Page 1
Wed.
Fri.
6.8-.10, 12,13 Introducing Potential Energy, Gravitational
6.11, 14-17 Visualizing Electric and Rest Energy
RE 6.c
RE 6.d,e
Mon.
Tues.
Things Engineers and Physicists Do
EP6, HW6: Ch 6 Pr’s
58, 59, 91, 99(a-c),
105(a-c)
motion is neither created nor destroyed, but transferred via interactions.
all
f
i
syssysif rdFmcmc
22 Energy Work
WE
Accounting for Interactions internal to the system – change in potential energy.
Page 2
233213311221321 WWWWWWEEE
extextnet
extextnet
WWWrFFFrFE
WWWrFFFrFE
32313332313332
23212223212222
1
2
3
extextnet WWWrFFFrFE 13121113121111
12F
13F
31F
21F
32F 23F
1r
3r
2r
extextext WWW 321
extsystemernal WWEEE int321
ExtF1
ExtF 2
ExtF 3
System
Energy and internal interactions
Page 3
System 1
2
3
12F
13F
ExtF1
31F
21F
32F
ExtF 2
23F
ExtF 3
1r
3r
2r
Unlike Internal Forces, Internal Work’s don’t necessarily cancel.
For Example…
233213311221internal WWWWWWW
Energy and internal interactions
Page 4
Work on/by Pair of Particles
1
2
21F
12F
12r
2r1r
12212,1 WWW
1r
2r
2121212,1 rFrFW
1221 FF
By Reciprocity
2211212,1 rFrFW
2121 rrF
21212,1 rFW
Page 5
1
2
3
12F
13F
31F
21F
32F 23F
1r
3r
2r
extsystemWWEEE internal321
ExtF1
ExtF 2
ExtF 3
System
Energy and internal interactions
extsystemWWEEE internal321
internalWU system
Potential Energy
extsystemsystem WUEEE 321
extsystemsystem WE
Page 6
Say we have an isolated system, so there are no external
interactions. If the sum of particle energies increases by 50 J,
what must be the change in the system’s potential energy?
Page 7
Work on/by
Pair of Particles
1
2
21F
12F
12r
2r1r
1r
2r
21212,1 rFW
Potential Energy Change of
Pair of Particles
21212,1 rFU
Potential Energy is shared by members of a system
Page 8
1
2
3
12F
13F
31F
21F
32F 23F
1r
3r
2r
ExtF1
ExtF 2
ExtF 3
System
Energy and internal interactions
extsystemsystem WUEEE 321
systemU 2,1U 3,2U 1,3U
3,23,12,1321 UUUEEEEsystem
Page 9
Say you have a system of 4
particles; in the system’s total
energy expression, how may pair-
wise potential energy terms (like
U1,2) are there?
(it may help to write them out)
a) 2
b) 4
c) 6
d) 8
e) 10
f) 12
g) 14
h) 16
Write the energy of a system containing four particles, including the relativistic
particle energies E1 etc. plus the potential energy pairs U12 etc.
A spacecraft travels from near the Earth
toward the Moon. How many gravitational
potential energy terms Ug are there in the
Energy Principle?
System: Earth, Moon, spacecraft
a) 1
b) 2
c) 3
d) 6
e) 0
Page 10
A thrown ball
heads straight up.
SYSTEM: Ball
What is the work done by the
surroundings?
1) 0
2) mgy
3) –mgy
4) something else
A thrown ball heads straight up.
SYSTEM: Ball + Earth
What is the work done by the
surroundings?
1) 0
2) mgy
3) –mgy
4) something else
A thrown ball heads straight up.
SYSTEM: Ball + Earth
What is the change in
potential energy?
1) 0
2) mgy
3) –mgy
4) something else
Page 11
You drop a metal ball from 1 m up. How fast is it going just before it hits the
ground?
System= ball+Earth extsystemWE
Earth & Ball Revisited
𝑦
Active members of
environment = none
(neglecting air’s resistance)
yyrballˆ
ymgF EarthBallˆ
m1y
EbEbbE rdFU
,
0,.., bEbEbrestErestbE UKKEEENot changing
ibfbb KKK .. Pretty sure v<<c, so 2
21 mvK
02
21 ymgmv f
ymgmv f 2
21
ygv f 2 ms
m 18.92 2 sm4.4
?fv
initial
final
0
EbEbbE rFU
,
Constant force, so…
ymgU bE ,Pause and Consider:
does sign make sense?
0,, bEbbE UKE
Page 12
𝑒𝑛𝑒𝑟
𝑔𝑦
Visualizing Energy
𝐹 𝐸 = 𝐹𝑜𝑟𝑐𝑒 𝑜𝑓 𝐸𝑎𝑟𝑡ℎ′𝑠 𝑝𝑢𝑙𝑙
𝐹 𝐴 = 𝐹𝑜𝑟𝑐𝑒 𝑜𝑓 𝐴𝑖𝑟′𝑠 𝑑𝑟𝑎𝑔
𝑣 = 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝐾 = 𝑏𝑎𝑙𝑙′𝑠 𝐾𝑖𝑛𝑒𝑡𝑖𝑐 𝐸𝑛𝑒𝑟𝑔𝑦
𝑈 = 𝑏𝑎𝑙𝑙 & 𝐸𝑎𝑟𝑡ℎ′𝑠 𝑠ℎ𝑎𝑟𝑒𝑑 𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑛𝑒𝑟𝑔𝑦
𝑈 + 𝐾
Page 13
Thinking about Potential Energy
Work is something you do
Potential is something you have
Example: Work-Study Contract
Energy of Configuration
The Physics is in the change in Potential
Have vs. Do
Page 14
A ball of mass 0.1 kg is dropped
from rest near the Earth.
The ball travels downward 2 m,
speeding up.
SYSTEM: Ball
What is the work done by
the surroundings?
a) 0
b) + 1.96 J
c) – 1.96 J
A ball of mass 0.1 kg is dropped
from rest near the Earth.
The ball travels downward 2 m,
speeding up.
SYSTEM: Ball + Earth
What is the work done by
the surroundings?
a) 0
b) + 1.96 J
c) – 1.96 J
Page 15
A cart on a track connected by a string,
over a pulley to a weight that hangs off the
edge of the table.
SYSTEM: cart + weight + Earth
How many energy terms (rest, kinetic, potential) are
in the initial work-energy relation? Include ones that
probably aren’t changing.
a) 2
b) 3
c) 4 d) 5
e) 6
f) 7
g) 8
h) 9
i) 10
Page 16
0 wcwEcEwcEwrestcrestErest UUUKKKEEE ,,,...
Fun with near-Earth Gravitation System= Earth + cart + weight
Active surrounds = negligible
(ignoring friction and energy
invested in spinning up the pulley)
gssurroundinbEbE WE ,,0
02
212
212
212
21 ygmvmvmvmvm wiwfwicfc
02
212
21 ygmvmmvmm wiwcfwc
022
ygmm
mv
wc
wf
ygmm
mv
wc
wf
2Pause:
reasonable dependences?
If it starts from rest, how quickly is the 0.54-kg cart
moving once the 0.20-kg weight has fallen 0.50m?
sm
sm
f mkgkg
kgv 615089
20540
2022 ...
..
.
check:
units work?
initial
initial
final
final
my 500.y
x
kgmc 540.
kgmw 200.
Page 17
Fun with near-Earth Gravitation System= Earth + ball
Active surrounds = negligible
a) If it is thrown and caught at the same elevation, how are
the initial and final speeds related?
0 bEbEbrestErest UKKEE ,..
gssurroundinbEbE WE ,, 0
022 ibfb vv ..
02
212
21 ifbibbfbb yygmvmvm ..
0222 ifibfb yygvv ..
They’re the same.
b) Say at a certain instant, the ball is moving 6 m/s and at an
elevation of 10 m. How fast is it going when it is at 5 m?
smmmsmsmv fb /././. 2181058926 22
ifibfb yygvv 22
..
c) If it were thrown straight up and it peaked at around 5.1m
above the launch height (in the ball park of what happened in
lab), what must have been it’s launch / initial speed?
0222 ifibfb yygvv ..
ifib yygv 2. smmsm 1015892 2 ./.
Page 18
Newton’s Universal Law of Gravitation
Force
12r
1m
2m
12F
21F
122
12
21
rr
mmG ˆ
12F
22
1110676kg
mNG
.
12r 12r
12
12
r
r
Page 19
Newton’s Universal Law of Gravitation
Potential Energy
ir .12
1m
2m
iF .12
iF .21
122
12
21
rr
mmG ˆ12F
fr .12
fF .21
fF .12
ifr
mmG
r
mmGU
.21
21
.21
212,1
f
i
r
rr
rdFU
21
21212,1
f
i
r
rr
rdrr
mmGU
21
21212
21
212,1
ˆ
f
i
r
rrr
drmGm
21
2
21
2121
21
212,1
r
mmGU
At any particular separation
Warning: the physics is in
the change not the value
Page 20
Rocket to the Moon
Emr
Em
mm21
212,1
r
mmGU
Err
rm
rmr
System= Earth + rocket + Moon
Active members of environment = negligible
Ev
mv
rv
0,,,... rMMErErMErestMrestrrestE UUUKKKEEE
0,,,, extrmErmE WENot or negligibly changing
0,, rMrEr UUK
02
212
21
EM
RM
M
RM
E
RE
ME
REiRfR
R
mmG
R
mmG
R
mmG
R
mmGvmvm
ER
MR
Initial: on Earth Surface
Final: on Moon Surface
Kr
Ur,E+ Ur,M
Kr+Ur,E+ Ur,M
04_02.py
Page 21
Rocket to the Moon
Emr
Em
mm21
212,1
r
mmGU
Err
rm
rmr
System= Earth + rocket + Moon
Active members of environment = negligible
Ev
mv
rv
0,,,... rMMErErMErestMrestrrestE UUUKKKEEE
0,,,, extrmErmE WENot or negligibly changing
0,, rMrEr UUK
02
212
21
EM
RM
M
RM
E
RE
ME
REiRfR
R
mmG
R
mmG
R
mmG
R
mmGvmvm
ER
MR
Initial: on Earth Surface
Final: on Moon Surface
Kr
Ur,E+ Ur,M
Kr+Ur,E+ Ur,M
Page 22
Rocket not quite to the Moon Em
Err
rm
System= Earth + rocket
Active members of environment = negligible
Ev
rv
0 rErErestrrestE UKKEE ,..
0 extrErE WE ,,
Not or negligibly changing
0 rEr UK ,
ER Initial: on Earth Surface
Final: …
f
REfR
E
REiR
r
mmGvm
R
mmGvm 2
212
21
Kr+Ur,E remains constant
0 irEfrEirfr UUKK .,.,..
frEfrirEir UKUK .,..,.
Stalled
f
RE
E
REiR
r
mmG
R
mmGvm 2
21
f
E
E
Ei
r
Gm
R
Gmv
22
E
Eescape
R
Gmv
2
frlim
Escape Speed
Insensitive to
direction
Page 23
A huge asteroid smacks into the leading edge of the Earth – stopping
the Earth’s orbit. Subsequently, the Earth falls straight into the sun!
With what speed would the Earth hit the Sun’s surface?
System= Earth + Sun
Active environment = none
Em
kgmS
301099.1
0. iEv
?. fEvmr iSE
11
. 105.1
SEfSE RRr .
m81002.7 extsystemWE 0
0,.., SESESrestErestSE UKKEEE
Not changing
0,, SEESE UKE
iEfEE KKK .. 2
.21
fEEvm
iES
sE
fES
sE
ES
sEES
r
mmG
r
mmG
r
mmGU
..
02
.21
, ESi
sE
ESf
sEfEESE
r
mmG
r
mmGvmE
ESiESf
sfErr
Gmv11
2.
smmm
kgkgNm /1014.6105.1
1
1002.7
11099.1/1067.62 5
118
302211
Page 24
System: comet + star
As a comet travels away from a star, how does the
kinetic energy and potential energy of the system
change?
K U
a) increase decrease
b) increase increase
c) decrease increase
d) decrease decrease
e) no change no change
Page 25
Wed.
Fri.
6.8-.10, 12,13 Introducing Potential Energy, Gravitational
6.11, 14-17 Visualizing Electric and Rest Energy
RE 6.c
RE 6.d,e
Mon.
Tues.
Things Engineers and Physicists Do
EP6, HW6: Ch 6 Pr’s
58, 59, 91, 99(a-c),
105(a-c)
motion is neither created nor destroyed, but transferred via interactions.
all
f
i
syssysif rdFmcmc
22 Energy Work
WE
Accounting for Interactions internal to the system – change in potential energy.
Page 26
Different Initial Speeds / kinetic Energies, Different Paths (orbit noncircular, with energy vs position.py)
Page 27
04_potential_energy_well.py