work energy and power - national chiao tung...

p. 1

交通大學李威儀

Work

Energy

And Power

功，能量及功率

p. 2

交通大學李威儀黃河壺口瀑布

QUIZ #2

(1) An xx-kg man ascends a 10 m high staircase in yy sec. What is

his horsepower? ( g = 9.8 m/sec2 , 1 hp = 746 W )

p. 3

交通大學李威儀黃河壺口瀑布

QUIZ #2

(2) 請問：一個原點固定於地球表面上的參考座標是不是一個慣性座標？

如果不是，請問為什麼不是？

p. 4


p. 5


甚麼是 “能量” ?

p. 6


• W = F cos q • s ( Joule, a scalar )

= F • s or F • Dx

F : force , s : displacement , W : work

1 J = 1 N • 1 m = 1 ( kg m2 ) / s2

F

s

F

F cosq

q

要有出力才有“功”勞

造成位移才有“功”勞

常力所作的功 ( Work Done by a Constant Force )

p. 7


• W = F • s = ( Fx i + Fy j + Fz k ) • (Dx i + Dy j + Dz k )

= Fx Dx + Fy Dy + Fz Dz

W : mechanical work ( different from chemical work )

• F s W = 0

• negative work

FAB = – FBA , W = F • s

Won A by B ( negative work ) = – Won B by A ( positive work )

fk (friction on B)

A

B

s FBA

FAB

p. 8


• work done by kinetic friction

Wf = fk s cos180o = – fk s

• kinetic friction can also do positive work

fk (friction on B)

A

B

s FBA

FAB

B

A fk ( fk moves block A to the right also )

p. 9


DW = F • Dx = Fx • Dx

W = DW = Fx1 Dx + Fx2 Dx + …… + Fxn Dx

= Fxi Dx

as Dx 0

W = Fx dx or more generally W = F • ds

i =1

n

b

a

b

a

x

Fx

a b Dx

Fx1

Fx2

Fx3

等速率圓周運動

非常力所作的功 ( Work Done by Variable Force in 1-Dim )

WEM04VD2.MOV

p. 10


http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

p. 11


F cause v change

v cause x change

W cause ? change

in one-dim. , const. force :

W = Fx Dx = ma Dx

v2 = vo2 + 2a Dx

a Dx = ½ (v2 – vo2 )

W = ½ mv2 – ½ mvo2

define kinetic energy ( K.E. ) : Ek = ½ mv2 ( joule, scalar )

Wnet = DEk : Work – Energy Theorem ( 功能定律 )

( Work-Energy Theorem still valid in 3-dim. space and with non-constant

forces. )

功與動能

動能 ( Kinetic Energy )

WEM04VD1.MOV

p. 12


垂直於運動方向的力沒有作功

( 沒有改變物體的動能 )

等速率圓周運動

WEM04VD2.MOV

p. 13


Ex.

F

F cosq q

N

mg

m = 4 kg , q = 53o

F = 3 N , Dx = 2 m

vo = 3 m/s , mk = 1/8

Find (1) DEk , (2) vf

f

(1) Fy = 0 N = mg – Fsinq

DEk = WNet = WF + Wf

= Fx•s – mk N•s

= F cosq •s – mk ( mg – Fsinq )•s

= …… = 32 J

(2) D Ek = ½ mvf2 – ½ mvo

2 = 32 J , vo = 3 m/s

vf = 5 m/s

p. 14


p. 15


• An experiment by Galileo Galilei :

raise ball from C A , do work W

AC , gain K.E. = W, reach vmax

CB , K.E. = 0 ( A and B same height )

BCA, K.E. = 0

where does K.E. go at A and B ?

A B

C

位能 ( Potential Energy )

p. 16


vf vo

v = 0 , K.E. = 0

vo = vf

p. 17


vf vo

v = 0 , K.E. = 0

vo = vf

• potential energy ( P.E., Ep) : energy associated with the relative

positions of two or more interacting particles

Earth

p. 18


Example :

do work ( W ) to raise the apple

Changes the apple-earth

relative position

Changes the apple-earth

system’s potential ( or

the apple’s potential )

vmax

Fext

earth surface

release the apple

stored P.E. K.E.

If one raises the apple at const. speed ( K.E. unchanged )

Wext = D ( P.E. ) = Epf - Epo

p. 19


Wext

changes particles’ relative position

( pendulum and earth )

P.E. changed

K.E. ( 1/2 mvmax )

positive work increase in P.E. ( Ep )

Only changes in P.E. are important

Freedom to assign Ep = 0 configuration

Once the initial configuration for Ep=0 is defined :

The P.E. of a system is the external work needed to bring the

particles from the Ep=0 configuration to the given position at a

const. velocity ( or speed ).

2 A B

C

earth

如何計算一個系統中的位能 ?

p. 20


• Gravitational Potential Energy (near the earth’s surface)

external work done to move m

from yi to yf at a const. velocity

( Wnet = Wext + Wg = 0 = DEk )

Wext = Fext • s , | Fext | = mg

( mg 為此 apple/earth 雙粒子系統內力 )

Wext = Fext • s , | Fext | = mg

= mg ( yf – yi ) = mgyf – mgyi = DEp

= Ep( yf ) – Ep( yi )

Define Ep = 0 at yi = 0 Ep( yf ) = mg yf Ep(y) = mg y

Fext

yi

yf

mg

earth surface

p. 21


• Spring Potential Energy

Hooke’s Law : Fsp = - k x , k : spring constant

x xf

kxf

F

p. 22


伽利略

看遠處的大東西

虎克

看身邊的小東西

http://micro.magnet.fsu.edu/optics/timeline/people/hooke.html

http://s593.photobucket.com/albums/tt16/shernap05_bucket/DPS/?action=view&

current=hooke.jpg&newest=1

p. 23


p. 24


Move the spring from x = 0 to xf at a constant speed :

W = F • ds = ½ k xf2

define Ep,sp = 0 at x = 0 Ep,sp = ½ k xf2

Ep,sp = ½ k x2

• Spring Potential Energy

Hooke’s Law : Fsp = - k x , k : spring constant

Fsp

b

a

x xf

kxf

F

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

p. 25


• Work changes

the velocity of the particle DEk

the relative positions of particles DEp

• Inversely :

Energy is a measure of the particle’s, or

the particle system’s, capability to perform work ( on others )

甚麼是 “能量” ?

p. 26


• mechanically conservative system ( 機械守恆系統 ) : no energy enters

or leaves the system ( e.g. heat, radiation, work by external force,

work done by internal chemical reaction )

能量守恆 ( Energy Conservation )

p. 27


• mechanically conservative system ( 機械守恆系統 ) : no energy enters

or leaves the system ( e.g. heat, radiation, work by external force,

work done by internal chemical reaction )

• In a mechanically conservative system :

Ek + Ep = const.

( Ek + Ep )initial = ( Ek + Ep )final

½ mv22 + mg y2 = ½ mv1

2 + mg y1 y1

y2

v2

v1

earth

A system of 2 particles ( apple & earth)

能量守恆 ( Energy Conservation )

p. 28


p. 29


There are other forms of energy : e.g. thermal energy

( Actually, in microscopic scale, thermal energy may also be a kind

of kinetic energy )

摩擦力與功能定律

A B

C

earth

WEA07VD1.MOV

p. 30


從位能

轉換為動能

動能

再轉換為電能

2003 三峽大壩

p. 31


The ball on which track will hit the finish line earlier ?

p. 32


• mechanical power

Pav = DW / Dt ( J / s ) , 1 J/s = 1 watt ( W ) , 1 hp = 746 W

lim DW/Dt = P = dW/dt

DW = F • Ds dW = F • ds

v = ds / dt

P = dW / dt = F • ds / dt = F • v

Ex. A tractor exert 3 x 104 N, moving at 5 m/s. What is its horsepower ?

Dt0

功率，機械功率 ( Mechanical Power )

P = F • v = 3 x 104 x 5 = 1.5 x 105 ( W ) 200 ( hp )

work energy and power - national chiao tung...

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