课程名称︰普通物理学甲 课程性质︰系定必修 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）︰100/01/13 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 : Final examination(13/1/2011) 1.Please prove that for an adiabatic (绝热) expansion (膨胀) of an ideal gas, the pressure p and the volume V obey pV^γ = a constant, where γ = Cp/Cv is the ratio of the molar specific heats (莫耳比热) for the gas. Cp is the molar specific heat for a constant pressure process and Cv is the molar specific heat for a constant volume process, respectively.(10%) 2.Please describe and prove the Doppler effect[We are not considering the Doppler effect for light here].(10%) 3.Determine the amplitude of the resultant wave when two sinusoidal string waves having the same frequency and travelling in the same direction are combined, if their amplitudes are 6.0 cm and 8.0 cm and they have phase constants of 0 and π/2 rad, respectively(10%).(Hint:use Phasors) ∞ 4.Please calculate ∫ e^(-x^2)x^2 dx (10%). 0 ∞ 5.Please prove(证明) that ∫ e^(-x^2)x^5 dx = 1 (10%). 0 6.Use the wave equation to find the speed of a wave given by y(x,t) = (2.00mm)[(20 m^-1)x-(4.0s^-1)t]^0.5 (10%) 7.Please find the general solution(通解) of the following equation: d^2y/dx^2 -4 dy/dx + 3y = 0 . (10%) 8.Let us consider n moles(莫耳) of an ideal gas doubles its volume in a free expansion(自由膨胀).Please use the Boltzmann's entropy equation S=k lnW and Stirlings approximation lnN!=N(lnN)-N when N(the number of ideal gas molecules) is large to prove(证明) that Sf-Si=nR ln2.[Hint: R is the gas constant, k=nR/N is the Boltzmann constant, and W is the number of microstates].(10%) 9.An ice at 0°C falls into water at 0°C. Its (the ice's) gravitational potential energy is fully transformed into heat. If 10% of the ice (by weight) is melted, what is its (the ice's) origin height counting from water level? (10%)[Hint:the heat of fusion of water is given by 333KJ/kg] 10.A pendulum of length L and mass M has a spring of force constant k connected to it at a distance h below its point of suspension (Fig.1). Find the frequency of vibration of the system for small values of the amplitude (smallθ). Assume the vertical suspension rod of length L is rigid, but ignore its mass. (10%) ███████████ ∕▏ █ ∕∣ █ ∕▕ █ ∕ ▕ h █ L∕θ▕ █ ∕ ▕ █ ∕ ▕ █ ∕ ▕~~~~~~~~~█ ┌─┐ ▕ k █ │Ｍ│ │ █ └─┘ ┌┴┐ █ │Ｍ│ █ └─┘ █ Figure 1 --

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