課程名稱︰普通物理學甲上
課程性質︰系定必修
課程教師︰梁啟德
開課學院：
開課系所︰材料系
考試日期（年月日）︰2008年1月14日
考試時限（分鐘）：130分鐘
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
1. Please prove that 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 = C_p/C_V is
the ratio of the molar specific heats (莫耳比熱) for the gas. C_p is the molar
specific heat for a constant pressure process and C_V is the molar specific
heat for a constant volume process, respectively (10%)
2. If the temperature of a rod of length L is raised by an amount of ΔT, its
length is found to increase by an amount ΔL = LαΔT, where α is a constant
called the coefficient of linear expansion. Please prove that the coefficient
of volume expansion β is given by β = 3α. (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. A model Stirling engine uses n = 8.10x10^-3 mol of ideal gas as a working
substance. It operates between a high-temperature reservoir at T_H = 95 ℃ and
a low-temperature reservoir at T_C = 24 ℃: the volume of its working
substance doubles (變成2倍) during the expansion stroke: and it runs at a rate
of 0.70 cycle per second. Assume that the engine is ideal. The ideal gas
constant R = 8.31J/mol/K (a)How much work does the engine do per cycle? (5%)
(b) What is the power of the engine? (5%)
5. A verticl (垂直) spring stretches 9.6 cm when a 1.3 kg block is hung from
its end. (a) Calculate the spring constant (5%). This block is then displaced
an additional 5.0 cm downward and released from rest. Find (b) the period (5%)
(c) the frequency (5%) and (d) the amplitude (5%) of the resulting Simple
Harmonic Motion (SHM).
6. A toy rocket moves at a speed of 242 m/s directly toward a stationary pole
while emitting sound waves at frequency f=1250Hz (a) What frequency f' is
sensed by a detector that is attached to the pole? (5%) (b) Some of the sound
reaching the pole reflects back to the rocket, which has an onboard detector.
What frequency f'' does it detect? (5%) (Hint: The speed of sound is 343 m/s.
Please consider the Doppler effect 都卜勒效應).
7. Use the wave equation to find the speed of a wave given by
y(x,t) = (2.00 mm)[(20 m^-1)x-(4.0 s^-1)t]^0.5 (10%)
8. A nylon guitar string has a linear density 7.20 g/m and is under a tension
of 150 N. The fixed supports are distance D= 90.0 cm apart. The string is
oscillating in the standing wave pattern with a wavelength of 60 cm. Calculate
the (a) speed (10%) and (b) frequency (10%) of the traveling waves whose
superposition gives this standing wave.
9. An ideal gas is carried through a thermodynamic cycle consisting of two
isobaric (等壓) and two isothermal (等溫) processes as shown in Fig. 1. Please
show that the net work done on the gas is the entire cycle is given by
W_net =-P_1(V_2-V_1)ln(P_2/P_1)
(10%)
P
| B C
P_2|_ _ _ _ ______>______
| \ \
| \ \
| ^ \
| \ \
| \ \
P_1|_ _ _ _ _ _ _\_____<______\
| A| |D
|______________|___________|__ V
V_1 V_2
Figure 1
10. A block of mass m is connected to two springs of force constant k_1 and
k_2 as shown in Figure 2. The block moves on a frictionless table after it is
displaced from equilibrium and released. Show that that block exhibits simple
harmonic motion with period T =2π(m/(k_1+k_2))^0.5
| _____ |
| k_1 | | k_2 |
|\/\/\/\/-| m |-\/\/\/\/\|
|_________|_____|__________|
Figure 2
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