A.8. Show with the aid of the Lorentz transformation (A.28) that the quantity c²1²-x² - y² – z² is an invariant, namely X c²₁² - (.x² + y²² + =²) = c²1¹² − (x^² + ¸‚µ‚²² + =²²)
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- 2.9. (a) Solve the integral ...| (dx .dx3N) 3N and use it to determine the "volume" the relevant region of the phase space of an extreme relativistic gas ( = pc) of 3N particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy E among this system of particles and show that, asymptotically, w0 = h³N. (b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8.Given: R(N)=f(N)g(N) where f(N) = 7.2N and g(N) = (1 - 7.9/N. If R(N) = H means that R'(N) = 0 and H = 10.5N, what is the value of N that causes R'(N) = 0?In special relativity, we introduced the momentum of a particle in a given Lorentz frame as p" = (E,p'), where E is the energy of the particle and p¹ is the relativistic 3-momentum. Explain why, in general relativity, the energy of a particle measured by an observer is given, irrespective of the coordinate system used, by -p Uobs, where p" is the momentum of the particle and uobs is the velocity of the observer in those coordinates.
- (b) Write a necessary condition for a transformation (q,p) to (Q,P) to be connonical. Prove that P-2(1+√qcosp)√q sinp:Q-log(1+√qcosp)The difference of the scalar potential squared and the modulus of the vector potential squared, Φ2 - |A|2, is Lorentz invariant (a Lorentz scalar). why the statement true?dx 4x 3. In so-called "natural units" (which is just a sneaky way to let us ignore a bunch of constants), the relativistic kinetic energy of a rigid body is given by the formula 1 КЕ — т V1 – v2 where m is the rest mass of the body and v is its relative speed. Alien scientists on a space station are observing an object falling into a black hole. As the object falls, it is disintegrating, losing mass at a rate of 3 (so its mass is changing at a rate of -3). How fast is the kinetic energy of the main part of the object changing when its mass is 20, its velocity is .7, and it is accelerating at a rate of .1 (remember that acceleration is the derivative of velocity with respect to time: a = dt 1Note that this formula does not make sense when v > 1. That is because in natural units, a speed of 1 corresponds to the speed of light, and nothing with positive rest mass can go that fast.
- SECTION 4-3 DeMorgan's Theorems 9. Apply DeMorgan's theorems to each expression: (a) A + B (e) A(B + C) (f) AB + CD (g) AB + CD (b) AB (c) A + B + C (d) ABC (h) (A + B)C + D) 10. Apply DeMorgan's theorems to each expression: (a) AB(C + D) (b) AB(CD + EF) (c) (A + B+C + D) + ABCD (d) (A + B+ C + D)(AB CD) (e) AB(CD + EF)(AB + CD) 11. Apply DeMorgan's theorems to the following: (a) (ABC)(EFG) + (HIJ)(KLM) (b) (A + BC + CD) + BC (c) (A + B)(C + D)(E + F)(G + H) SECTION 4-4 Boolean Analysis of Logic Circuits 12. Write the Boolcan expression for cach of the logic gates in Figure 4-55. FIGURE 4-55 D B. (a) (b) (c) (d)Consider an electron orbiting around a proton with an orbital radius of R=8.48⋅10−10R=8.48⋅10-10 m. What is the orbital frequency of the electron motion? Use me=9.11×10−31me=9.11×10-31 kg, e=1.6×10−19e=1.6×10-19 C, and k=9×109k=9×109 Nm2/C2. The frequency, f0 = Units . By how much would this frequency increase (assume the same orbital radius) if an external magnetic field of B = 0.5 T is applied to the system along the the electron axis of rotation? The increase in the frequency , Δf = Units .I need help on Question 7!
- A galaxy G is moving away radially with speed with respect to an observer O. The relation between X, the wavelength of light emitted at G, and λo, the wavelength observed at O, is 入。 λ = λe λε 1+B 1- B' = where ẞ v/c (c is the speed of light). For ẞ < 1 find a power series expansion of the above formula up to and including terms of order ẞ³.Your video analysis of the motion of a marble gives it position in frame 23 as (x23, 923) = (0.134 m, 0.120 m) and its position in frame 24 as (x24, Y24) = (0.122 m, 0.112 m). You esti- mate that you can measure the x and y positions with uncertainty +0.003 m. The frame rate of the video is 30 frames/s, which means the time interval between frames is At = 0.033 333 s. The uncertainty of the frame rate of a video camera is VERY small. For the sake of this prob- lem, use (At) = 1 x 10-6 s. The mass of the marble is (2.031 +0.001) x 10-2 kg. Calculate the following quantities: 1. The value of the velocity component U = Xi+1Xi At and its uncertainty, Yi+1 Yi = and its uncertainty, At 2. The value of the velocity component vy 3. The value of the velocity magnitude v = 4. The value of the momentum component p 5. The value of the momentum component Py = muy and its uncertainty, and 6. The value of the kinetic energy K = =mv² = m(v²+v²) and its uncertainty. v2+2 and its uncertainty, = mv, and…