114年 - [非官方正解]114 國立臺灣大學_碩士班招生考試：普通物理學#125985

1. Two particles, each with a mass of 2 kg and initially at rest, attract each other gravitationally. If the initial distance between them is 1 m, what is the speed of each particle when they are 0.5 m apart? Assume the gravitational constant Nm²/kg². (A)(B)(C)(D) (E)

2. Two small balls, each of mass 1 kg, move in opposite directions with velocities of 4 m/s and collide elastically. If they collide off-center, causing one ball to spin post-collision, what is the angular momentum of the system about the point of the collision? (A) 0 kg-m²/s (B) 1 kg-m²/s (C) 2 kg-m²/s (D) 4 kg-m²/s (E) 8 kg-m²/s 

3. A particle of mass m moves in a circular path of radius R under the influence of a central potential U(r)=k/r² where k is a constant. What is the Lagrangian of the system in terms of polar coordinates with angular displacement θ? (A)(B) (C) (D)(E)

4. A rigid body rotates about a fixed axis with an angular velocity ω. The body consists of two particles, each of mass m, at distances r₁ and r₂ from the axis. What is the total kinetic energy of the system? (A) (B) ½m(ωr₁)² + ½m(ωr₂)² (C) (D)(E) mω(r₁ + r₂) 

5. A source of sound moves towards a stationary observer at speed v, emitting a frequency f₀. If the speed of sound is c, what frequency does the observer hear? (A) f₀ (B) f₀c/(c-v) (C) f₀c/(c+v) (D) f₀(c-v)/c (E) f₀(c+v)/c

6. Unpolarized light passes through two polarizing filters. The first is oriented vertically, and the second is rotated 30° relative to the first. What fraction of the intensity of the light that passes through the first filter also passes through the second filter? (A) 1/2 (B) 3/4 (C)(D) 3/8 (E)

7. What is the approximate minimum diameter of a focused spot for a laser beam with a wavelength of 0.5 µm? (A) 0.005 μm (B) 0.05 μm (C) 0.5 µm (D) 0.1 µm (E) 1 μm 

8. During an adiabatic compression of an ideal gas, which of the following statements is true? (A) The temperature of the gas decreases. (B) The pressure of the gas remains constant. (C) The internal energy of the gas decreases. (D) The temperature of the gas increases. (E) The entropy of the gas increases. 

9. An ideal gas undergoes an isothermal expansion, doubling its volume. If the initial pressure is P₀ and the initial volume is V₀, calculate the final pressure of the gas. (A) P₀/4 (B) P₀/2 (C) P₀ (D) 2P₀ (E) 0 

10. A cylinder containing a gas at 400 K is fitted with a piston, allowing the gas to do 200 J of work during an isobaric expansion. Calculate the change in internal energy if 50 J of heat is lost to the surroundings. (A) 150 J (B) -50 J (C) -250 J (D) 50 J (E) 200 J

11. What is the entropy change when 18 grams of water at 100°C and 1 atm is completely converted to steam at the same temperature and pressure? (The latent heat of vaporization of water is 2260 kJ/kg) (A) 12.6 J/K (B) 25.2 J/K (C) 75.6 J/K (D) 109.2 J/K (E) 252.0 J/K 

12. A point charge q is placed at the center of a non-conducting spherical shell of inner radius r₁ and outer radius r₂. What is the electric field at a distance r from the center where r₁<r<r₂? (A) (B)(C) 0 (D)(E)

13. A rectangular loop of wire lies in the xy-plane with current I flowing in a clockwise direction when viewed from the positive z-axis. The loop dimensions are a along the x-axis and b along the y-axis. A uniform magnetic field B₀ along the positive z-direction is applied in the region. Using Ampere's Law, determine the net magnetic field at the center of the loop. (A) B₀ along the positive z-direction (B) B₀ along the positive x-direction (C) B₀ along the negative y-direction (D) B₀ along the negative z-direction (E) 0 

14. In a region where there is an electric field E and a magnetic field B, a charged particle is observed to travel straight without deflection. What is the speed of the particle if the fields are perpendicular to each other? (A) E+B (B) B/E (C) EB (D) 1/(EB) (E) E/B 

15. A solenoid with a core of relative permeability μr has 300 turns, a length of 0.5 meters, and carries a current of 2 A. If the permeability of free space is μ₀, what is the magnetic field inside the solenoid? (A) 600μ₁μ₀/0.5 (B) 300μ₁μ₀/0.5 (C) 600μ₁μ₀/2 (D) 0.5μ₁μ₀/600 (E) 0.5μ₁μ₀/300

第二大題(SECTION 2):填充題 (Fill-in-the-Blank)

16. A 2 kg block slides down a 30° incline with an acceleration of 2 m/s². The coefficient of kinetic friction between the block and the incline is   (16)  

17. A particle moves in a potential U(x)=U₀[1-exp(-x²/a²)], where U₀ and a are constants. At x=0, the force acting on the particle is   (17)  

18. Light in an optical fiber undergoes total internal reflection. If the index of refraction of the fiber is 1.5 and that of the surrounding air is 1.0, the critical angle for the light traveling in the fiber is   (18)  

19. A Carnot engine operates between two heat reservoirs at temperatures of 400 K and 800 K. The maximum efficiency of the engine is   (19)   

20. A circuit consists of a resistor R=10 Ω and an inductor L=1 H in series connected to a direct circuit source of 5 V. The time constant of the circuit is   (20)