90年林敏聰 大一普通物理學 考古題
total 150 points
1. Please write down the three laws of thermodynamics with brief
interpretation.
(15%)
請寫出熱力學三大定律並作簡單的解釋。
r
2. Please prove the relation, pV = a(常數) constant,during an adiabatic process of an ideal gas,where r=Cp/Cv(p,v下標),the
ratio of the molar specific heats for the gas.
(5%) r
請證明此關係式 pV =a(常數)，在一個理想氣體的絕熱過程，r=Cp/Cv(p,v下標)，是氣體莫耳比熱的比率。
3. A solid cylinder is attached to a horizontal massless spring so that it can
roll without slipping along a horizontal surface. The spring constant k is
3.0 N/m. If the system is released from rest at a position in which the
spring is stretched by 0.25m,(原卷有附圖)find
(a)the translational kinetic energy
(b)the rotational kinetic energy of the cylinder as it passes through the
equilibrium position.
(c)Show that inder these conditions the center of mass of the cylinder
executes simple harmonic motion. What is its period?
(15%)
一個實心均勻的圓柱體繫在一個水平沒什麼質量的彈簧上,如此以至於它(圓柱體)可以滾而不會沿著水平面滑動。又這個彈黃的k (彈性係數)是 3N/m。
這個系統由靜止釋放，而繩子的原長是0.25m，則:
(a) 平移動能?
(b) 當圓柱體通過平衡位置時，圓柱體的轉動動能是多少?
(c) 當這個圓柱體的質心在做簡諧運動時，它的週期為何?
4. (a)Please derive the entropy change:DS=S-S=nRln(Vf/Vi)+nCvln(Tf/Ti) (D為
delta的大寫,打不出來;f,i,v均為下標) for all reversible processes that
take the gas from state i to state f.
請推論”熵”變化: DS=S-S=nRln(Vf/Vi)+nCvln(Tf/Ti) (D為delta的大寫,打不出來;f,i,v均為下標)對所有可逆反應過程，且氣體由狀態i變成狀態f。
(b)Please use this relation to calculate the change in the entropy for a
free expansion process from V to 2V. Please also give the reason that
you may do in this way.
請用上面的關係求體積V 到 2V ，此這個自由膨脹過程計算”熵”的變化。 也請說明你為什麼要透過這個方法求的原因。
(c)Derive this increase of entropy with statistical mechanics (using the
Boltsmann's entropy equation S=klnW,where k is the Boltzmann's const
,W the multiplicity of the configuration).
(You can use the Stirling's Approximation:lnN!=NlnN-N.)
用統計力學推論”熵”的增加 (使用 Boltsmann's 的”熵等式”: S=klnW，k是Boltzmann常數，W架構的多樣性)
(你可以使用:the Stirling's Approximation:lnN!=NlnN-N.)
(each10%)
5. Please express (in string tension T(tou打不出來) and mass length density
u(miew打不出來)) and derive the equation for the wave speed v on stretched
string from Newton's second law.
請表達(在繩子張力 T 和線密度u中和從牛頓的第二定律得到伸開的繩子上波速V的等式)。
(10%)
6.An apparatus that liquefies helium is in a room maintained at 300K. If the
helium in the apparatus is at 4.0K,what is the minimum ratio of the heat
delivered to the room to heat removed from the helium?
(15%)
一個裝置保持在300K的低溫環境之下保持氦(He)的液化。如果氦(He)所處的環境變成了4.0K,把氦(He)由300K的環境移到4.0K的環境的熱的最小比率是多少?
7. Please construct the plots of P versus V,T versus S,and S versus Einternal
(internal下標)for the isothermal expansion and isobaric expansion
thermodynamic process.
(15%)
請建構一個恆溫膨脹和等壓膨脹熱力學過程之中:P相對於V的圖表,T相對於S ,和S相對於Einternal(下標)。
8.One mole of an ideal gas are expanded from V1 to V2 =3V1 (1,2下標). If the
expansion is isothermal at temperature 300K,find
(a)the work done by the expanding gas and
(b)the change in its entropy.
(c)If the expansion is reversibly adiabatic instead of isothermal,what is
the change of entropy of the gas?
(15%)
一個1mole的理想氣體從V1膨脹到V2 =3V1( 1,2 個下標)。如果擴張在溫度為300K的恆溫過程,則:
(a) 膨脹氣體所對外做的功? 和
(b)”熵”的變化?
(c)如果膨脹是可逆絕熱而不恆溫過程, 氣體”熵”的變化是多少?
9.What is the entropy change
(a)for any reversible CLOSED cycle and
(b)for the irreversible process whose final T and V are the same the
initial ones.
(20%)
“熵”的變化是多少在下列所述之環境:
(a) 任何一個可逆的封閉循環 和
(b)不可逆轉過程, 最後的T和V和一開始的相同。
10. Consider a damped simple harmonic motion with the total force EF=-kx-bv
(E表Sigma打不出來),where k is force constant of the spring,x the
displacement,v the velocity,b the damping constant. Please write down and
solve the (differential) equation of motion from the Newton's Second Law.
(10%)
用總力把減弱的簡諧運動看作 EF=- kx - bv , k是彈簧中的"力常數", x是位移, v是速度, b減弱的常數。請寫出和解出牛頓第二定律運動的微分方程式。
11.Good luck!!!and Happy NEW YEAR!
祝好運!!!新年快樂!
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