14. In quantum mechanics, every dynamic variable is represented as an operator. Consider the position representation, which ones of the following statements are correct?
(A) the momentum operator along x-axis is p_x = m (∂x/∂t)
(B) the momentum operator along x-axis is p_x = −im (∂x/∂t)
(C) the momentum operator along x-axis is p_x = −iℏ (∂/∂x)
(D) the angular momentum operator is L = −iℏ (r × ∇)
(E) the angular momentum operator is L = iℏ (∇× r)

1. Consider a cylindrical tank filled with water with 2.5-m depth. The water emerges from a small hole at the center of the bottom. The diameter of the hole is 1 mm. The cross-section area of the vertical water stream decreases as it falls. Find the diameter of the water stream at the position 7.5 m from the hole. (A) 0.23 mm(B) 0.36 mm(C) 0.50 mm(D) 0.58 mm(E) 0.71 mm

2. Three identical cylinders are stacked on a table as shown in the figure. Find the minimum value of the coefficient of static friction between the cylinder and the table surface. (A)(B)(C)(D)(E)

3. A cube rotates around one of its edges as shown in the figure. The mass of the cube is M and the side length is L. Find the moment of inertia of the cube about that axis. (A) (1/6)ML²(B) (1/3)ML²(C) (1/2)ML²(D) (2/3)ML²(E) (5/6)ML²

4. A man plays a Yo-Yo ball as shown in the figure. The mass of the ball is m, and the length of the string is L. The ball moves in a circle along a vertical plane. What is the difference of the string tension at the lowest point (T₂) and the highest point (T₁)? (A) T₂ − T₁ = 2mg(B) T₂ − T₁ = 3mg(C) T₂ − T₁ = 4mg(D) T₂ − T₁ = 5mg(E) T₂ − T₁ = 6mg

5. A parallel plate capacitor is composed of two circular plates of radius r = 40 mm separate by a distance d = 1 mm. The capacitor is charged that the potential difference between the plates changes at 10⁴ volt/sec. Find the displacement current in the capacitor?(A) 3.0 × 10⁻⁷ A(B) 3.5 × 10⁻⁷ A(C) 4.0 × 10⁻⁷ A(D) 4.5 × 10⁻⁷ A(E) 5.0 × 10⁻⁷ A

6. Consider a thermodynamic engine based on ideal gas. The operating cycle is shown in the pressure(P)-volume(V) diagram. The a→b and c→d are isochoric (constant volume) processes, and the b→c and d→a are adiabatic processes. Which one of the following statements is wrong? (A) During a→b, the heat flows from the engine to the surrounding.(B) During b→c, the temperature of the gas decreases.(C) During c→d, the internal energy of the gas decreases.(D) During d→a, the internal energy of the gas increases.(E) During d→a, the engine does negative work on the surrounding.

7. Continued from the last question, find the efficiency of this engine.  (Note: adiabatic process of ideal gas: PVᵞ= constant, γ = Cp/Cv, Cv: molar specific heat at constant volume, Cp: molar specific heat at constant pressure.)(A)(B)(C)(D)(E)

8. As shown in the right figure, a circuit is composed of a DC voltage source (V), two resistors (R₁ and R₂), one inductor (L), and a switch (S). Initially S is closed and the circuit is at steady condition, then S is opened at time t = 0. Find the current through the inductor as a function of time at t ＞ 0. (A)(B)(C)(D)(E)

9. A standing light wave is composed by two counter-propagating plane waves. The electric field is written as Estanding(r, t) = E₁(r, t) + E₂(r, t), where E₁(r, t) = E₀ cos(kz − ωt) ex and E₂(r, t) = E₀ cos(−kz − ωt) ex. Find the magnetic field of the standing wave.(A) B(r, t) = (2E₀/c) cos(kz) cos(ωt) ez(B) B(r, t) = (2E₀/c) cos(kz) cos(ωt) ey(C) B(r, t) = (2E₀/c) sin(kz) cos(ωt) ey(D) B(r, t) = (2E₀/c) cos(kz) sin(ωt) ey(E) B(r, t) = (2E₀/c) sin(kz) sin(ωt) ey

10. The setup of photoelectric effect experiment is shown in the following figure. The anode is irradiated by light with intensity I and frequency f. The electric potential between the anode and cathode is controlled by a variable power supply with voltage V. The resulted current A is measured by an ampere meter. If we use two difference light sources with the same intensity I and different frequencies f₁ and f₂, where f₁ ＜ f₂, which one of the following current-voltage (A-V) diagrams is correct?(A)(B)(C)(D)(E)

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