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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 read through the Section Particle Decays and Annihilations Slides a-h (see slide key on Course Assignments Web page)

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Conservation Laws

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Conservation laws in Physics can give explanations as to why some things occur and other do not. Three very important Conservation Laws are: I. Conservation of Energy II. Conservation of Momentum III. Conservation of Charge

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Energy Conservation (I) There are many forms of energy. For now, well focus on two types 1. Kinetic Energy (KE) – Energy of motion KE = ½ mv 2 if v is much less than c (v << c) 2. Mass Energy E = mc 2 m = mass c = speed of light = 3x10 8 [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)!

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Conservation of Energy (II) AB vAvA vBvB Total Energy (after decay) = E A + E B = (KE A +m A c 2 ) + (KE B +m B c 2 ) Suppose D decays into 2 particles A and B, what is the energy of the system afterward? D Total Energy (initially) = E D = m D c 2 Since energy must be conserved in the decay process, m D c 2 = (KE A +m A c 2 ) + (KE B +m B c 2 )

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Conservation of Energy (III) m D c 2 = (KE A +m A c 2 ) + (KE B +m B c 2 ) 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 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 EAEA EBEB EDED Before Decay After Decay

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Conservation of Energy (IV) m D c 2 = (KE A +m A c 2 ) + (KE B +m B c 2 ) Since m A and m B must be larger than zero, and v A 2 >0 and v B 2 >0, the KE can only be positive (KE cannot be negative!) m D c 2 m A c 2 + m B c 2 m D m A + m B This is also true if particle D has KE>0 also! KE A = ½ m A v A 2 KE B = ½ m B v B 2 > 0 If I subtract off the KE terms from the RHS * of the top equation, I will no longer have an equality, but rather an inequality: and dividing both sides by c 2,

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Conservation of Energy (V) MDc2MDc2 MAc2MAc2 MBc2MBc2 KE A KE B MBc2MBc2 MDc2MDc2 MAc2MAc2 LHS = RHS LHS > RHS

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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? D A B m A =0.2 kg m B =0.1 kg I D A m A =0.2 kg m B =0.4 kg B II D A m A =0.1 kg IV B m B =0.1 kg D A B m A =0.49 kgm B =0.0 kg III

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Energy Conservation (VII) Bam q 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] = [eV])

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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) Its equal to the mass of q B) It must be less than 0.5 TeV C) It must be less than 1 [TeV] D) Its equal but opposite in direction to that of the t particle

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Momentum Conservation (I) Momentum (p) = mass x velocity = mv p = mv Momentum has a direction, given by the direction of v m1m1 v1v1 p 1 = m 1 v 1 m2m2 v2v2 p 2 = -m 2 v 2 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.

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Momentum Conservation (II) m1m1 v1v1 m2m2 v2v2 Consider a head-on collision of two particles What is the total momentum before the collision ? A) m 1 v 1 +m 2 v 2 B) m 1 v 2 -m 2 v 1 C) zero D) (m 1 +m 2 )(v 1 +v 2 ) If m 1 = m 2, what can be said about the total momentum? A) its zero B) its positive C) its negative D) cant say? If m 1 = m 2 and v 1 > v 2, what can be said about the total momentum? A) its zero B) its positive C) its negative D) cant say?

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Momentum Conservation (III) If m 1 v 2, what can be said about the total momentum? A) its zero B) its positive C) its negative D) cant say? If m 1 = m 2 and v 1 = v 2 (in magnitude), what can be said about the total momentum? A) its zero B) its positive C) its negative D) cant 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) cant say? m1m1 v1v1 m2m2 v2v2 Consider a head-on collision of two particles

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Momentum Conservation (IV) D A B mAmA mBmB I vAvA vBvB Consider a particle D at rest which decays into two lighter particles A and B, whose combined mass is less than D. If m A > m B, 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 m A = m B, what can be said about the magnitudes of the velocities of A and B? A) v A >v B B) Equal and Opposite C) v B >v A D) Same direction but different magnitudes

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Momentum Conservation (V) Can m A +m B exceed m D ? A) Not enough data B) Yes, if v A and v B are zero C) No D) Yes, if v A and v B 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 D A B mAmA mBmB I vAvA vBvB

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Momentum Conservation (VI) n p mPmP e meme Can this process occur? a)No, momentum is not conserved b)Yes, since m n is larger than the sum of m P and m e c)No, energy cannot be conserved d)Yes, but only between 8 pm and 4 am Consider a neutron, n, which is at rest, and then decays. m p +m e < m n The observation that momentum was not conserved 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.

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n p mPmP e meme Discovery of the Neutrino The observation that momentum conservation appeared to be violated 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.

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Charge Conservation The total electric charge of a system does not change. Consider the previous example of neutron decay: n p + e + Charge Can these processes occur? p + p p + n Charge NO p + e + n Charge YES n + n p + p Charge NO

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Summary of Conservation Laws Total Energy of an isolated system is conserved D A + B cannot occur if m A +m B > m D 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 explain why some processes occur and others do not.

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