课程名称︰普通物理学甲上 课程性质︰必修 课程教师︰梁启德 开课学院：工学院 开课系所︰机械系 考试日期（年月日）︰99/01/14 考试时限（分钟）：13:20~15:10 是否需发放奖励金：是 谢谢 （如未明确表示，则不予发放） 试题 : 1. A large block P executes horizontal simple harmonic motion as it slides across a frictionless surface with a frequency f = 1.50 Hz. Block B rests on it as shown in Fig. 1, and the coefficient of static friction between the two is μs = 6.00. What maximum amplitude of oscillation (最大振幅) can the system have if block B is not to slip (不滑动) ? (10%) ▋ μs ┌─┐ ▋ ↘│Ｂ│ ▋ ┌┴─┴┐ ▋﹌﹌﹌│ Ｐ │ ▆▆▆▆▆▆▆▆▆ Figure 1 2. A 1.00-mol sample of a monatomic ideal gas is taken through the cycle shown in Fig. 2. At point A, the pressure, volume, and temperature are Pi, Vi, and Ti, respectively. In terms of the gas constant R and Ti, find (a) the total energy entering the system by heat per cycle, (5%) (b) the total energy leaving the system by heat per cycle (5%), and (c) the efficiency of an engine operating in this cycle (5%). P Q2 │ ↓ 3Pi├ － ┬─→┐ │Q1→↑ →│Q3 │ │ ↓ ↓ Pi├ － ┼←─┤ │ ： Q4 ： │ ： ： └──┴──┴──V Vi 2Vi 3. Maxwell's speed distribution (速率分布) law is given by p(v) = M 1.5 2 ( -Mv^2 / 2RT ) 4π(───) v e , where p(v) is the probability 2πRT distribution function, M is the molar mass of the gas, R is the gas constant, T is the gas tmperature, v is the molecular speed, respectively. ∞ ∫vp(v)dv 0 It is known that the average speed of the gas is given by ─────, and ∞ ∫p(v)dv 0 ∞ 2 ∫ v p(v)dv 0 the average of the square of the speed is given by ──────. Please ∞ ∫p(v)dv 0 ∞ prove (证明) that (a) ∫p(v)dv = 1 (10%) (b) the average speed of the gas 0 8RT 0.5 is given by (───) (5%) (c) the average of the square of the speed is πM 3RT x x given by ── (5%). [Hint: de / dx = e ] M 4. Please prove (证明) that the general solution (通解) of the following equation kx + ma = 0. Where k is the spring constant, m is the mass of the block attached to the spring, x is the displacement of the block, 2 2 a = d x / d t is the linear acceleration of the block, and t is time, is given by C1 cos[(k/m)^1/2] + C2 sin[(k/m)^1/2]. Here C1 and C2 are iθ -iθ constants. (15%) [Hint: e = cosθ + i sinθ and e = cosθ - i sinθ, where i = √-1] 5. A solid sphere (实心球) of mass m and radius r rolls without slipping (没有 滑动) along the track shown in Fig. 3. It starts from rest with the lowest point of the sphere at height h above the bottom of the loop of radius R, much larger than r. What is the minimum value (最小值) of h (in terms of R) such that the sphere completes the loop? (10%) [Hint: I = 2/5 mr^2 ; you need to consider both translational energy and rotational kinetic energy] ▕╲.m ▕↑╲ ▕ h ╲ ╭─╮ ▕│ ╲│ R│ ▕↓ ╲─╯ ▇▇▇▇▇ Figure 3 6. A toy rocket moves at a speed of 242 m/s directly toward a stationary pole while emitting sound waves at frequency f = 1250 Hz. (a) What frequency f' is sensed by a detector that is attached to the pole? (10%) Some of the sound reaching the pole reflects back to the rocket, which has an onboard detector, what frequency f" does it detect? (10%) [Hint: The speed of sound is given by 343 m/s] 7. A hypodermic syringe contains a medicine having the density of water (Fig. 4). The barrel of the syringe has a cross-sectional area A = 2.50 ×10^(-5) m^2, and the needle has a cross-sectional area a = 1.00 ×10^(-8) m^2. In the absence of a force on the plunger, the pressure everywhere is 1 atm. A force of magnitude 2.00 N acts on the plunger, makng medicine squirt horizontally from the needle. Determine the speed of the medicine as it leaves the needle's tip. (10%) ▁▁▁▁▁▁▁▁▁▁▁▁▁ ▋ ▕ ╲ ▋ ▋ ▕ A ╲▁▁▁▁ F→ ████▋ ▕↙ a→▏ →v ▋ ▋ ▕ ╱▇▇▇▇ ▋ ▕ ╱ ▇▇▇▇▇▇▇▇▇▇▇▇▇ Figure 4 --

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