Physics, Page 1

Chapter 8. Potential EnergyChapter 8. Potential Energyand Conservation of Energyand Conservation of Energy

Potential Energy (U)

중력 U, 탄성 U, 전자기 U, …..

Conservative Force ( 보존력)

Energy Conservation (에너지보존)

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Work: Transfer of Energy by Force• WF = |F| |S| cosθ

Kinetic Energy (Energy of Motion) • K = 1/2 mv2

Work-Kinetic Energy Theorem:• ΣW = ΔK

Review

θ

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QuestionQuestionImagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball reach the bottom with the highest speed?

1. Dropping2. Slide on ramp (no friction)3. Swinging down4. All the same

ball A is the only one without any tension or friction acting on it

In all three cases, the work done by the gravitational force is the same since the change in vertical distance is the same

1 2 3

it has the most distance to acceleratehas tension and gravity as forces, causing the greater acceleration

correct

Wrong: all have same acceleration due to gravity

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QuestionQuestionImagine that you are comparing three different ways of having a ball move down through the same height. In which case does the ball get to the bottom first?

1. Dropping2. Slide on ramp (no friction)3. Swinging down4. All the same 1 2 3

correct

It has the most direct path.

They each have only gravitational force acting on them so they fall at the same speed. Since they are at the same height, they will hit at the same time.

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QuestionQuestionWhich of the following statements correctly define a Conservative Force (보존력):

1. A force is conservative when the work it does on a moving object is independent of the path of the motion between the object's initial and final positions.

2. A force is conservative when it does no net work on an object moving around a closed path, starting and finishing at the same point.

3. Both of the above statements are correct.

4. Neither of the above statements is correct.

correct

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Suppose the initial kinetic and potential energies of a system are 75J and 250J respectively, and that the final kinetic and potential energies of the same system are 300J and -25J respectively. How much work was done on the system by non-conservative forces?

1. 0J 2. 50J 3. -50J 4. 225J 5. -225J

Wnc = Ef - Ei=(KEf + PEf) - (KEi + PEi) =(300J -25J) - (75J + 250J) =275J - 325J= -50J

correct

The change in kinetic energy plus the change in potential energy equals the work done on the system by non-conservative forces

QuestionQuestion

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GalileoGalileo’’s Pendulum ACTs Pendulum ACTHow high will the pendulum swing on the other side now?

A) h1 > h2 B) h1 = h2 C) h1 < h2

h1 h2

m

Conservation of Energy

0= ΔK + Δ UKinitial + Uinitial = Kfinal+Ufinal0 + mgh1 = 0 + mgh2

h1 = h2

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Potential energyPotential energy

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일과일과 potential energypotential energy

ΔU > 0

W (< 0)

W (> 0)

ΔU < 0

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ΔU > 0

W < 0

ΔU < 0W > 0

ΔK < 0

ΔK > 0

평형점

(a) 수축시

(b) 이완시

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Conservative force — Work is independent of the path.Non-conservative force — Friction

보존력과보존력과 비보존력비보존력

W = 0

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Potential energy Potential energy 구하기구하기

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역학에너지역학에너지 보존보존

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Em = U + K = 일정

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(보기문제 8-3)

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(보기문제)

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E = U + K = 5 J

E = U + K = 4 J

(X2, X3, X4, 0~X0, X5~)

(X1)

(X2, X4)

(X3)

Potential energy Potential energy 곡선읽기곡선읽기

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외부외부 힘이힘이 계에계에 한한 일일

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(보기문제)

30o

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Energy is Conserved !Energy is Conserved !

• Energy is “Conserved” meaning it can not be created nor destroyed.–Can change form–Can be transferred

• Total Energy does not change with time.

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에너지에너지 보존보존

마찰력마찰력 ((비비 보존력보존력))이이 있는있는 경우경우

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(보기문제)

h2

ΔE

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Skiing Example (no friction)A skier goes down a 78 meter high hill with a variety of slopes. What is the maximum speed should can obtain if she starts from rest at the top?

Conservation of energy:

Ki + Ui = Kf + Uf

½ m vi2 + m g yi = ½ m vf

2 + m g yf

0 + g yi = ½ vf2 + g yf

vf2 = 2 g (yi-yf)

vf = sqrt( 2 g (yi-yf))

vf = sqrt( 2 x 9.8 x 78) = 39 m/s26

0 = Kf-Ki + Uf - Ui

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Pendulum ExampleAs the pendulum falls, the work done by the string is

1) Positive 2) Zero 3) Negative

How fast is the ball moving at the bottom of the path?

W = F d cos θ. But θ = 90 degrees so Work is zero.

h

Conservation of Energy

0= ΔK + Δ U0 = Kf - Ki + Uf- Ui

Ki + Ui = Kf+Uf0 + mgh = ½ m v2

f + 0vf = sqrt(2 g h)

30

“Why mass doesn't matter when you drop something?”

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θ

h

i

j

Total Energy

mghmvUKE g +=+= 202

1

At y = 0

mghmvmvE +== 202

1221

jsinvicosvv θ+θ= 000

θ= cosvv:i x 0

gtsinvv:j y −θ= 0

gghsinvsinv

t

gttsinvhy

2

022

00

221

0

+θ+θ=

=−⋅θ+=

ghsinvvy 2220 +θ−=

ghvv 220 +=

θ++θ=+= 220

220

22 2 cosvghsinvvvv yx

ghv 220 +=

(보기문제)

v at y = 0?

vo

에너지보존법칙이용

등가속운동법칙이용

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m2m1

θR

Total Energy of m1 ffii UKUKE +=+=1

( ) ( ) ( )oifif gRmvmUUKK θcos1 12

121 +−−=⇒−−=−

( ) ( )ogmRvmamgmTF θθ cos120 1

2

111 −===−==

( )ogmT θcos231 −=

gmT 2>If , it lift up m2 . In case, 12 2mm =

( ) gmgmgm o 121 2cos23 =>− θ

21cos <oθ Therefore, If θο > 60°, it lifts up m2 .

(보기문제)

What is the minimum of the initial angle θοfor lifting m2 (= 2m1) at θ = 0?

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(a) No Friction

tE

0

30sin

221

21

221

+=

==

+=

f

ii

mv

mgdmgd

mgymv

glv f =(b) Friction coefficient μk

HUKUKE ffiit ++=+=Heat LossH = fk·d = μkmg cos30°·d

mgcosθ

mgdmvmgd kf μ232

21

21 +=

( )gdv kf μ31−=

(보기문제)vf ? (a) without friction, (b) with friction (μk)

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SummarySummary

Conservative Forces• Work is independent of path• Define Potential Energy U

– Ugravity = m g y– Uspring = ½ k x2

Work – Energy Theorem

internalEUKEW total Δ+Δ+Δ=Δ=