Physics 202. Professor P. Q. Hung. 311B, Physics Building. Physics 202 p. 1/2

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1 Physics 202 p. 1/2 Physics 202 Professor P. Q. Hung 311B, Physics Building

2 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.

3 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

4 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.

5 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.

6 Physics 202 p. 5/2 Momentum in Special

7 Physics 202 p. 6/2 Momentum in Special : Example An electron, which has a mass of kg, moves with a speed of c. Find its relativistic momentum and compare this value with the momentum calculated from the classical expression.

8 Physics 202 p. 7/2 Momentum in Special : Example p = mv 1 v2 c 2 = kg.m/s. ( kg)( m/s) =

9 Physics 202 p. 7/2 Momentum in Special : Example p = mv 1 v2 c 2 = kg.m/s. ( kg)( m/s) = The classical result is p = mv = ( kg)( m/s) = kg.m/s. A 50% smaller than the relativistic result.

10 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.

11 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

12 Physics 202 p. 9/2 Momentum in Special

13 Physics 202 p. 10/2 Relativistic Energy γ = 1. 1 v2 c 2

14 Physics 202 p. 10/2 γ = 1. 1 v2 c 2 Relativistic Energy Notice: For v c, one has v 2 1 v2 2 c 2 c 2 K mc 2 (1 + 1 v 2 2 c ) mc 2 = mv2. The classical result!

15 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.

16 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.

17 Physics 202 p. 12/2 Relativistic Energy From Eq. (4), one also finds: v c = pc E

18 Physics 202 p. 12/2 Relativistic Energy From Eq. (4), one also finds: v c = pc E Some units: 1 ev = joule. 1 kev = 10 3 ev 1 MeV = 10 6 ev 1 GeV = 10 9 ev 1 T ev = ev

19 Relativistic Energy: Examples Examples: 1) The deuteron H 2 consists of a neutron and a proton bound together. Its rest mass is MeV. The rest masses of the proton and neutron are MeV and MeV respectively, and whose sum is MeV > Rest mass of the deuteron. Therefore the deuteron cannot spontaneously decay into a proton and a neutron. The difference between the two: MeV MeV = 2.23 MeV is the binding energy of the deuteron MeV must be added in order to break up the deuteron. Physics 202 p. 13/2

20 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.

21 Physics 202 p. 15/2 a) For the electron: Kinetic energy of both: K = 10 MeV

22 Physics 202 p. 15/2 a) For the electron: Kinetic energy of both: K = 10 MeV γ = 1 + K mc 2 = = 20.6 One cannot use the classical non-relativistic approximation here.

23 Physics 202 p. 15/2 a) For the electron: Kinetic energy of both: K = 10 MeV γ = 1 + K mc 2 = = 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 = MeV/c.

24 a) For the electron: Kinetic energy of both: K = 10 MeV γ = 1 + K mc 2 = = 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 = MeV/c. p = γmv = (γmc2 )v c 2 v c = pc MeV γmc = MeV = Physics 202 p. 15/2

25 Physics 202 p. 16/2 a) For the proton: γ = 1 + K mc 2 = classical, non-relativistic approximation might be good.

26 Physics 202 p. 16/2 a) For the proton: γ = 1 + K mc 2 = classical, non-relativistic approximation might be good. 1 2 mv2 = 10 MeV v c

27 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;

28 Physics 202 p. 18/2 General relativity

29 Physics 202 p. 19/2 General relativity

30 Physics 202 p. 20/2 General relativity

31 Physics 202 p. 21/2 General relativity

32 Physics 202 p. 22/2 General relativity

33 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.

34 Physics 202 p. 24/2 General relativity Example: For a black hole with a mass comparable to that of the Earth, R S = 2( N.m 2 /kg 2 )( kg) ( m/s) 9 mm 2

35 Physics 202 p. 25/2 General relativity

36 Physics 202 p. 26/2 General relativity

37 Physics 202 p. 27/2 General relativity

38 Physics 202 p. 28/2 General relativity

39 Physics 202 p. 29/2 General relativity

Part A-type questions

PHYS306: lecture 8 th February 008 Part A-type questions. You toss an apple horizontally at 8.7 m/s from a height of.6 m. Simultaneously, you drop a peach from the same height. How long does each take