21,22
HW2 9/18/2008
LL #5
Weihao Ge Irakli YK
u
u
u
L = T ( y, , )
F t = 6my (CM) u t F s = F =T 2 u = y a cos
1 v= u 10
L* = T * ( y, , )
L* p = p ( , y ) = 0 ?
L* = T * ( y, , )
py = L* y m u 2 2 T = ? mu py
=
v=
9y 10l
=
v=
1 u 10
9y 10l
1 u 10
1
9/18/2008
HW2 #7
Is it possible to express Qu and Qw in terms of Fr and F only ?
F r = F q
i i i i i, j
ri
j
q j = Q j q j
j
Q j = Fi
i
ri q j
Qu = Fr
r + r F u u
HW2 HW2 #7
t=
d 2 E U eff ( ) ml 2
U eff =
=
Mz d sin 2 E U ( ) l 2m eff
M z2 mgl cos = E 2ml 2 (1 cos 2 )
Motion of planets:
p = 1 + e cos r p latus rectum (not momentum) = e eccentricity = 1 + where U = 2 EM 2 m 2 and < 0; > 0 M2 m
r
=
r
How the equation of orbit will change if we change sign of (or )? If > 0 (repulsive force) how we can find the region of the finite motion? When the trajectory of motion in attractive Coulomb field is a hyperbola and when a parabola?
2
9/18/2008
What is called perihelion and aphelion of the orbit?
Additional integral of motion for Coulomb central field (LL end of 15).
I = [v M ] + r r = const
There are integrals of motion: (1) total energy; (2) angular momentum; and why now there is (3) if for the description of motion you need only two coordinates (r,) two and equations of motion?
Particles decay (spontaneous decay) in CMS:
Eint 0 = Eint1 +
2 p0 p2 + Eint 2 + 0 2m1 2m2
v01 , v02 are fixed in CMS
= Eint 0 Eint1 Eint 2 > 0 =
2 2 p0 1 1 p0 + = 2 m1 m2 2m
disintegration energy (usually known)
reduced mass
Laboratory reference system LS:
v ( LS ) = V (CMD in LS) + v0 (CMS )
v 2 + V 2 2vV cos = v0 v = v(cos ) in Lab v0 and V are fixed
3
9/18/2008
CMS and LS angles are related by
v = V + v0
What is min and max velocity in the Lab?
Distribution functions
in CMS the distribution of secondary particles is isotropic
dN = N
d 0 4
d =
dN = const d 0
d = d d cos dN N d 0 = d 0 4 d 0
dA R2
d
dN
0
d 0 = N =
dN dN d d cos = const d 0 d cos
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9/18/2008
Angular distribution in CMS
dN d0 2 sin 0 d 0 d cos 0 = = = N 4 4 2
dN N = = const d cos 0 2
angular distribution in CMS
Kinetic energy distribution in Lab (e.g. for final particle 1)
2 from v 2 = v0 + V 2 + 2v0V cos 0 d cos 0 = cos
d (v 2 ) 2v0V
dT1 dN = N 2m1v0V
or
dN N = = const dT1 2m1v0V
energy distribution in LS
Lets plot this uniform distribution
dN dT
Tmin =
N 2mv0V
T
Tmax
Tmax
m1 (v01 V ) 2 2 m = 1 (v01 + V ) 2 2
Tmin
Check that integral of this distribution = N
? What will happen if LS system will coincide with CMS system?
5