[試題] 99上 梁啟德 普通物理學甲 期末考
課程名稱︰普通物理學甲
課程性質︰系定必修
課程教師︰梁啟德
開課學院：工學院
開課系所︰機械系
考試日期（年月日）︰100/01/13
考試時限（分鐘）：13:20~15:10
是否需發放獎勵金：是 謝謝
（如未明確表示，則不予發放）
試題 :
Final examination(13/1/2011)
1.Please prove that for an adiabatic (絕熱) expansion (膨脹) 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 (莫耳比熱) 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(證明) 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(自由膨脹).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(證明) 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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