L5 Work Energy - uni-leipzig.de · U (y) - ? l U 0 = Mg(L 1-L 2) + mgL 1 l2 + y2 = d 2 DL=l-d U y = 2MgDL-mgy +U 0 1 2 4 2 2 m M m M y eq d-= 0 5 10 15 20 25-140-120-100-80-60-40-20

Experimental Physics - Mechanics - Energy and Work 1

Experimental Physics EP1 MECHANICS

- Energy and Work -

Rustem Valiullin

https://bloch.physgeo.uni-leipzig.de/amr/


Different forms of energy

• Kinetic • Potential• Mechanical• Electromagnetic• Chemical• Internal (heat)• … many more

1 Joule = 1 kg m2 s-2

Dimension: M L2 T-2

Amalie Emmy Noether

The invariance of physical systems with respect to time translation (in other words, that the laws of physics do not vary with time) gives the law of conservation of energy.


Work

qcosFdssdFdW =×=!!

F!

S!

q

[ ] 2

2

smkg

=W

ïî

ïí

ì

=-==

=pq

pqq

,2/,00,

Fds

FdsdW

ò ò== sdFWdW !!

dzdydxsd kji ++=!

S!

sd!


Kinetic energy

F!

S!

mò=B

A

sdFW !!

dtvdtvdmsd

dtvdmW

B

A

t

t

B

A

!!

!!

òò ==

( ) 2

21

21 dvvvdvdv =×=

!!!!

dtdtdvmW

B

A

t

tò=

2

2

22

22AB mvmvW -=

energykinetic2

2

-=mv

Ek

[ ] 2

2

smkg

=kE


WEk =D

Work- kinetic energy theorem

F!

S!

mfv!

FSsdFW ò ==!!

222

220

2ff mvmvmv

W =-=

mFSv f2

=

Regenerative brake


0v!

S!

m

kf! fv

!

Work done by kinetic friction

dtdtdvmsd

dtvdmW

B

A

t

t

B

Af òò ==

2!!

22

20

2 mvmvW f

f -=

ò -==B

Akkf SfsdfW !!

SfE kk -=D

SfEE kinitialkfinalk -= ,,


Work done by spring

òò -==final

initial

final

initial

x

x

x

xs dxkxFdxW )(

kxF -= Hook’s law

2

0

2

0 22)( f

xx

xs xkxkdxkxW

ff

i

-=-=-= ò+

=

20

20

22)( f

xxxs xkxkdxkxW

ffi

=-=-= ò=

222

222 if

x

xs xkxkxkW

f

i

--=-=


Work done by gravitation

ò ==B

Ag mgSsdgmW qsin!!

gm!

v!

S gm! v!

ïî

ïíì

=

-=

mghWmghW

dg

ag

,

,

ℎ


Power

tWPav D=

dtdWP =

vFdtFds

dtsdF

dtsFdP !!

!!

!!!!×=+==

)(

For constant force

[ ] 3

2

smkgWatt ==P

1 Watt = 1 J/s1 Horsepower = 1 hp = 746 W1 kilowatt-hour = 1 kWh = 3.6´106 J


Physics of bicycle

F!

f!

D! v, km/h P, W

20 9040 ?

amDf !""=-

2kvDf ==

For the constant velocity case

3kvfvP ==3

2

1

2

1÷÷ø

öççè

æ=vv

PP

1 Horsepower = 1 hp = 746 W


Taking account of friction

0cos =+- qmgFn

nkk Ff µ=

qµ cosmgLFLW k==

nF!

gm!

kf!

F!

q


Ø Work is energy transferred to/from an object by

applying a force to this object.

Ø Kinetic energy is associated with the motion of an

object. Change in the kinetic energy is due to work done

on the object.

Ø Frictional, gravitational and spring forces

can do work.

Ø Power is the rate at which work is done

on an object.

To remember!

Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 13

Conservative forces

sdFdW !!×= F

!sd!

ò == 0veconservati sdFW !!

Contour integral: integration over a closed curve or loop

1

2

a

b

c

Ø Position-dependent forces

ïî

ïíì

+==

+==bc

ba

WWWWWW

2112

2112

00 ca WW 1212 =Þ

dUsdFdW -=×=!!


Field forces

kdEmvdvdtdtdvmsdFdW =÷÷

ø

öççè

æ==×=

2

2!!

grgr dUmghdmgdhdW -=-=-= )(

spsp dUkx

dkxdxdW -=÷÷ø

öççè

æ-=-=

2

2

CC dUrqkqddr

rqkqdW -=÷

øö

çèæ-== 21

221


Equilibrium

2

2kxUsp = dxdU

F -=

-10 -5 0 5 10

-2

-1

0

1

2

3

U(x)

x

F(x)

-10 -5 0 5 10

-10

-5

0

5

U(x)

x

-10 -5 0 5 10

-1

0

1

2

3

4

5

U(x)

x

F(x)


Mapping the potential field

M

M

m

d2

1L2L

y

?)( -yU

l

1210 )( mgLLLMgU +-=

222 dyl =+

dlL -=D

02 UmgyLMgUy +-D= 22 412Mm

Mmdyeq-

=0 5 10 15 20 25

-140

-120

-100

-80

-60

-40

-20

U(y

), J

[arb

. u.]

y, m

M = 3 kgm = 5 kgd = 5 m

yeq = 7.54 m


Conservation of mechanical energy

kEUsdFW D=D-=×= ò!!

total

0=D+D UEk

constUEE k =+º

Mechanical energy

å+= inc FFF!!!

ò åòò +×=×= sdFsdFsdFW inc!!!!!!

total

kincinctotal EUWWWW D=D-=+= åå

EEUW kinc D=D+D=å Work done by nonconservative forces is equal to change of mechanical energy.


A mass on a spring under gravitation

å =+= 0ikinitial UEE

021

21 22 =+-= kymgymvE

-yeq

0

y

-ym

kmgykymgy mm20

21 2 =Þ=+-

0=+-= eqkymgdydU

kmgyeq =

0 5 10 15 20 25 30 35

-10

0

10

20

U(y)

y

eqy my


A mass on a spring: dynamics (no gravity!)

0

a

-a

y

myUEvyUmvE ))((2)(

2

2 -=Þ+=

dtmyUE

dym

yUEdtdyv 1

))((2))((2

=-

Þ-

==

2)(

2kyyU = ?=E2

2kaE =

dtmk

yady

=- 22 òò =

--

ty

a

dtmk

yady

0

'

22

ayy

aay /'arcsin

2/3

' sinp

q=-

tmk

ay

aadaay

=-=-ò 2

3'arcsin

sin

cos/'arcsin

2/3222

pq

qq

p

( )0sin q+×= tmkay


Physics of pendulum

L

gm!h

'L

0q

T

0+=mghEi 20 2mvEf +=

LLh =+ ' 0cos' qLL =

)cos1(22 02 q-== gLghv

)cos1(2 02 q-==- gLmvmgT

s dtdvmmg =qsin qLs =

dtdL

dtdsv q

==dtd

ddv

dtdv q

q= q

qsing

Lv

ddv

=


The virial theorem

ijF

i

j

Virial: −"#∑%&"

' ∑(&"' )%( * +%

− 12.%&"

')% * +%

/0 = −12.%&"

'.(&"

')%( * +%

Ensemble average

/0 = −12.%&"

'.(&"

')%( * +% Time average

2(4) = 1678

9

2 4 :4


The virial theorem: gravitational field

2ij

jiij r

mGmF =

ijF

i

jij

jiij r

mGmU -=

1a!

amF !!= R

mvRGmMF

2

2 ==R

GmMmv22

2

=

RGmMU -=

UEk -=2

!" = −12'()*

+',)*

+-(, . /(


The virial theorem: Selected examples

−12 −$% % = 12$%

' = ()*

+, = ()*

()* =∫./()*%0%∫./ %0%

= $4 2

'

UEk -=2 +, = ()*gravity spring

( ∝ %4 +, =526(

+,−?

+, =126(

+, =1289ℎ


Ø Work done by conservative forces along a closed

trajectory is zero.

ØUnder action of conservative forces mechanical energy

of the object does not change.

Ø Mechanical energy is sum of kinetic and potential

energies.

Ø Work done by nonconservative forces

equals the change of mechanical

energy.

To remember!

Documents

L5 Work Energy - uni-leipzig.de · U (y) - ? l U 0 = Mg(L 1-L 2) + mgL 1 l2 + y2 = d 2 DL=l-d U y = 2MgDL-mgy +U 0 1 2 4 2 2 m M m M y eq d-= 0 5 10 15 20 25-140-120-100-80-60-40-20