momentum for n particles: why bother introducing one more definition? most general form of the...
Post on 21-Dec-2015
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Momentum vmp
N
iitotal pP
1
For N particles:
Why bother introducing one more definition?
Fdt
pd
Most general form of the Newton’s law:Valid when mass is changingValid in relativistic physics
N
iexti
total Fdt
Pd
1,
Total momentum is conserved if the sum of external forces equals to zero!
Nuclear reactions• Fission: decay of heavy nuclei into lighter fragments
•Fusion: synthesis of light nuclei into a heavier nucleus
Energy released per proton is ~10-20 MeV!!
Energy is released in fission reaction if the mass of an initial nucleus is larger that the sum of masses of all final fragments
MU > MRb + MCs + 3 mn
Rubidium and Cesium are more tightly bound, or have larger binding energy than Uranium.
It is energetically favorable for Uranium to split.
When is the energy released in fission reactions?
M = MU – (MRb + MCs + 3 mn)
Energy released E = M c2
Proton-proton cycle: four hydrogen nuclei fuse to form one helium nucleus
Hydrogen Fusion
Einstein’s relation: E = mc2
!04 mmm Hep
J103.4MeV8.26 122 cmE
Energy released in one reaction:
(Binding energy)
kg10048.0 27m
Hans Bethe 1939
0.007, or 0.7% of the rest energy of protons (4mpc2) is released
This is 107 times more efficient than chemical reactions!
600 million tons of hydrogen are fused every second on the Sun!
How much hydrogen should be fused per second to provide the Sun’s luminosity?
W104sec1
007.0 262 cm
L
Nuclear fusion efficiency:0.7% of the hydrogen mass is converted into radiation in the p-p cycle
Matter-antimatter annihilation has even higher efficiency: 100% !!
kg106007.0
104 112
26
c
m
There is more than enough nuclear fuel for 1010 years!
years10310104 10
5612
Lt
Does nuclear fusion provide enough energy to power the Sun?
Assume 1056 protons in the core:
10,000 years
Neutrino have very small mass, no electric charge, and they almost do not interact with matter
How do we know? Neutrino!!
Neutrino image of the Sun
Center of Mass
i
N
iicm rm
Mr
1
1
N
iimM
1
i
N
iicm xm
Mx
1
1i
N
iicm ym
My
1
1
Motion of the Center of Mass
N
iiicm vm
Mv
1
1
N
iiixcm xvm
Mv
1
1
N
iiiycm yvm
Mv
1
1
i
N
iicm am
Ma
1
1
N
iicm FaM
1
N
iicm FaM
1
The center of mass of a system moves as if all of the mass of the system were concentrated at that point and as if all of the forces were acting at that point
For internal forces
external
N
iexternalicm FFaM
1
jiij FF
Only external forces affect the motion of the center of mass
Momentum is a vector!
vmp
cmi
N
ii
N
iitotal vMvmpP
11
N
iiicm vm
Mv
1
1
cmtotal vMP
Vector equation!
dt
Pd
dt
vdMFaM totalcm
externalcm
externaltotal Fdt
Pd
If ,0externalF
0dt
Pd total
ConstPtotal
Conservation of Momentum
If there is no net external force acting on a system, then the total momentum of the system is a constant
ConstPtotal
)()( afterPbeforeP
True in X and Y directions separately!
externalx
totalx Fdt
dP external
y
totaly Fdt
dP
,0externalxFIf only then consttotal
xP
Problem Solving
For Conservation of Momentum problems:
1. BEFORE and AFTER
2. Do X and Y Separately
From B. Dutta’s talkOctober 2008
Perfectly elastic collision
A collision in which the total kinetic energy after the collision is the same than that before the collision is called an elastic collision.
A B
Av
Inelastic collision
A collision in which the total kinetic energy after the collision is less than that before the collision is an inelastic collision.
A
Av
B
0BvBEFORE
AFTERA B
Vafter?