physics 2d lecture slides jan 15
TRANSCRIPT
Relativistic Momentum and Revised Newton’s Laws Need to generalize the laws of Mechanics & Newton to confirm to Lorentz Transform
and the Special theory of relativity: Example : p mu=
1 2Before
v1’=0
v2’21
After V’
S’
S
1 2
Before v v 21
After V=0
P = mv –mv = 0 P = 0
' ' '1 2
' '1 21 2 2
1 122 2
2
2
2
'
' 'before after
20, , '
2
11 1
1
1
, 2 2
'
p p
afterbeforemvp mv m
v v v v v V vv v V vv v v v V
vvc
vvcc c c
p mV mv
− − − −= = = = =
−= +
= −−− − +
=
≠
+= = −
Wat
chin
g an
Inel
astic
Col
lisio
n be
twee
n tw
o pu
tty b
alls
Definition (without proof) of Relativistic Momentum
21 ( / )mup muu c
γ= =−
With the new definition relativistic momentum is conserved in all frames of references : Do the exercise
New ConceptsRest mass = mass of object measuredIn a frame of ref. where object is at rest
2
is velocity of the objectNOT of a referen
11 ( / )
!ce frame u
u cγ =
−
Nature of Relativistic Momentum
21 ( / )mup muu c
γ= =−
With the new definition of Relativistic momentum
Momentum is conserved in all frames of references
mu
Good old Newton
Relativistic Force & Acceleration
Relativistic Force And
Acceleration
21 ( / )mup muu c
γ= =−
( )
( )
( )
3 / 2 22 2
2 2 2
3 / 2
2
3 / 2
2
2
2
1 ( / )
: Relativistic For
1 2
ce
( )( )
1 ( / )
Since A
21 ( / ) 1 (
ccel
/ )
1 (
e
)
a
/
r
d du dusedt dt du
m mu u duFc dtu c u c
mc mu mu duF
dudt
dp d muFdt dt u c
mFu c
dtc u c
=
− − = + × − −
− + = −
= = −
= −
3 / 22
tion a =
Note: As / 1, a 0 !!!!Its harder to accelerate when
,
Fa =
you get closer to speed of light
1 ( / )m
d
c
ut
u c
u
d
⇒
→
−
→
Reason why you cant quite get up to the speed of light no matter how hard you try!
A Linear Particle Accelerator
V
+- F
E
E= V/dF=eE
3/ 2 3/ 22 2
2 2
Charged particle q moves in straight line
in a uniform electric field E with speed u
accelarates under f F=qE
a 1 =
orce
larger
1
the potential difference V a
du F u qE udt m c m c
= = − −
cross
plates, larger the force on particle
d
q
Under force, work is done on the particle, it gains
Kinetic energy
New Unit of Energy
1 eV = 1.6x 10-19 Joules
PEPPEP--II accelerator schematic and tunnel viewII accelerator schematic and tunnel view
A Linear Particle Accelerator
3/ 22eEa= 1 ( / )m
u c −
Magnetic Confinement & Circular Particle Accelerator V
2
2
ClassicallyvF mrvqvB mr
=
=BF
B
r
2
2
( )
(Centripetal accelaration)
dp d mu duF m quBdt dt dt
du udt rum quB mu qBr p qBr
r
γ γ
γ γ
= = = =
=
⇒ == ⇒ =
Test of Relativistic Momentum In Circular Accelerator
2
1 ( / )mup muu c
qBrm
mu qB
u
rγ
γ
γ
=
=
=
=−
Relativistic Work Done & Change in Energy
x1 , u=0
X2 , u=u
2 2
1 1
3 / 22 2
2 2
3 / 220
2
22
3 / 2 1 / 22 20
2 2
2 2
substitute i
. .
, , 1 1
n W
(change in var x u 1
1 1
W rk d
)
o
x x
x x
u
u
dpW F dx dxdt
dummu dp dtpdtu u
c cdumdtWuc
mudu m
udt
mccW
c
c
c
mmcu u
γ
= =
= ∴ = − −
∴ =
−
= = − =
− −
→
−
∫ ∫
∫
∫
2
2
2
2
one is change in Kinetic energy KK = or Total Ener E= gy
mc mcKmc mcγ
γ −
= +
Why Can’s Anything go faster than light ?
( )
2
2 222 21/ 2 1/ 22 2
2 2
2 22 4 2
22 2
2
2
(Parabolic in Vs )
1 2Non-relativistic case: K =
1 ( 1)
1 1
2
1
uK Ku c
mc mcK mc K mcu uc c
u m c K mcc
mc mc
Kmu um
−
−
= − ⇒ + = − −
⇒ − = +
⇒
−
+
=
=
⇒
Lets accelerate a particle from rest, particle gains velocity & kinetic energy
Relativistic Kinetic Energy & Newtonian Physics2
12 22
2
2
2
22
2
22
Relativistic KE =
1When , 1- 1 ...smaller terms2
1so [1 ] (classical form recovered)1
22
u uu cc c
uK mc
mc
mcc
mc
mu
γ−
−
<< ≅ − +
≅ − − =
2 2
2
For a particle
Total Energy of a Pa
at rest, u = 0
Total Energy E=
r
m
ticle
c
E mc KE mcγ= = +
⇒
Relationship between P and E2 2 2 4
2 2 2 2 2 2
2 2 2 2 4 2 2 2 2 2 2 2 2
2 2 2 42 2
2
2 2
2
2 2 2 2
2
22
2
2
2 4
( )
= ( ) ( )
........important relati( on
For
)
1
p
u
E mc
p mu
E p c mc
E m c
p c m u c
E p c m c m u c m c um c m cc u c m
c uu
c
c
γ
γ
γ
γ
γ
γ γ
=
=
⇒ =
⇒ =
⇒ − = − = −
+
−−
=
− =−
=
2 2 2 2 4
EE= pc or p = (light has momentum!)c
Re
articles with zero rest mass like photon (EM
lativistic Invariance :
waves)
: In all Ref Frames
Rest M
E p c m c− =
ass is a "finger print" of the particle
Mass Can “Morph” into Energy & Vice Verca
• Unlike in Newtonian mechanics• In relativistic physics : Mass and Energy are the same
thing• New word/concept : Mass-Energy• It is the mass-energy that is always conserved in every
reaction : Before & After a reaction has happened• Like squeezing a balloon :
– If you squeeze mass, it becomes (kinetic) energy & vice verca !• CONVERSION FACTOR = C2
Mass is Energy, Energy is Mass : Mass-Energy Conservation
be
2
f
2
ore after
2 22
2
2
2
2
2
2
2
2 2 1
Kinetic energy has been transformed
E E
into mass increase
2 2 - 21
1 1
mc mc Mc
K m
u uc
cM M
mM m
m
ucc
c c u
=
+ = ⇒
− −
∆ = = =
= >
−
−
2
2
2
mc
c
−
Examine Kinetic energy Before and After Inelastic Collision: Conserved?
S
1 2
Before v v 21
After V=0
K = mu2 K=0
Mass-Energy Conservation: sum of mass-energy of a system of particles before interaction must equal sum of mass-energy after interaction
Kinetic energy is not lost, its transformed into more mass in final state
Conservation of Mass-Energy: Nuclear Fission
22 22 31 2
1 2 32 2 21 2 32 2 21 1 1
M cM c M cMcu u uc c
M M
c
M M= +
−
> ++
−
+⇒
−
M M1 M2M3+ + Nuclear Fission
< 1 < 1 < 1
Loss of mass shows up as kinetic energy of final state particlesDisintegration energy per fission Q=(M – (M1+M2+M3))c2 =∆Mc2
1423692
10
-28
355
9092+ +3 n
m=0.177537u=2.947
U Cs R
1 10 165.4 MeV
b
kg∆ × =
→
What makes it explosive is 1 mole U = 6.023 x 1023 Nuclei !!