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.

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

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

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

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

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?

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

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

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

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

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

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

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

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

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

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.

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 ./.

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

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

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

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

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.