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.