momentum for n particles: why bother introducing one more definition? most general form of the...

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?