maf1102_7-energi potensial & hukum kekekalan energi
TRANSCRIPT
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 1/10
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
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 2/10
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
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 3/10
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
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 4/10
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
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 5/10
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
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 6/10
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 .
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 7/10
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 8/10
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 9/10
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
8/8/2019 MAF1102_7-Energi Potensial & Hukum Kekekalan Energi
http://slidepdf.com/reader/full/maf11027-energi-potensial-hukum-kekekalan-energi 10/10
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