PHY 1371Dr. Jie Zou1 Chapter 39 Relativity (Cont.)

PHY 1371Dr. Jie Zou2 Outline Lorentz Velocity Transformation Equation Relativistic linear momentum and the relativistic form of Newton’s laws Relativistic energy Equivalent of mass and energy

PHY 1371Dr. Jie Zou3 Lorentz velocity transformation equation Consider S to be the stationary frame of reference, and S’ to be the frame moving at a velocity v along x-axis relative to S. Lorentz velocity transformation equation: u x ’ and u x are the speed of an object measured in the S’ and S, respectively Note: To obtain u x in terms of u x ’, replace v by –v and interchange the roles of u x and u x ’.

PHY 1371Dr. Jie Zou4 Example 39.7 Two spaceships A and B are moving in opposite directions. An observer on the Earth measures the speed of ship A to be c and the speed of ship B to be c. Find the velocity of ship B as observed by the crew on ship A.

PHY 1371Dr. Jie Zou5 Relativistic linear momentum and the relativistic form of Newton’s laws Relativistic linear momentum: u is the velocity of the particle and m is its mass. When u << c, the relativistic definition for p reduces to the classical expression. Relativistic force F: Under relativistic conditions, the acceleration a of a particle decreases under the action of a constant force, in which case a  (1-u 2 /c 2)3/2. It is impossible to accelerate a particle from rest to a speed u>=c.

PHY 1371Dr. Jie Zou6 Example Linear momentum of an electron: An electron, which has a mass of 9.11 x kg, moves with a speed of 0.750c. Find its relativistic momentum and compare this value with the momentum calculated from the classical expression.

PHY 1371Dr. Jie Zou7 Relativistic energy Relativistic kinetic energy: It can be shown that when u << c, relativistic K reduces to the classical expression of K = (1/2)mu 2. Rest energy E R = mc 2. Total energy E =  mc 2 = Kinetic energy + Rest Energy = K + mc 2. Or This is Einstein’s famous equation about mass-energy equivalence.

PHY 1371Dr. Jie Zou8 Equivalence of mass and energy The relationship E = K + mc 2 shows that mass is a form of energy. A small mass corresponds to en enormous amount of energy. m: invariant mass. Relations between E and p: E 2 = p 2 c 2 + (mc 2 ) 2 When particle is at rest: p = 0 and E = mc 2. For particles having 0 mass, e.g. photons, m = 0 and E = pc. Conservation of energy and mass: The energy of a system of particles before interaction must equal the energy of the system after interaction, where energy of the ith particle is given by the expression:

PHY 1371Dr. Jie Zou9 Example Binding energy of the Deuteron: A deuteron, which is the nucleus of a deuterium atom, contains one proton and one neutron and has a mass of u. This total deuteron mass is not equal to the sum of the masses of the proton and neutron. Calculate the mass difference and determine its energy equivalence, which is called the binding energy of the nucleus.

PHY 1371Dr. Jie Zou10 Homework Ch. 39, P. 1285, Problems: #31, 38, 53.