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Experimental Physics - Mechanics - Energy and Work 1
Experimental Physics EP1 MECHANICS
- Energy and Work -
Rustem Valiullin
https://bloch.physgeo.uni-leipzig.de/amr/
Experimental Physics - Mechanics - Energy and Work 2
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.
Experimental Physics - Mechanics - Energy and Work 3
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!
Experimental Physics - Mechanics - Energy and Work 4
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
Experimental Physics - Mechanics - Energy and Work 5
WEk =D
Work- kinetic energy theorem
F!
S!
mfv!
FSsdFW ò ==!!
222
220
2ff mvmvmv
W =-=
mFSv f2
=
Regenerative brake
Experimental Physics - Mechanics - Energy and Work 6
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 -= ,,
Experimental Physics - Mechanics - Energy and Work 7
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
--=-=
Experimental Physics - Mechanics - Energy and Work 8
Work done by gravitation
ò ==B
Ag mgSsdgmW qsin!!
gm!
v!
S gm! v!
ïî
ïíì
=
-=
mghWmghW
dg
ag
,
,
ℎ
Experimental Physics - Mechanics - Energy and Work 9
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
Experimental Physics - Mechanics - Energy and Work 10
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
Experimental Physics - Mechanics - Energy and Work 11
Taking account of friction
0cos =+- qmgFn
nkk Ff µ=
qµ cosmgLFLW k==
nF!
gm!
kf!
F!
q
Experimental Physics - Mechanics - Energy and Work 12
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 -=×=!!
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 14
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
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 15
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)
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 16
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
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 17
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.
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 18
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
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 19
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
p
( )0sin q+×= tmkay
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 20
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
=
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 21
The virial theorem
ijF
i
j
Virial: −"#∑%&"
' ∑(&"' )%( * +%
− 12.%&"
')% * +%
/0 = −12.%&"
'.(&"
')%( * +%
Ensemble average
/0 = −12.%&"
'.(&"
')%( * +% Time average
2(4) = 1678
9
2 4 :4
Experimental Physics - Mechanics - Energy and Work 2222
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'()*
+',)*
+-(, . /(
Experimental Physics - Mechanics - Energy and Work 23
The virial theorem: Selected examples
−12 −$% % = 12$%
' = ()*
+, = ()*
()* =∫./()*%0%∫./ %0%
= $4 2
'
UEk -=2 +, = ()*gravity spring
( ∝ %4 +, =526(
+,−?
+, =126(
+, =1289ℎ
Experimental Physics - Mechanics - Potential Energy and Conservation of Energy 24
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!