7. A small ball of mass m starts from rest at the top of a solid sphere of radius r and slides down its frictionless surface (see Fig. 3). At what angle θ will the small ball leave the sphere? (9) ___
Calculate the speed v of small ball when it leaves the sphere. (10) ___

1. What is the change in entropy of 500 g of water as its temperature increases from 10 °C to 25 °C?(1) ___ J/K ? (The specific heat of water is 4.19 kJ/(kg·K).)

2. The plates of a parallel-plate capacity carry surface charge density σ C/m². A dielectric slab of thickness t and dielectric Constant κ is inserted into the parallel plate capacitor with plates of area A, separated by distance d, as shown in Fig.1. Assume that the battery is disconnected before the slab is inserted.What is the total potential difference between the plates? (2) ___ V.What is the capacitance C₁ between the plates? (3) ___ nF.If the dielectric slab is replaced by a metal sheet with the thickness t, compared to the capacitance C₁, how is the capacitance affected? (increase? decrease? or keep constant?) (4) ___(Take d = 1 cm, t = 0.3 cm, σ = 2 nC/m², κ = 5, A = 40 cm², ε₀=8.85×10⁻¹² F/m.)

3. A long, straight wire carries a constant current I. A metal rod of length L moves at velocity v relative to the wire, as shown in Fig. 2. What is the potential difference between the ends of the rod? (5) ___

4. A dipole with a dipole moment p is pivoted freely at its midpoint. It lies in a uniform field. If its moment of inertia about the center is I, what's the angular frequency ω of the dipole oscillations for small angular displacements? (6) ___

5. A Cu strip of thickness 0.75 cm carrying a current of 10 A is set normal to a magnetic field. The Hall potential difference is 1.2 μV. What is B? (7) ___ T. (Take n = 8.5 × 10²⁸ electrons/m³)

6. A long solenoid of length l and cross-sectional area A has N turns. Find its self-inductance. (8) ___(Assume that the field is uniform throughout the solenoid.)

8. (a) Determine the coordinate for the center of mass (CM) of a uniform cone of height h and radius R, as shown in Fig. 4: (11) ___(b) If this cone is not uniform with a z-dependent density ρ = ρ₀·z, determine its CM. (12) ___

9. A guitar player tunes the fundamental frequency of a guitar string to 440 Hz. What will be the fundamental frequency if she then increases the tension in the string by 15%? (13) ___ (Hz).What will it be if, instead, she decreases the length along which the string vibrates by sliding her finger from the tuning key one-fourth of the way down the string toward the bridge at the lower end? (14) ___ (Hz)

10. Two masses, m₁ = 1.0 kg and m₂ = 2.0 kg, are connected by a rope that hangs over a pulley. The pulley is a uniform cylinder of radius 0.1 m and mass 0.5 kg. Initially m₁ is on the ground and m₂ rests 2.5 m above the ground (see Fig. 5). If the system is released, taking the gravitational acceleration g = 10 m/s², what is the acceleration of m₂ just before it strikes the ground? (15) ___. What is the tension T₂ before m₂ strikes the ground? (16) ___

（二）每題10分，請寫出步驟。(1) Two moles of an ideal diatomic gas (γ = 7/5) operate in the cycle of Fig. 6, where T₀ = 400 K, Tc = 200 K, and P₀ = 100 kPa. Find: (a) V₀ = ___ m³? (5%) and (b) the work done from a to b? Wa→b = ___? 

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