Physics 2D Lecture Slides Lecture 9 : Jan 19th 2005 Vivek Sharma UCSD Physics Definition (without proof) of Relativistic Momentum � � mu � With the new definition relativistic = = � p mu momentum is conserved in all � 2 1 ( / ) u c frames of references : Do the exercise New Concepts Rest mass = mass of object measured In a frame of ref. where object is at rest 1 � = � 2 1 ( / ) u c u is velocity of the object NOT of a referen ce frame ! 1
Relativistic Force & Acceleration � � � � � dp d mu d du d � � � = = = F use � mu � � � dt dt dt dt du � 2 1 ( / ) u c � � = = � p mu � � � 2 1 ( / ) u c � � m mu 1 2 u du � � = + � F ( )( ) ( ) � 3/2 2 � 2 c dt � 2 � 1 ( / ) u c 1 ( u c / ) 2 � � � � Relativistic 2 � 2 + 2 mc mu mu du � � = � F ( ) 3/2 � dt 2 � 2 Force c 1 ( u c / ) � � � � And m du � � = � F : Relativistic For ce ( ) � 3/2 Acceleration dt � 2 1 ( / ) u c � � � � du Since A ccel e r a tion a = , [rate of change of v elocity ] dt � � F 3/2 � � � 2 � a = 1 ( / ) u c � � Reason why you cant m � � � quite get up to the speed Note: As / u c 1, a 0 !!!! of light no matter how It s harder to accelerate when you get hard you try! closer to s peed of l ight Linear Particle Accelerator : 50 GigaVolts Accelating Potential � � eE 3/2 � � � 2 a= 1 ( / ) u c � � m PEP-II accelerator schematic and tunnel view PEP-II accelerator schematic and tunnel view 2
Relativistic Work Done & Change in Energy � x � x � dp � 2 2 � � = = W F dx . . dx X 2 , u=u dt x x 1 1 du � m mu dp dt = � = p , 3/ 2 dt 2 � � u u 2 � � 1 1 � � c 2 2 c x 1 , u=0 � � substitute in W , du m dt ud t u � � = � W (change in var x u) 3/ 2 � � 2 u 0 � 1 � � c 2 � � Relativistic Work Done & Change in Energy X 2 , u=u u 2 mudu mc � = = � 2 W mc 3/2 1/2 � � � � 2 2 u u 0 � � 1 1 � � � � x 1 , u=0 2 2 c c � � � � = � 2 � 2 mc m c Work done is change in energy (KE in this case) � 2 � 2 � 2 = + 2 K = m c mc or Total Energy E= m c K m c 3
But Professor… Why Can’s ANYTHING go faster than light ? 2 � � � � 2 = mc 2 ( ) mc 2 � � 1/2 � mc 2 � K + mc 2 K = � � 1/2 � � � � 1 � u 2 1 � u 2 � � � � � � � � c 2 c 2 � � � � � � � � � 1 � u 2 � = m 2 c 4 K + mc 2 � 2 � � � � � c 2 � � � u = c 1 � ( K mc 2 + 1) � 2 (Parabolic in u Vs K mc 2 ) As u � c , Kinetic Energy K � � � Need to do infinite amount of work on the particle to rev it up to the speed of light! Non-relativistic case: K = 1 2 K 2 mu 2 � u = m Relativistic Kinetic Energy Vs Velocity 4
A Digression: How to Handle Large/Small Numbers • Example: consider very energetic particle with very large Energy E + 2 E mc K K � = = = + 1 mc 2 mc 2 mc 2 1/2 � � u 1 = � 1 • Lets Say γ = 3x10 11 , Now calculate u from � � � c 2 � � • Try this on your el-cheapo calculator, you will get u/c =1, u=c due to limited precision. • In fact u ≅ c but not exactly!, try to get this analytically 1 1 � = = In Quizzes, you are � � + � � � 2 (1 )(1 ) 1 Expected to perform u � � + � = Such simple Since = 1, 1 2 c approximations 1 � � � � 2 1 1 � � � = = � � 24 = � 1 5 10 , u c � 2 2 � = u 0.999 999 999 999 999 999 999 995c !! Such particles are routinely produced in violent cosmic collisions When Electron Goes Fast it Gets “Fat” � = + 2 2 Total Energy E = mc K mc = � 2 E mc v � � � � As 1, c � Apparent Mass approaches New Concept Rest Mass = particle mass when its at rest 5
Relativistic Kinetic Energy & Newtonian Physics � � 2 2 Relativistic KE K = mc mc � Remember Binomial Theorem � � � nx n(n-1) � � + for x << 1; (1+x) = n (1+ x +small 2 er terms) � � 1! 2! 1 � � � 2 2 u 1 u 2 � < < � + + When u c , 1- 1 ...smaller ter ms � � 2 2 c 2 c � � 2 1 u 1 � 2 + � 2 = 2 s o K mc [1 ] m c m u (classical form reco vered) 2 2 c 2 = � = + 2 2 Total Energy of a Part icle E mc KE mc � 2 For a partic le at rest, u = 0 Total Energy E= c m E=mc 2 ⇒ Sunshine Won’t Be Forever ! Q: Solar Energy reaches earth at rate of 1.4kW per square meter of surface perpendicular to the direction of the sun. by how much does the mass of sun decrease per second owing to energy loss? The mean radius of the Earth’s orbit is 1.5 x 10 11 m. r • Surface area of a sphere of radius r is A = 4 π r 2 • Total Power radiated by Sun = power received by a sphere whose radius is equal to earth’s orbit radius 6
E= mc 2 ⇒ Sunshine Won’t Be Forever ! Total Power radiated by Sun r = power received by a sphere with radius equal to earth-sun orbit radius( r in figure) Earth Earth P P = = � = � � � sun 2 3 2 11 2 P A 4 r (1.4 10 W / m )(4 )(1.5 10 ) incident incident � � lost earth sun earth sun A A sun = � 26 P 4.0 10 W lost � 26 So Sun loses E = 4.0 10 J of rest energy per second � 26 E 4.0 10 J = = � 9 I ts mass decreases by m = 4.4 10 k g per se c!! 2 � 8 2 c (3.0 1 0 ) � 30 If the Sun's Mass = 2.0 1 0 kg So how long with the Sun last ? One day the sun will be gone and the solar sy stem will not be a hosp itable place for l i f e E = � mc 2 � E 2 = � 2 m 2 c 4 Relationship between P and E p = � mu � p 2 c 2 = � 2 m 2 u 2 c 2 � E 2 � p 2 c 2 = � 2 m 2 c 4 � � 2 m 2 u 2 c 2 = � 2 m 2 c 2 ( c 2 � u 2 ) = m 2 c 2 ( c 2 � u 2 ) = m 2 c 4 c 2 � u 2 ( c 2 � u 2 ) = m 2 c 4 1 � u 2 c 2 E 2 = p 2 c 2 + ( mc 2 ) 2 ........important relation For particles with zero rest mass like photon (EM waves) E= pc or p = E c (light has momentum!) Relativistic Invariance : E 2 � p 2 c 2 = m 2 c 4 : In all Ref Frames Rest Mass is a "finger print" of the particle 7
Mass Can “Morph” into Energy & Vice Verca • In Newtonian mechanics: mass and energy separate concepts • In relativistic physics : Mass and Energy are the same thing ! • New word/concept : MassEnergy , just like SpaceTime • It is the mass-energy that is always conserved in every reaction : Before & After a reaction has happened • Like squeezing a balloon : Squeeze here, it grows elsewhere – If you “squeeze” mass, it becomes (kinetic) energy & vice verca ! • CONVERSION FACTOR = C 2 • This exchange rate never changes ! Mass is Energy, Energy is Mass : Mass-Energy Conservation Examine Kinetic energy Before and After Inelastic Collision: Conserved? K=0 S K = mu 2 Before V=0 1 2 v v 1 2 After Mass-Energy Conservation: sum of mass-energy of a system of particles before interaction must equal sum of mass-energy after interaction = E E befor e after 2 2 mc mc 2 m + = 2 � = > Mc M 2 m 2 2 2 u u u Kinetic energy is not lost, � � � 1 1 1 2 2 2 c c c its transforme d into Kinetic energy has been transformed int o mass increase more mass in final state � � � � 2 2 K 2 mc � � � = = = � 2 M M - 2 m mc � � 2 2 c c 2 u � � � 1 2 � c � 8
Creation and Annihilation of Particles γ µ - µ + Sequence of events in a matter-antimatter collision: � + � + � � � µ + µ + e e Relativistic Kinematics of Subatomic Particles Reconstructing Decay of a π Meson � + � µ � + The decay of a stationary happens quickly, � � µ + is invisible, has m 0; leaves a trace in a B field P ν , Ε ν µ + 2 mass=106 MeV/c , KE = 4.6 MeV � + What was mass of the fleeting ? ν Energy Conservati on: π + At rest = + � = + + E E E m c 2 ( m c 2 ) 2 p c 2 2 p c µ + � µ � � µ µ � = Momentum Conservation : p p µ � � 2 = 2 2 + 2 2 + m c ( m c ) p c p c � µ µ µ P µ , , Ε µ 9
Relativistic Kinematics of Subatomic Particles Energy Conservation: E � = E µ + E � � m � c 2 = ( m µ c 2 ) 2 + p µ 2 c 2 + p � c P ν , Ε ν Momentum Conservation : p µ = p � � m � c 2 = ( m µ c 2 ) 2 + p µ 2 c 2 + p µ c ν 2 c 2 = E µ 2 � ( m µ c 2 ) 2 = ( K µ + m µ c 2 ) 2 � ( m µ c 2 ) 2 also p µ 2 + 2 K µ m µ c 2 π + = K µ At rest 2 c 2 � µ + Substituting for p µ m � c 2 = 2 c 4 + K µ 2 + 2 K µ m µ c 2 + 2 + 2 K µ m µ c 2 m µ K µ Put in all the known #s P µ , , Ε µ � m � c 2 = 111 MeV + 31 MeV = 141 MeV � m � = 141 MeV / c 2 Detecting Baby Universes : Need a “Camera” 10
Two Faced Particle : Beauty With Strangeness (Bs) Sometimes Matter Sometimes Antimatter My Discovery (1993): Beauty With Strangeness 11
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