107年 - 107 國立臺灣大學_碩士班招生考試試題：普通物理學#139134

一、填空題
1. A particle is at rest at t = 0 and starts moving along the x axis from the origin. Based on the following figure, the displacement at t = 9 s is ___(1)___ (m).

2. A particle is at rest at t = 0 and starts moving along the x axis from the origin. Based on the following figure, the displacement at t = 9 s is ___(2)___ (m).

3. A particle is moving along a circular path with radius r. If its angular velocity and angular acceleration are ω and α, respectively, its acceleration is ___(3)___ in the radial direction and is ___(4)___ in the tangential direction.

4. What is the maximum power transfer of a spring with force constant k to an attached object in a simple harmonic oscillation with amplitude A and frequency f?___(5)___

5. The following metal objects, a disk, a solid sphere, a hollow sphere, and a loop have the same radius and mass. Assuming they are released from the same point on a ramp at rest and start rolling down, which one will take the shortest time to reach the bottom of this ramp? ___(6)___

6. Write down the Kepler's 3^rd law ___(7)___, the law of periods, for a planet circulating around its sun (mass M) with a radius r. Use T and G to denote its period and the gravitational constant, respectively.

7. A planetary nebula is a cloud of mainly hydrogen gas with density ρ of 1000.0 molecules per cubic centimeter, and a temperature T of 10,000 K. What is the pressure of the gas in the nebula? ___(8)___ (Pa)

8. A Carnot engine takes 5000 J of heat from a thermal reservoir with a temperature T_H = 500 K and discards heat to a thermal reservoir with a temperature T_L = 325 K. How much work does the Carnot engine do in this process? ___(9)___ (J)

9. A weightlifter snatches a barbell with mass m = 180.0 kg and lifts it up 1.25 m vertically. If we consider the weightlifter to be a thermodynamic system, how much heat must he give off if his internal energy decreases by 4000 J? ___(10)___ (J)

10. 0.5 mole of N₂ is confined in a volume of 1.0 m³ and undergoes free expansion to a new volume of 2.0 m³. What is the change in entropy of the gas?___(11)___ (J/K)

11. A spherically symmetrical but non-uniform charge distribution is given by
ρ(r) = ρ₀(1 − r/R) for r ≤ R; ρ(r) = 0 for r ＞ R, where ρ₀ = 10 μC/m³ and R = 0.250 m. What is the electric field produced by this charge at r = 0.500 m? ___(12)___ (N/C)

12. Suppose a thundercloud with a width of 2.0 km and a length of 3.0 km hovers at an altitude of 500 m over a flat area. The cloud carries a charge of 160 C, and the ground has no charge. What is the potential difference between the cloud and the ground? ___(13)___ (V)

13. Suppose the unpolarized light with intensity I₀ is initially incident on the first of two polarizer in a line. The first polarizer has a polarizing direction that is vertical. The second polarizer has a polarizing angle of 45° with respect to the vertical. What is the intensity of light after passing through the two polarizers? ___(14)___ (I₀)

二、計算題
1. A baseball player knows the fact that when he hits a baseball on the so called "sweet spot", the ball can get a maximum momentum to fly away. Strangely, the baseball player can feel almost no force exerted from this perfect hit. Based on the above no force exerted statement and using a uniform stick with length L to model a baseball bat, find the sweet spot on this stick. You should assume the baseball player holding the bat at its one end and the swing of the bat is like a rotation about this end.

2. A long solenoid has 220 turns/cm and carries a current I = 1.5 A; its diameter D is 3.2 cm. At its center, we placed a 130-turn closely packed coil C of diameter d = 2.1 cm. The current in the solenoid is reduced to zero in a steady rate in 0.025 s. What is the magnitude of emf that is induced in coil C while the current in the solenoid is changing?