1

Ch10 Energy講者： 許永昌 老師

2

ContentsA “Natural Money” called Energy

但是， money 事實上是“無定的”。Kinetic Energy and Gravitational Potential

EnergyElastic Force and Elastic Potential Energy

Elastic CollisionsEnergy Diagrams

Molecular Bonds

3

A “Natural Money” called Energy ( 請預讀 P267~P268)

Key Points:Transform:

L 與 S 可互相交換而不會損失 , i.e. L+S=W, W is a

constant.Transfer:

Income I & Expenditure E

Before After (for system)

Wi (I & E) Wf=Wi+I-E

4

Kinetic Energy and Gravitational Potential Energy ( 請預讀 P269~P277)

Owing to Newton’s 2nd Law: Fnet=ma,

For a block on a slide,

We get DK+DUg=0,

Kinetic energy is associated with the motion of a particle. K= ½ mv2 0.

Potential energy is associated with its position. Different type of interactions are related to different type

potential.

21, where .

2net

dvF dr m dr K K mv

dt

Fnet

N

v, dr

FG

g g, where .

net GF dr F dr mgdy

U U mgy

g g .f f i iK U K U

x x y y z zA B A B A B A B

5

ExercisesEnergy bar chart for a ball slide down a

slides

A block slides down a frictionless ramp of height h. It reaches velocity v at the bottom. To reach a velocity of 2v, the block would need to slide down a ramp of height a. 1.41h b. 2h c. 3h d. 4h e. 6h

h

1. y=0 m

2. y=0 m

6

Law of Conservation of Mechanical Energy ( 請預讀 P277)

Law of Conservation of Mechanical EnergyDEmech=DK+DUmech=0

It is not always true. E.g. Consider a box that is given a shove and then

slides along he floor until it stops. DUmech=0, DK<0, i.e. DEmech<0.

It will be discussed in Ch 11: work and Ch17:

thermodynamics.In general, as figure 10.2 (P269) shown,

E=K+UEnergy into system Energy out of system

( 千萬不要用背的，因為它是有條件的 )

, , & 0f f f

i i i

x x x

net x x xx x xF dx f dx K f dx

7

HomeworkStudent Workbook

10.110.210.310.410.6

8

Summary ILaw of Conservation of Mechanical Energy

E=K+UEnergy into system Energy out of system

0

We get

0 if all the forces are conservative forces.

f

inet f

c ci

cc

K ma drF ma

U F dr

K U

由於位能與 force 有關，我們會發覺利用interaction diagram 可以幫助我們分析位能與作工。

9

Action: spring ( 請預讀 P278~P284)

Purpose: The significance of spring constant.The mathematical formula of a spring force.

Actors and Objects:One student.Two springs.

Actions:How would you characterize the difference of

the springs?The mathematical formula of the spring force.

10

Spring forcesHooke’s Law: Fspring on object, x=-kDx=-k(x-xe)

k: spring constant.xe: equilibrium position.

Questions:If the spring is stretched, is the displacement Dx

positive or negative?Is x the same as Dx ?What direction does the force point?Is the force component Fx positive or negative?How does the sign of Fx compare with the sign of Dx?

What role dose the minus sign play in Hooke’s law?

x̂

0

xe

x

C

B

EE

as

if

11

Elastic Potential Energy ( 請預讀P280~P284)

Situation: The elastic potential energy for a ball attached to a spring.System: ball Elastic Potential energy: U= ½ kDx2.

Because:

Energy bar chart:

2

s ,

1

2

ff f

i ii

xx x

s x e ex xx

U F dx k x x dx k x x x̂

0

xe

x

Stretched or compressed, At position0 e

i si f sf

x xv

K U K U

0+ +=

12

ExercisesEx.10.15 You need to make a spring scale for

measuring mass. You want each 1.0 cm length along the scale to correspond to a mass difference of 100 g. What should be the value of the spring constant?

Compare the compressed length of these two situations:

A 10 kg runaway grocery cart runs into a spring with spring constant 250 N/m and compresses it by 60 cm. What was the speed of the cart just before it hit the spring?

Wall

13

HomeworkStudent Workbook

10.1010.1110.1210.1310.1510.16

14

Summary IIHooke’s Law 告訴我們的是 spring 的受力與伸長量的關係：Fspring on object, x=-kDx=-k(x-xe)

如果假設 spring 本身的質量 =0 ，則彈簧兩端接的物體所受的力就可看作好像 force pair ，該力的形式如上式。這個 force pair 所對應的位能為 U= ½ kDx2 ，即下

圖虛線所表示。 C

B

as

if

15

Elastic Collisions ( 請預讀 P284~P288)

A collision in which mechanical energy is conserved is called a perfectly elastic collision.Conditions:

Momentum conservation: (Owing to Dt 0, )

Energy conservation: (Just for elastic collision) K1i+K2i=K1f+K2f.

Q: The system we chose here is (1) blue dash line (2) green dash line (3) blue dot line

00

t

ext totF dt P

1 2 1 2i i f fp p p p

1 2

16

Elastic CollisionsFor 1D elastic collision:

Conditions: m1v1xi+m2v2xi=m1v1xf+m2v2xf

½m1v21xi+ ½m2v2

2xi= ½m1v21xf + ½m2v2

2xf

Solve:

1 1 1 1 1 2 2 2 2 2

1 1 1 2 2 2

1 1 2 2

1 1 1 1 2 2

Eq. (2) becomes ------(3)

Eq. (1) becomes ------(4)

Eq. (3)(5)

Eq. (4)

Eq. (4)+ Eq. (5) 2

xi xf xi xf xi xf xi xf

xi xf xi xf

xi xf xi xf

xi x

m v v v v m v v v v

m v v m v v

v v v v

m m v m m v

1 2 2

1 2 12 1 2

1 2 1 2

2 1 22 1 2 1

1 2 1 2

2

2Eq. (4)+ Eq. (5)

i xf

xf xi xi

xf xi xi

m m v

m m mv v v

m m m m

m m mm v v v

m m m m

( 請自己推一次，我就不在課堂推了 )

17

HomeworkStudent Workbook

10.17

18

Energy diagrams ( 請預讀 P288~P291)

They are important not only for their later use in electricity and in quantum physics but also because they help students think about how energy is TRANSFORMED between kinetic and potential as a particle moves.

E.g.

19

Action: Energy DiagramPurpose: Get the quantities from an energy

diagramTurning points: (?=0)Stable equilibrium (?=0, ??>0)Unstable equilibrium (?=0, ??<0)The speeds.

Actions:When TE=1 J, …When TE=2 J, …When TE=2.5 J, …Plot its Kinetic energy versus x graph when

TE=2.5 J.

0 1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

(PE)

E (J)

x (m)

20

HomeworkStudent Workbook P10-7~P10-8Textbook

10.1710.3110.3810.4210.5310.60

一樣，請將 terms and notation 製作成名詞卡片。