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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.

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announcements
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 laws1
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
Energy Conservation (I)

There are many forms of energy.For now, we’ll focus on two types

1. Kinetic Energy (KE) – Energy of motion

KE = ½ mv2 if v is much less than c (v << c)

2. Mass Energy

m = mass

c = speed of light

= 3x108 [m/sec]

E = mc2

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

D

A

B

vA

vB

Conservation of Energy (II)

Total Energy (initially)

= ED

= mDc2

Suppose D “decays” into 2 particles A and B, what is the energy of the system afterward?

Total Energy (after decay)

= EA + EB

= (KEA+mAc2) + (KEB+mBc2)

Since energy must be conserved in the “decay” process,

mDc2 =(KEA+mAc2) + (KEB+mBc2)

conservation of energy iii

ED

EA

EB

Before

Decay

After Decay

Conservation of Energy (III)

mDc2 =(KEA+mAc2) + (KEB+mBc2)

  • Important points here:
  • This equation DOES NOT say that kinetic energy is conserved
  • This equation DOES NOT say that mass is conserved
  • This equation states that the total energy is conserved Total energy before decay = Total energy after decay
conservation of energy iv

KEA = ½ mAvA2

> 0

KEB = ½ mBvB2

> 0

Conservation of Energy (IV)

mDc2 =(KEA+mAc2) + (KEB+mBc2)

Since mA and mB must be larger than zero, and vA2>0 and vB2>0, the KE can only be positive (KE cannot be negative!)

If I subtract off the KE terms from the RHS* of the top equation, Iwill no longer have an equality, but rather an inequality:

mDc2 mAc2 + mBc2

and dividing both sides by c2,

This is also true if particle D has KE>0

also!

mD mA + mB

conservation of energy v

KEA

KEB

MBc2

MAc2

MDc2

MBc2

MAc2

MDc2

Conservation of Energy (V)

LHS = RHS

LHS > RHS

energy conservation vi

I

D

A

B

mB=0.1 kg

mA=0.2 kg

II

D

B

A

mA=0.2 kg

mB=0.4 kg

III

D

A

B

mA=0.49 kg

mB=0.0 kg

IV

D

mA=0.1 kg

A

mB=0.1 kg

B

Energy Conservation (VI)

Consider some particle (call it “D”) at rest which has a mass of 0.5 kg

D

Which of the following reactions do you think can/cannot occur?

energy conservation vii

Fig. A

t

Bam

q

q

t

Energy Conservation (VII)

A particle (q) and an anti-particle (q) of equal masseach having1 [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 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
Energy Conservation (VIII)
  • What is the total energy in the collision ?A) 0 B) 2 [TeV] C) 1 [TeV] D) 0.5 [TeV]
momentum conservation i

m1

p1 = m1v1

v1

v2

m2

Note that particles moving in opposite directions have momenta which are opposite sign!

p2 = -m2v2

Momentum Conservation (I)

Momentum (p) = mass xvelocity = mv

p = mv

Momentum has a direction, given by the direction of v

Momentum Conservation: In any process, the value of the total momentum is conserved.

momentum conservation ii

Consider a head-on collision of two particles

m1

m2

v1

v2

Momentum Conservation (II)

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

Consider a head-on collision of two particles

m1

m2

v1

v2

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?

momentum conservation iv

I

D

A

B

vA

vB

mB

mA

Momentum Conservation (IV)

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

I

D

A

B

vA

vB

mB

mA

Momentum Conservation (V)
  • 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
  • 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
momentum conservation vi

n

neutron  proton + electron + neutrino

(n  p + e + n )

When the neutrino is included, in fact momentum is conserved.

Momentum Conservation (VI)

mP

p

Consider a neutron, n,which is at rest, and then decays.

mp+me < mn

n

me

e

  • Can this process occur?
    • No, momentum is not conserved
    • Yes, since mn is larger than the sum of mP and me
    • No, energy cannot be conserved
    • Yes, but only between 8 pm and 4 am

The observation that momentum was not conserved in neutron decay lead to theprofound hypothesis of the existence of a particle called the neutrino

discovery of the neutrino

mP

p

n

n

me

e

When the neutrino is included, in fact momentum is conserved.

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 + n )

charge conservation

Consider the previous example of neutron decay:

n  p + e + n

Charge 0 +1 -1 0

Charge Conservation

The total electric charge of a system does not change.

Can these processes occur?

p + p  p + n

Charge +1 +1 +1 0 NO

p + e  n + n

Charge +1 -1 0 0 YES

n + n  p + p

Charge 0 0 +1 +1 NO

summary of conservation laws
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.