MAF1102_7-Energi Potensial & Hukum Kekekalan Energi

8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi

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7. Potential Energy and Energy Conservation

7-1. Potential Energy

Conservative force Potential Energy

d F W gg

Conservative force — Work is independent of the path.

Non-conservative force — Friction

Example 1. Gravitational Potential Energy : Relative Quantity

mghU g : Work against to the gravitational force

ggigf

i f

i f

U U U mgymgy

j y y j mg



Example 2. Elastic Potential Energy in a Spring

xk f S

xd f dW SS

Work done by Spring

22

2

1

i f

x

x

x

x SS x xk dxkx xd f W f

i

f

i

2

2

1 kxU S SiSf SS U U U W



7-3. Conservative forces and Potential Energy

For a certain conservative forcecF

Work done bycF

cicf c x x cc U U U xd F W f

i

f

i

x

x ccicf c xd F U U U

7-4. Conservation of Mechanical Energy

Total Mechanical Energy E

U K E

f f ii U K U K

Conservation of Mechanical Energy in any isolated system

dt

dU

dt

dK

dt

dE 0

vF amvdt

dv

mvmvdt

d

dt

dK

mvK x

2

2

12

2

1

vdx

dU

dt

dx

dx

dU

dt

dU

0vdxdU vF

dt dE x

dxdU F x



In General

k ˆ

z

U j

y

U i

x

U U F

k ˆ F j F i F z y x

Ex-1. Gravitation

mgzU g k ˆ mgk ˆ

z

U F

g

g

Total Energy

mgzmvU K E g

2

2

1

Ex-2. Spring

2

2

1

kxU s iˆ

kxiˆ

x

U

F s

s

2

2

12

2

1 kxmvU K E s

z

2

21 kzmgzU U U sg

k ˆ kzmgk ˆ

z

U F

mgzkzmvU K E 2

2

12

2

1



h

i

j

Total Energy

mgmvU K E g

2

02

1

At y = 0mghmvmv E

2

02

12

2

1

j sinvi cosvv000

cosvv:i x 0

gt sinvv: j y 0

g

ghsinvsinvt

gt t sinvh y

2

0

22

00

2

2

1

0

ghsinvv y 222

0

22

0

22

0

222 cosvghsinvvvv y x

ghvv 22

0

ghv 22

0



m2m1

R

ExampleTotal Energy of m1

f f ii U K U K E 1

cosgRmvm

U U K K i f i f

11

2

12

1

cosgm R

vmamgmT F 120

1

2

111

cosgmT 231

gmT 2 If , it lift up m2 .

In case, 12 2mmgmgmcosgm

121223

2

1cos

If 60°, it lifts up m2 .







7-7. Gravitational Potential Energy in general

r ˆ

r

mGM F eG 2

dr r

mGM sd F W f

i

f

i

r

r

er

r GG2

i

e

f

e

r r

r r

e

r

mGM

r

mGM

r

mGM f

i

i f U U U

i

e

f

ei f

r

mGM

r

mGM U U

Assume U i = 0 at r i =

r mGM r U e

For r = Re, r + r = Re + r

r R R

r mGM

Rr RmGM RU r RU U

ee

e

ee

eee

11

r mgr R

mGM e

e2 mghU



In general, the potential energy;

r

mGmr U 21

m1

m2

m3

r 12 r 23

r 3131

13

23

32

12

21

r

mm

r

mm

r

mm

GU t

132

13

13

122

12

21

13121r ˆ

r

mmr ˆ

r

mmGF F F t

7-8. Energy Diagram and Stability of Equilibrium

2

2

1 kxU s

00 xF

kx

dx

dU F

s

ss

& Minimum at x = 0

Stable Equilibrium

f 0 xF s & Maximum at x

Unstable Equilibrium

U

MAF1102_7-Energi Potensial & Hukum Kekekalan Energi

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