:
1.,U” is the function of states.
2,The relativity of potential energy:
Wab=Ua- Ub—— The difference of potential
energy (“U”) has absolute meaning,but to
assign a particular U value to an object makes
sense only if the reference,U” value is given.
Reference of,U” is arbitrary to choose.
Only changes in potential energy are meaningful,U
is a relative quantity.（势能是相对量）
U?
7&8 Work and Energy
A location where the gravitational potential
energy is zero must be chosen for each
problem.
重力势能零势点的选取
often the Earth’s surface（地面）
may be some other point suggested by the
problem（任意点以使解题方便）
弹性势能零势点的选取
Reference configuration,the spring is at its relaxed
length and the block is at x i=0,Then the elastic
potential energy U i=0.
7&8 Work and Energy
★ Summary,Potential Energy p173
1,A potential energy is always associated with a
conservative force,and the difference in
potential energy between two points is defined
as the negative of the work done by that force.
UUUrdFW if
f
i
c o n s e r )(
2,The choice of where U=0 is always arbitrary
and can be chosen wherever it is most convenient.
3,Potential energy is associated with the
interaction of bodies in system.
7&8 Work and Energy
fi c o n s e rc o n s e r sdFWU
4,Relation between Force and Potential (P173)
or pr c o n s e rP rFU?
d
Calculate U or?U when the force is given;1.
0?orU
where
7&8 Work and Energy
If we consider a 1D system in which U depends only on
x,U =U(x),A graph can then be drawn of U versus x,
xxFxUxxx if )()(,ces i n
dx
dUxF)(
The component of F in the x-direction,F(x),
is the negative of the slope of the U versus x
curve.
Finding Force Analytically (由势能求保守力 )2.
7&8 Work and Energy
For general case,U=U(x,y,z);
x
UF
x?
;
y
UF
y?
,
z
UF
z?
- Uk
z
Uj
y
Ui
x
UkFjFiFF
zyx
A conservative force equals to minus gradient
of the potential energy related to the force 保守力沿某一给定的 l 方向的分量等于与此保守力相应的势能函数沿 l 方向的空间变化率 (梯度 ).
Partial derivative
偏微分
7&8 Work and Energy
§ 8-3 Mechanical Energy and Its Conservation (p174)
1.General Form of Work-Energy Principle(p182),
Where W int is the sum of work done by internal force(内力 );
Wext is the sum of work done by external force（外力）,
1m
2m
im
exiF?
iniF?
0kk0kk
inex EEEEWW
i
i
i
i
i
i
i
i
对质点系，有
0kk
inex
iiii EEWW
对第 个质点，有i
推广到有 个质点的质点系i
7&8 Work and Energy
)()( 0p0kpkinncex EEEEWW
机械能
pk EEE
0kk
inex EEWW
非保守力的功
in
nc
in
c
inin WWWW
i
i
0pp0pp
in
c )( EEEEW
i
i
i
i
0inncex EEWW
质点系的功能原理 Work-Energy Principle
质点系机械能的增量等于外力和非保守内力作功之和,
Mechanical energy
7&8 Work and Energy
A crate of mass 180kg is released that was being
held at rest at the top of a 3.7m-long-ramp
inclined 39o to the horizontal,= 0.28 (a) How
fast is the crate moving as it reaches the bottom
of the ramp? (b) How far will it subsequently
slide across the floor? (c) Do the answers to them
change if we halve the mass of the crate?
k?
c o sc o s mgNfmgN kkk
C
N f
mg?
A
B
0?U
0?v
s
(a)Solution:
7&8 Work and Energy
)(4.5/)cos( s i n mds kk
From the algebraic form of the results,it can be
seen that the answers do not depend on mass.
s i n21c o s 2 m g dmvm g dk
)s/m(5.5)co s( s i n2 kgdv
Answer,(c)
)c o s( s i n210 2 kk m g dmvm g s
(b)
ifm e cn o n c o n EEWEW.i,e
C
N f
mg?
A
B
0?U0?v
s
d
7&8 Work and Energy
2.The Law of Conservation of Energy(P174-181):
If only conservative forces do work,mechanical
energy Conserves —— both kinetic energy and
potential energy transfer each other.
机械能守恒定律当
0inncex WW 0EE?
时,有
)()( 0p0kpkinncex EEEEWW
功能原理机械能守恒定律 只有保守内力作功的情况下，
质点系的机械能保持不变,
7&8 Work and Energy
In an isolated system where only conservative
forces cause energy changes,the kinetic energy
and potential energy can change,but their sum,
the mechanical energy E mec of the system,
cannot change.
pk EE )( 0pp0kk EEEE
7&8 Work and Energy
* 宇宙速度牛顿的,自然哲学的数学原理,插图，抛体的运动轨迹取决于抛体的初速度
7&8 Work and Energy
设 地球质量,抛体质量,地球半径,Em ERm
v?h
``````
解 取抛体和地球为一系统,当物体离开地球飞去时，只有保守力做功，所以这一系统的机械能 E 守恒,
1） 人造地球卫星 第一宇宙速度第一宇宙速度,是在地面上发射人造地球卫星所需的最小速度,1v
表示地面上发射人造地球卫星所需的最小速度,
1v
v
表示物体离开地球高度为 h时的速度,
7&8 Work and Energy
v?h
``````
)(
2
1
E
E2
1 R
mmGmE v
)(
2
1
E
E2
hR
mmGm
v
2
E
E
E
2
)( hR
mmG
hR
m
v
由牛顿第二定律和万有引力定律得
7&8 Work and Energy
解得
hR
Gm
R
Gm
E
E
E
E
1
2v
2
E
E
R
Gmg
)2(
E
E
E1 hR
RgR
v
地球表面附近 hRE 故 E1 gR?v
7&8 Work and Energy
v?h
``````
m / s109.7 31v
计算得第一宇宙速度
0
)(2 E
E?
hR
GmmE
0?E
7&8 Work and Energy
我国 1977年发射升空的东方红三号通信卫星
7&8 Work and Energy
2） 人造行星 第二宇宙速度设 地球质量,抛体质量,地球半径,Em ERm
第二宇宙速度,是 抛体脱离地球引力所需的最小发射速度,也称为逃逸速度。 2v
E取抛体和地球为一系统系统机械能 守恒,
v
表示物体远离地球时的速度。
p
2
E
E2
2
2
1
)(
2
1
Em
R
mm
GmE vv
取无穷远处为势能零点,逃逸速度是应为最小值，
这和无穷远处速度为零相对应。
7&8 Work and Energy
E
E
E
2 2
2 gR
R
Gmv
第二宇宙速度
0?E
0)(
2
1
E
E2
2 R
mmGmE v
``````
v?
h
k m / s2.112?v
计算得
0)(
2
1
pk
E
E2
2 EER
mmGmE v
7&8 Work and Energy
3） 飞出太阳系 第三宇宙速度第三宇宙速度,是 抛体脱离太阳引力所需的最小发射速度,3v
v?h
设 地球质量,抛体质量,地球半径,Em ERm
太阳质量,抛体与太阳相距,
Sm SR
7&8 Work and Energy
取地球为参考系,由机械能 守恒得
2
E
E2
3 2
1)(
2
1 v'v m
R
mmGm
取抛体和地球为一系统,抛体 首先要 脱离地球引力的束缚,其相对于地球的速率为,
v'
取太阳为参考系,抛体 相对于太阳的速度为,
3'v 地球相对于太阳的速度E3 '' vvv则如 与 同向,有E ' vv E3 '' vvv
7&8 Work and Energy
要 脱离太阳引力，机械能至少为零
0)(
2
1
pk
S
S2
3 EER
mmGmE v'
21
S
S
3 )
2(
R
Gm?v'则由于 与 同向,
则抛体与太阳的距离 即为地球轨道半径设地球绕太阳轨道近似为一圆，
E3' v v
SR
则
2
S
SE
S
2
E
E R
mm
G
R
m?
v
21
S
S
E )( R
mG?v
7&8 Work and Energy
1-21
E
E2
3 s,4 k m16)2( R
mGv'v计算得第三宇宙速度
2
E
E2
3 2
1)(
2
1 v'v m
R
mmGm取地球为参照系
v?h E3 vv'v'
21
S
S ))(12(
R
Gmv'
计算得
7&8 Work and Energy
Summary for Chapter Seven and Eight
See P161 and 190
7&8 Work and Energy
Homework,
p164:31; P164:36; P165:45; P166:65
Homework,P192:6;p192:7; p197:72
Chapter 7 & 8
Energy and Conservation Law
1,Work and Energy
2,Kinetic Energy （动能） & Work- Energy Principle
3,Conservative（保守） and Nonconservative
Forces
4,Potential Energy（势能）
5,Conservation of Energy
7&8 Work and Energy
New words
Conservative and non-conservative forces（保守力与非保守力）
Potential energy（势能）
Conservation of mechanical energy
（机械能守恒）
Conservation of energy（能量守恒）
7&8 Work and Energy
§ 7-1&7-3 Work and Power (P147-156)
Work W Provides a link between force and energy.
1,Work done by a constant force(P148)
Φ Φ xFrFrFW x )c o s(?
力的 空间累积 效应,
WrF,
Work is product of the
magnitude of the
displacement of the force
parallel to the displacement.
7&8 Work and Energy
if
4321 FFFFF
321
321
WWW
rdFrdFrdFrdFW
The work done by several forces = algebraic
sum of the work done by the individual force.
2 Work done by several forces
合力的功 = 分力的功的代数和
7&8 Work and Energy
3 work done by a general variable force(变力 P152-156)
a
b
1.无限分割路径；
The path is divided into short intervals,
2.以直线段代替曲线段；
The intervals could be treated as straight lines.
3.以恒力的功代替变力的功；
During each small interval,the force is
approximately constant.
4.将各段作功代数求和；
The total work done is the sum
of all terms.
2F
2?
1F
1?
r?
7&8 Work and Energy
111 c o s?rFW
222 c o s rFW?
iii rFW?c o s
nnn rFW?c o s+）
i
n
i
WW
1 ii
n
i
rF c o s
1
7&8 Work and Energy
0?r?Let
ii
n
ir
rFW
c o slim
10
c o sF d rba b
a rdF
BABA rFrFW dc o sd
co sF
Ar Brrd
r
o
When a particle moves from A to B along a
curve path,the total work done by the force
equals the integral from A to B.
7&8 Work and Energy
a n d, kFjFiFF zyx, kdzjdyidxrd
f
i
f
i
f
i
z
z
z
y
y
y
x
x
x
B
A
zyx
B
A
dzFdyFdxF
dzFdyFdxFrdFW )(
4 Special expression in scalar product(p154)
zFyFxFW zyx ddd
zyx WWWW
7&8 Work and Energy
5 Work is a scalar quantity,no direction,
0d,900 W?
sFrFW dc o sdc o sd
0d,1 8 090 W?
rFW dd
0dd90 WrF
F?
r?d?
iF
1dr?
ir
d B*
*
i?
1?
A 1F?
positive work;
negative work.
zero work;
7&8 Work and Energy
6 Geometrical Representation of Work:
The work done by a force
equals the area under the F
versus x curve —— 变力曲线与位移轴在极限 x1,x2 之间所包围的面积,
co sF
Ar Brrd
r
o
Example 7-5 and 7-6
7&8 Work and Energy
质量为 10kg 的质点，在外力作用下做平面曲线运动，该质点的速度为 jit 164 2v
解 24dd ttxxv ttx d4d 2?
16dd tyyv ty 16?
ttmF xx 80dd v 0dd?tmF yy v
J 1 20 0d3 2021 3 tt
在质点从 y = 16m 到 y = 32m 的过程中，外力做的功。求例
,开始时质点位于坐标原点。
时16?y 1?t
时32?y 2?t
yFxFW yx dd
7&8 Work and Energy
F?
L
缓慢拉质量为 m 的小球，
解 0s in θTF
0c o s mgθT
θmgF ta n?
sθθmg dc o st a n
x
y
θ
G?
T?
0c o s1 θm g L
例
=?0 时，求已知用力 F? 保持方向不变F?
作的功。F?
00 c o st a nθ θθθL m g d
sθFrFW dc o sd
7&8 Work and Energy
P164:37
7&8 Work and Energy
c o sFvvFdt rdFdtdWP
Instantaneous power:
The instantaneous power P is defined as the
rate at which work is performed.
The SI unit of power is the watt,
1 W = 1 J/s
1 horsepower=1 hp = 746 W
§ 8-8 Power (功率 ) (P188)
7&8 Work and Energy
§ 7-2 Kinetic Energy and the work-Energy
Priciple (P156-p160)
Kinetic energy(动能 ) K is energy associated with
the state of motion of an object,（ The faster the
object moves,the greater is its kinetic energy.
p156）
When the object is stationary,its kinetic energy is zero.
● SI unit of energy is joule (J)
1 Kinetic Energy (动能 ) P156
2
2
1 vmK?
7&8 Work and Energy
A particle moves from a to b along an arbitrary path,
and may vary from point to point along the path.sd?
F?
2 Work Energy Principle (动能定理 p157-158)）
当外力移动物体从 a到 b过程中，力对物体作功，将外力分解为切向分力和法向分力
a
bFnF
F?
sFrFrFW ddd tt
t
mF
d
d
t
v?
2
1
2
2 2
1
2
1dd
d
d 2
1
2
1
vvvvv v
v
v
v
mmms
t
mW
7&8 Work and Energy
动能（ 状态 函数 ）
m
pmE
22
1 22
k v
k1k2 EEW
动能定理 Work-energy principle
合外力对质点所作的功,等于质点动能的增量,
Net work done on an object equals to the
change in its kinetic energy,P157
7&8 Work and Energy
This principle is not a new independent law.
Rather,it has been derived from the
definitions of work and kinetic energy using
Newton’s second law,p158
7&8 Work and Energy
Descriptions:
(1) Work is the measurements for the change in
the kinetic energy (“K”) of a body,
if
if
KKW
KKWIf
,0
,0
——Body gains the kinetic energy.
—Body loses,K” & does work outside.
(2) Both work and,K” are scalars,They have
same units and dimensions;,K” only depends on
the speed of initial and final states,but work
depends on the real process(动能是状态量,功是过程量 ).
Example 7-10
7&8 Work and Energy
3 Work Energy Principle for a system with more
particles
如果研究的对象为质点系，动能定理又如何表示？以最简单的两个质点组成的质点系为研究对象。
两个质点质量为 m1、
m2,受外力 F1,F2，内力为 f12,f21，初速度为
v10,v20,末速度为 v1,v2,
位移为,21,rr
1?
'1?
2?
'2?21f
12f
1F
2F
1r?
2r?
1m
2m
110 vv?
220 vv?
7&8 Work and Energy
对 m1,m2 应用质点动能定理，
10111 kk EEWW 内外
20222 kk EEWW 内外由于 m1,m2 为一个系统，将上两式相加：
内外 i
n
ii
n
i
WW
11
011 ki
n
iki
n
i
EE
ki
n
i
k EE
1
令 为质点系的动能，
内外 i
n
ii
n
i
WW
11 0kk
EE k
7&8 Work and Energy
1m
2m
im
exiF?
iniF?
内力功外力功
0kk0kk
inex EEEEWW
i
i
i
i
i
i
i
i
对质点系，有
0kk
inex
iiii EEWW
对第 个质点，有i
推广到有 个质点的质点系i
7&8 Work and Energy
质点系 动能定理
0kk
inex EEWW
work-kinetic energy theorem should be,The
sum work done on a system by all external and
internal forces equal to the change in kinetic
energy of the system.
7&8 Work and Energy
EXAMPLE:
A simple pendulum of mass m=1kg
is released from position?o=30o,It
moves in a circular arc of radius
R=1m in a vertical plane,Use the
work-kinetic energy theorem to
determine the speed of the
pendulum bob at the angle?=10o.
Solution:
sTsGsFW n e t ddd
G?,constant T?,variable
sT d sGW n e t d
o?
R?
O
ds
d
G?
T?
7&8 Work and Energy
)c o s( c o s 0 m g R
KKKW ifn e t
2
0 2
1)c o s( c o s
fg mvmg RW
smgRv /53.1)co s( co s2 0
Gn e t smgsgmW?
)(
CBn e t sGsGW dd o?
R?
A
BB’
C’’C’
O
C
G?
T?'' Cm g B?
7&8 Work and Energy
§ 8-1 Conservative and Nonconservative Forces
1,Work Done by and Its Features
gF
Considering a body near the
surface of Earth,from a to b
(P168-170)
0d zmgW
)( AB m g zm g z
kmgP
zmgrPW B
A
z
z
B
A
dd
A
B
Az
Bz
mg
o
x
y
z
7&8 Work and Energy
The Feature of the gravity:
Wg on a particle depends only on the initial
and final position (or height),and does not
depend on the real path taken by the particle.
1.
0 m g hm g hW a e b c a
b c aa e ba e b c a WWW
,)( m g hyymgW fiae b
m g hyymgW fibc a )(
Wg on a particle moving around a closed path=0,2.
x
y
o
mg
yi
f
a
b
c
d
h
e
7&8 Work and Energy
Based on Hooke’s law:
2,Work Done by and Its Features (P172):
sF
have the same features!
gs Fa n dF
ikxF
Ax Bx
F?
x
o
0d xkxW
B
A
B
A
x
x
x
x
xkx
xFW
d
d
)
2
1
2
1( 22
AB kxkxW
7&8 Work and Energy
3,Conservative & Nonconservative Forces
Conservative Forces,The work done by the force
on an object moving from one point to another
depends only on the initial and final positions and
is independent of the particular path taken.
结论：万有引力、重力、弹性力做功都与路径无关。
保守力,力所作的功与路径无关,仅决定于相互作用质点的 始末 相对 位置,
7&8 Work and Energy
A
B
C
D
物体沿 闭合 路径运动 一周时,保守力对它所作的功等于零,
Conservative Forces:
The net work done by the force on
an object moving around any
closed path is zero,P169
另外一种表述：
7&8 Work and Energy
A
B
C
D
0dl rF
BD AAC Bl rFrFrF d d d
A
B
C
D
AD BAC B rFrF d d
AD BAC B rFrF d d
These two definitions are equivalent and powerful
because they allow us to simplify difficult
problems when only conservative forces are
involved.
7&8 Work and Energy
4 Work done by friction 摩擦力的功 p170
1M 2M
v?
F?
mgF
摩擦力方向始终与质点速度方向相反结论,摩擦力的功，不仅与始、末位置有关，而且与质点所行经的路径有关,
摩擦力 在这个过程中所作的功为
2
1
dc o sM
LM
sFW?
m g sW
7&8 Work and Energy
If work done by force is dependent on the path
L 0rdFthe is called nonconservative force.F?
A nonconservative force is also called dissipative
force (耗散力,P182),such as a friction force,
sfE kth
The work done by frictional force equals to the
increase in thermal energy.
In general,
7&8 Work and Energy
§ 8-2 Potential Energy (P170-173)
Potential energy U is energy that can be
associated with the configuration (or
arrangement) of a system of objects that exert
forces on one another.
Gravitational potential energy
Elastic potential energy
Such as:
If the configuration of the system changes,then
the potential energy of the system can also
change.
7&8 Work and Energy
we have discussed the
work done by gravitational
force:
)( if yymgW
1,Gravitational potential energy（重力势能）
mg
rd
y
yi
yt
x0
p
h
= -mgh
m g yU g?
Then,
gif
x
x
s Um g xm g ydxFW
f
i
)(
7&8 Work and Energy
2
2
1 kxU
s?
The work done by conservative force = minus
change of,U” related to the conservative force
(保守力作功等于该过程始末两状态势能增量的负值 ).
2,Elastic potential energy（弹性势能）
UkxkxdxFW if
x
x
s
f
i
)
2
1
2
1
( 22
this work does not depend on the path too.
define
3,potential energy in general（势能）,
UUUrdFW if
f
i
c o n s e r )(
7&8 Work and Energy
Descriptions:
1.,U” is the function of states.
2,The relativity of potential energy:
Wab=Ua- Ub—— The difference of potential
energy (“U”) has absolute meaning,but to
assign a particular U value to an object makes
sense only if the reference,U” value is given.
Reference of,U” is arbitrary to choose.
Only changes in potential energy are meaningful,U
is a relative quantity.（势能是相对量）
U?
7&8 Work and Energy
A location where the gravitational potential
energy is zero must be chosen for each
problem.
重力势能零势点的选取
often the Earth’s surface（地面）
may be some other point suggested by the
problem（任意点以使解题方便）
弹性势能零势点的选取
Reference configuration,the spring is at its relaxed
length and the block is at x i=0,Then the elastic
potential energy U i=0.
7&8 Work and Energy
★ Summary,Potential Energy p173
1,A potential energy is always associated with a
conservative force,and the difference in
potential energy between two points is defined
as the negative of the work done by that force.
UUUrdFW if
f
i
c o n s e r )(
2,The choice of where U=0 is always arbitrary
and can be chosen wherever it is most convenient.
3,Potential energy is associated with the
interaction of bodies in system.
7&8 Work and Energy
fi c o n s e rc o n s e r sdFWU
4,Relation between Force and Potential (P173)
or pr c o n s e rP rFU?
d
Calculate U or?U when the force is given;1.
0?orU
where
7&8 Work and Energy
If we consider a 1D system in which U depends only on
x,U =U(x),A graph can then be drawn of U versus x,
xxFxUxxx if )()(,ces i n
dx
dUxF)(
The component of F in the x-direction,F(x),
is the negative of the slope of the U versus x
curve.
Finding Force Analytically (由势能求保守力 )2.
7&8 Work and Energy
For general case,U=U(x,y,z);
x
UF
x?
;
y
UF
y?
,
z
UF
z?
- Uk
z
Uj
y
Ui
x
UkFjFiFF
zyx
A conservative force equals to minus gradient
of the potential energy related to the force 保守力沿某一给定的 l 方向的分量等于与此保守力相应的势能函数沿 l 方向的空间变化率 (梯度 ).
Partial derivative
偏微分
7&8 Work and Energy
§ 8-3 Mechanical Energy and Its Conservation (p174)
1.General Form of Work-Energy Principle(p182),
Where W int is the sum of work done by internal force(内力 );
Wext is the sum of work done by external force（外力）,
1m
2m
im
exiF?
iniF?
0kk0kk
inex EEEEWW
i
i
i
i
i
i
i
i
对质点系，有
0kk
inex
iiii EEWW
对第 个质点，有i
推广到有 个质点的质点系i
7&8 Work and Energy
)()( 0p0kpkinncex EEEEWW
机械能
pk EEE
0kk
inex EEWW
非保守力的功
in
nc
in
c
inin WWWW
i
i
0pp0pp
in
c )( EEEEW
i
i
i
i
0inncex EEWW
质点系的功能原理 Work-Energy Principle
质点系机械能的增量等于外力和非保守内力作功之和,
Mechanical energy
7&8 Work and Energy
A crate of mass 180kg is released that was being
held at rest at the top of a 3.7m-long-ramp
inclined 39o to the horizontal,= 0.28 (a) How
fast is the crate moving as it reaches the bottom
of the ramp? (b) How far will it subsequently
slide across the floor? (c) Do the answers to them
change if we halve the mass of the crate?
k?
c o sc o s mgNfmgN kkk
C
N f
mg?
A
B
0?U
0?v
s
(a)Solution:
7&8 Work and Energy
)(4.5/)cos( s i n mds kk
From the algebraic form of the results,it can be
seen that the answers do not depend on mass.
s i n21c o s 2 m g dmvm g dk
)s/m(5.5)co s( s i n2 kgdv
Answer,(c)
)c o s( s i n210 2 kk m g dmvm g s
(b)
ifm e cn o n c o n EEWEW.i,e
C
N f
mg?
A
B
0?U0?v
s
d
7&8 Work and Energy
2.The Law of Conservation of Energy(P174-181):
If only conservative forces do work,mechanical
energy Conserves —— both kinetic energy and
potential energy transfer each other.
机械能守恒定律当
0inncex WW 0EE?
时,有
)()( 0p0kpkinncex EEEEWW
功能原理机械能守恒定律 只有保守内力作功的情况下，
质点系的机械能保持不变,
7&8 Work and Energy
In an isolated system where only conservative
forces cause energy changes,the kinetic energy
and potential energy can change,but their sum,
the mechanical energy E mec of the system,
cannot change.
pk EE )( 0pp0kk EEEE
7&8 Work and Energy
* 宇宙速度牛顿的,自然哲学的数学原理,插图，抛体的运动轨迹取决于抛体的初速度
7&8 Work and Energy
设 地球质量,抛体质量,地球半径,Em ERm
v?h
``````
解 取抛体和地球为一系统,当物体离开地球飞去时，只有保守力做功，所以这一系统的机械能 E 守恒,
1） 人造地球卫星 第一宇宙速度第一宇宙速度,是在地面上发射人造地球卫星所需的最小速度,1v
表示地面上发射人造地球卫星所需的最小速度,
1v
v
表示物体离开地球高度为 h时的速度,
7&8 Work and Energy
v?h
``````
)(
2
1
E
E2
1 R
mmGmE v
)(
2
1
E
E2
hR
mmGm
v
2
E
E
E
2
)( hR
mmG
hR
m
v
由牛顿第二定律和万有引力定律得
7&8 Work and Energy
解得
hR
Gm
R
Gm
E
E
E
E
1
2v
2
E
E
R
Gmg
)2(
E
E
E1 hR
RgR
v
地球表面附近 hRE 故 E1 gR?v
7&8 Work and Energy
v?h
``````
m / s109.7 31v
计算得第一宇宙速度
0
)(2 E
E?
hR
GmmE
0?E
7&8 Work and Energy
我国 1977年发射升空的东方红三号通信卫星
7&8 Work and Energy
2） 人造行星 第二宇宙速度设 地球质量,抛体质量,地球半径,Em ERm
第二宇宙速度,是 抛体脱离地球引力所需的最小发射速度,也称为逃逸速度。 2v
E取抛体和地球为一系统系统机械能 守恒,
v
表示物体远离地球时的速度。
p
2
E
E2
2
2
1
)(
2
1
Em
R
mm
GmE vv
取无穷远处为势能零点,逃逸速度是应为最小值，
这和无穷远处速度为零相对应。
7&8 Work and Energy
E
E
E
2 2
2 gR
R
Gmv
第二宇宙速度
0?E
0)(
2
1
E
E2
2 R
mmGmE v
``````
v?
h
k m / s2.112?v
计算得
0)(
2
1
pk
E
E2
2 EER
mmGmE v
7&8 Work and Energy
3） 飞出太阳系 第三宇宙速度第三宇宙速度,是 抛体脱离太阳引力所需的最小发射速度,3v
v?h
设 地球质量,抛体质量,地球半径,Em ERm
太阳质量,抛体与太阳相距,
Sm SR
7&8 Work and Energy
取地球为参考系,由机械能 守恒得
2
E
E2
3 2
1)(
2
1 v'v m
R
mmGm
取抛体和地球为一系统,抛体 首先要 脱离地球引力的束缚,其相对于地球的速率为,
v'
取太阳为参考系,抛体 相对于太阳的速度为,
3'v 地球相对于太阳的速度E3 '' vvv则如 与 同向,有E ' vv E3 '' vvv
7&8 Work and Energy
要 脱离太阳引力，机械能至少为零
0)(
2
1
pk
S
S2
3 EER
mmGmE v'
21
S
S
3 )
2(
R
Gm?v'则由于 与 同向,
则抛体与太阳的距离 即为地球轨道半径设地球绕太阳轨道近似为一圆，
E3' v v
SR
则
2
S
SE
S
2
E
E R
mm
G
R
m?
v
21
S
S
E )( R
mG?v
7&8 Work and Energy
1-21
E
E2
3 s,4 k m16)2( R
mGv'v计算得第三宇宙速度
2
E
E2
3 2
1)(
2
1 v'v m
R
mmGm取地球为参照系
v?h E3 vv'v'
21
S
S ))(12(
R
Gmv'
计算得
7&8 Work and Energy
Summary for Chapter Seven and Eight
See P161 and 190
7&8 Work and Energy
Homework,
p164:31; P164:36; P165:45; P166:65
Homework,P192:6;p192:7; p197:72
