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Announcements
For the lab this week, you will have 2 lab session to complete it.They will be collected after week 1 and redistributed the following week.
Pick up HW assignments: Due next Wednesday.
Read page 7-13 in Schwarz. Do exercises 1-6. In http://particleadventure.org/particleadventure/frameless/startstandard.html read through the Section “Particle Decays and Annihilations” Slides a-h (see slide key on Course Assignments Web page)
Conservation Laws
Conservation Laws
Conservation laws in Physics can give explanations as to why some things occurand other do not.
Three very important Conservation Laws are:
I. Conservation of Energy
II. Conservation of Momentum
III. Conservation of Charge
Energy Conservation (I)There are many forms of energy.
For now, we’ll focus on two types1. Kinetic Energy (KE) – Energy of motion
KE = ½ mv2 if v is much less than c (v << c)
2. Mass Energy
E = mc2
m = massc = speed of light = 3x108 [m/sec]
That is, mass is a form of energy, and the “conversion” is to just multiply the mass by a constant number (the speed of light squared)!
Conservation of Energy (II)
A BvA vB
Total Energy (after decay) = EA + EB = (KEA+mAc2) + (KEB+mBc2)
Suppose D “decays” into 2 particles A and B, what is the energy of the system afterward?
DTotal Energy (initially)
= ED
= mDc2
Since energy must be conserved in the “decay” process,
mDc2 = (KEA+mAc2) + (KEB+mBc2)
Conservation of Energy (III)mDc2 = (KEA+mAc2) + (KEB+mBc2)
Important points here:
1) This equation DOES NOT say that kinetic energy is conserved
2) This equation DOES NOT say that mass is conserved
3) This equation states that the total energy is conserved
Total energy before decay = Total energy after decay
EA EBED
Before Decay After Decay
Conservation of Energy (IV)mDc2 = (KEA+mAc2) + (KEB+mBc2)
Since mA and mB must be larger than zero, and vA2>0 and vB
2>0, the KE can only be positive (KE cannot be negative!)
mDc2 mAc2 + mBc2
mD mA + mB
This is also true if particle D has KE>0
also!
KEA = ½ mAvA
2 KEB = ½ mBvB
2> 0 > 0
If I subtract off the KE terms from the RHS* of the top equation, Iwill no longer have an equality, but rather an inequality:
and dividing both sides by c2,
Conservation of Energy (V)
MDc2
MAc2 MBc2
KEA KEBMBc2
MDc2
MAc2
LHS = RHS
LHS > RHS
Energy Conservation (VI)
D
Consider some particle (call it “D”) at rest which has a mass of 0.5 kg
Which of the following reactions do you think can/cannot occur?
DA BmA=0.2 kg mB=0.1 kg
I
DAmA=0.2 kg mB=0.4 kg
BII
DA mA=0.1 kgIV
B mB=0.1 kg
DA BmA=0.49 kg mB=0.0 kg
III
Energy Conservation (VII)
Bamq q
Fig. A
t
t
A particle (q) and an anti-particle (q) of equal mass each having 1 [TeV] of energy collide and produce two other particles t and t (of equal mass) as shown in Fig. A. (1 [TeV] = 1012 [eV])
Energy Conservation (VIII) What is the total energy in the collision ? A) 0 B) 2 [TeV] C) 1 [TeV] D) 0.5 [TeV]
What is total energy of the t and t (individually)? A) 0 B) 2 [TeV] C) 1 [TeV] D) 0.5 [TeV]
What can be said about the mass energy of the “t” particle ? A) It’s equal to the mass of “q” B) It must be less than 0.5 TeV C) It must be less than 1 [TeV] D) It’s equal but opposite in
direction to that of the t particle
Momentum Conservation (I)Momentum (p) = mass x velocity = mv p = mv
Momentum has a direction, given by the direction of v
m1 v1p1 = m1v1
m2
v2p2 = -m2v2
Note that particles moving in opposite directions have momenta which are opposite sign!
Momentum Conservation: In any process, the value of the total momentum is conserved.
Momentum Conservation (II)
m1 v1
m2v2
Consider a head-on collision of two particles
What is the total momentum before the collision ? A) m1v1+m2v2 B) m1v2-m2v1 C) zero D) (m1+m2)(v1+v2)
If m1= m2, what can be said about the total momentum? A) it’s zero B) it’s positive C) it’s negative D) can’t say?
If m1= m2 and v1 > v2, what can be said about the total momentum? A) it’s zero B) it’s positive C) it’s negative D) can’t say?
Momentum Conservation (III)
If m1< m2 and v1 > v2, what can be said about the total momentum? A) it’s zero B) it’s positive C) it’s negative D) can’t say?
If m1= m2 and v1 = v2 (in magnitude), what can be said about the total momentum? A) it’s zero B) it’s positive C) it’s negative D) can’t say?
In this previous case, what can be said about the final velocities of particles 1 and 2 ? A) their zero B) equal and opposite C) both in the same direction D) can’t say?
m1 v1
m2v2
Consider a head-on collision of two particles
Momentum Conservation (IV)
DA BmA
mB
IvA vB
Consider a particle D at rest which decays into two lighter particles A and B, whose combined mass is less than D.
If mA > mB, answer the following questions: What can be said about the total momentum after the decay? A) Zero B) Equal and Opposite C) Equal D) Opposite, but not equal
If mA= mB, what can be said about the magnitudes of the velocities of A and B? A) vA>vB B) Equal and Opposite
C) vB>vA D) Same direction but different magnitudes
Momentum Conservation (V)
Can mA+mB exceed mD ? A) Not enough data B) Yes, if vA and vB are zero C) No D) Yes, if vA and vB are in opposite
directions
Which statement is most accurate about the momentum of A ? A) Zero B) Equal to B C) Equal and opposite to B D) Opposite, but not equal
DA BmA
mB
IvA vB
Momentum Conservation (VI)
np mP
e me
Can this process occur?a) No, momentum is not conservedb) Yes, since mn is larger than the sum of mP and me
c) No, energy cannot be conservedd) Yes, but only between 8 pm and 4 am
Consider a neutron, n,which is at rest, and then decays. mp+me < mn
The observation that momentum was not conserved in neutron decay lead to theprofound hypothesis of the existence of a particle called the neutrino
neutron proton + electron + neutrino ( n p + e +
When the neutrino is included, in fact momentum is conserved.
np mP
e me
Discovery of the Neutrino
The observation that momentum conservation appeared to beviolated in neutron decay lead to the profound hypothesis of the existence of a particle called the neutrino
neutron proton + electron + neutrino ( n p + e +
When the neutrino is included, in fact momentum is conserved.
Charge ConservationThe total electric charge of a system does not change.Consider the previous example of neutron decay:
n p + e + Charge 0 +1 -1 0
Can these processes occur?
p + p p + nCharge +1 +1 +1 0 NO
p + e + nCharge +1 -1 0 0 YES
n + n p + pCharge 0 0 +1 +1 NO
Summary of Conservation Laws Total Energy of an isolated system is conserved D A + B cannot occur if mA+mB > mD
Total momentum of an isolated system is conserved - missing momentum in neutron decay signaled the existence of a new undiscovered particle
Total Charge of an isolated system is conserved - the sum of the charges before a process occurs must be the
same as after the process
We will encounter more conservation laws later which will help explainwhy some processes occur and others do not.
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