Physics 202 p. 1/2 Physics 202 Professor P. Q. Hung 311B, Physics Building
Physics 202 p. 2/2 Momentum in Special Classically, the momentum is defined as p = m v = m r t. We also learned that momentum is conserved. We also learned that F = p t. Momentum conservation is the consequence of zero external force.
Physics 202 p. 2/2 Momentum in Special Classically, the momentum is defined as p = m v = m r t. We also learned that momentum is conserved. We also learned that F = p t. Momentum conservation is the consequence of zero external force. Requirement: The laws of physics must be the same in all inertial frames. For instance, the total momentum should be
Physics 202 p. 3/2 Momentum in Special A detailed analysis reveals that, if we were to use p = m v, the momentum might be conserved in one inertial frame but not in another inertial frame. Should one give up momentum conservation? NO. Redefine the momentum.
Physics 202 p. 4/2 Momentum in Special Instead of t, one should use the proper time t 0 = 1 v2 c t. 2 The proper form for the momentum is p = m v 1 v2 c 2 For v c, one recovers the usual classical p = m v.
Physics 202 p. 5/2 Momentum in Special
Physics 202 p. 6/2 Momentum in Special : Example An electron, which has a mass of 9.11 10 31 kg, moves with a speed of 0.750 c. Find its relativistic momentum and compare this value with the momentum calculated from the classical expression.
Physics 202 p. 7/2 Momentum in Special : Example p = mv 1 v2 c 2 = 3.1 10 22 kg.m/s. (9.11 10 31 kg)(0.750 3 108m/s) 1 0.750 2 =
Physics 202 p. 7/2 Momentum in Special : Example p = mv 1 v2 c 2 = 3.1 10 22 kg.m/s. (9.11 10 31 kg)(0.750 3 108m/s) 1 0.750 2 = The classical result is p = mv = (9.11 10 31 kg)(0.750 3 10 8 m/s) = 2.05 10 22 kg.m/s. A 50% smaller than the relativistic result.
Physics 202 p. 8/2 Relativistic Energy What does the folkloric E = mc 2 mean? Start with motion in blueone dimension for simplicity. And also start the motion from rest. Work done = Change in kinetic energy. W = F dx = dp dt dx.
Relativistic Energy What does the folkloric E = mc 2 mean? Start with motion in blueone dimension for simplicity. And also start the motion from rest. Work done = Change in kinetic energy. W = F dx = dp dt dx. After some calculations, one finds W = K = mc2 1 v2 c 2 mc 2 mc2 1 v2 c 2 mc 2 Physics 202 p. 8/2
Physics 202 p. 9/2 Momentum in Special
Physics 202 p. 10/2 Relativistic Energy γ = 1. 1 v2 c 2
Physics 202 p. 10/2 γ = 1. 1 v2 c 2 Relativistic Energy Notice: For v c, one has 1 1 + 1 v 2 1 v2 2 c 2 c 2 K mc 2 (1 + 1 v 2 2 c ) mc 2 = 1 2 2 mv2. The classical result!
Physics 202 p. 11/2 Relativistic Energy There is one term which does not depend on the speed: mc 2 Rest Energy of the particle.
Physics 202 p. 11/2 Relativistic Energy There is one term which does not depend on the speed: mc 2 Rest Energy of the particle. Define the Total Energy of the particle as: E = γ mc 2 = K + mc 2 Using p = γ mv, one finds (squaring both and subtracting E 2 p 2 c 2 ): E 2 = p 2 c 2 + (mc 2 ) 2 For p 2 c 2 (mc 2 ) 2, one has E pc.
Physics 202 p. 12/2 Relativistic Energy From Eq. (4), one also finds: v c = pc E
Physics 202 p. 12/2 Relativistic Energy From Eq. (4), one also finds: v c = pc E Some units: 1 ev = 1.602 10 19 joule. 1 kev = 10 3 ev 1 MeV = 10 6 ev 1 GeV = 10 9 ev 1 T ev = 10 12 ev
Relativistic Energy: Examples Examples: 1) The deuteron H 2 consists of a neutron and a proton bound together. Its rest mass is 1875.58 MeV. The rest masses of the proton and neutron are 938.26 MeV and 938.55 MeV respectively, and whose sum is 1877.81 MeV > Rest mass of the deuteron. Therefore the deuteron cannot spontaneously decay into a proton and a neutron. The difference between the two: 1877.81 MeV 1875.58 MeV = 2.23 MeV is the binding energy of the deuteron. 2.23 MeV must be added in order to break up the deuteron. Physics 202 p. 13/2
Physics 202 p. 14/2 Relativistic Energy: Examples 2) An electron and a proton are each accelerated through a potential of 10 7 V. Find the momentum and speed of each.
Physics 202 p. 15/2 a) For the electron: Kinetic energy of both: K = 10 MeV
Physics 202 p. 15/2 a) For the electron: Kinetic energy of both: K = 10 MeV γ = 1 + K mc 2 = 1 + 10 0.51 = 20.6 One cannot use the classical non-relativistic approximation here.
Physics 202 p. 15/2 a) For the electron: Kinetic energy of both: K = 10 MeV γ = 1 + K mc 2 = 1 + 10 0.51 = 20.6 One cannot use the classical non-relativistic approximation here. The rest mass of the electron is 0.51 MeV K. Therefore p E/c = (mc 2 + K)/c = 10.51 MeV/c.
a) For the electron: Kinetic energy of both: K = 10 MeV γ = 1 + K mc 2 = 1 + 10 0.51 = 20.6 One cannot use the classical non-relativistic approximation here. The rest mass of the electron is 0.51 MeV K. Therefore p E/c = (mc 2 + K)/c = 10.51 MeV/c. p = γmv = (γmc2 )v c 2 v c = pc 10.51 MeV γmc = 2 20.6 0.51 MeV = 0.999 Physics 202 p. 15/2
Physics 202 p. 16/2 a) For the proton: γ = 1 + K mc 2 = 1 + 10 938 1 classical, non-relativistic approximation might be good.
Physics 202 p. 16/2 a) For the proton: γ = 1 + K mc 2 = 1 + 10 938 1 classical, non-relativistic approximation might be good. 1 2 mv2 = 10 MeV v c 0.146.
Physics 202 p. 17/2 General relativity Applies to accelerated frame of references and provides a theory of gravitation beyond that of Newton. Principle of equivalence: Experiments conducted in a uniform gravitational field and in an accelerated frame of reference give identical results. Some consequences: A gravitational field bends light. The stronger the field is the more bend one gets. Observations: Bending of light near the sun in 1919 by Eddington;
Physics 202 p. 18/2 General relativity
Physics 202 p. 19/2 General relativity
Physics 202 p. 20/2 General relativity
Physics 202 p. 21/2 General relativity
Physics 202 p. 22/2 General relativity
Physics 202 p. 23/2 General relativity Black holes: We mentioned that last semester. Heuristic derivation of the Schwarschild radius: Escape veolcity: c = 2GM R R S = 2GM c 2. Schwarschild radius of a black hole beyond which light cannot escape.
Physics 202 p. 24/2 General relativity Example: For a black hole with a mass comparable to that of the Earth, R S = 2(6.67 10 11 N.m 2 /kg 2 )(5.98 10 24 kg) (3 10 8 m/s) 9 mm 2
Physics 202 p. 25/2 General relativity
Physics 202 p. 26/2 General relativity
Physics 202 p. 27/2 General relativity
Physics 202 p. 28/2 General relativity
Physics 202 p. 29/2 General relativity