20. Two particles with the same mass m are moving towards each other with a relative velocity of 0.8c. What is the total energy of one of the particles as seen by an observer on the other particle?
(A)
(B)
(C) 1.8mc²
(D) 1.09mc²
(E) mc².

1. The fine structure constant is defined as α ≡ . The unit of α is the same as: (A) Length (B) Energy (C) Length²/Time (D) 1/Time (E) No unit

2. A student dropped a ball from the roof of a building. After 11.5 seconds, she heard it hitting the ground. Assuming that the speed of sound is 340 m/s, which of the following is closest to the height of the building? (A) 30 m (B) 100 m (C) 200 m (D) 500 m (E) 800 m

3. A solid sphere of mass m and radius r moves along a track with a circular loop of radius R1 as shown in following picture. It is released at height h. Assuming that the track has no friction, the ball thereforeslides down without rotation. Do not ignore the finite size of the sphere, what is the minimal value of h such that the sphere remains on the track at all times? (A) 2.5R − 1.5r (B) 2R − r (C) 2.5R (D) 2R + r (E) 2R − 1.5r

4. Same arrangement as in problem 3. If the sphere is released at a height h = 3R, what is the force exerted by the track at the top of the loop? (A) 2mg (B) mg (C) 0.5mg (D) mg(R + 3r)/(R − r) (E) mg(R + r)/(R − 4)

5. Same as problem 3, except now the track surface is rough and the sphere rolls without sliding. What is the minimal value of h such that the sphere remains on the track? (A) 2.5R + 0.7r (B) 2.7R + 1.7r (C) 2.7R − 1.7r (D) 2.5R − 1.5r (E) 2.5R + 1.5r

6. In a photoelectric effect experiment, a light ray of intensity I and frequency ν shines on a metal surface. The resulting photo current is j, and the highest energy of the photo electrons is E. Which of the following relations describes the photoelectric effect. (A) j ∝ I (B) j = hν + W (C) E = hI + W (D) E = hν + W (E) E ∝ I. Here h and W are constants.

7. A Carnot engine operated between temperatures TH and TL. What is the entropy gain per cycle? (A) 0 (B) kB N TH (C) (D) (E) kB N(TH − TL). Here kB is the Boltzmann constant, and N is the number of gas molecules.

8. If two objects are in thermal equilibrium, which of the following statements is correct? (A) Their temperatures are the same (B) Their entropy are the same (C) Their specific entropy are the same (D) Their internal energies are the same (E) Their internal energy densities are the same.

9. Consider a spring-mass complex as shown in the following figure. Assume that there is no friction, what is the force required to hold m₁ at a fixed position of x₁ = δ? (A) δ × (k₁ + k₂ + k₃) (B) δ × (C) δ × (D) δ × (k₁ + k₂) (E) none of above.

10. Same setup as in problem 9. Let k₁ = k₃ = k and m₁ = m₂ = m. The system oscillates with two eigenmodes. What are the corresponding eigenfrequencies? (A) (B) (C) (D) (E)

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