※ 引述《hydest ()》之銘言： : 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。 Sol: 因為dQ = dU + dW =>dQ = nCvdT + PdV = nCvdT + (nRT/v)dV Tf Vf =>△S = ∫(dQ/T) = ∫{nCvdT + PdV}/T = ∫ (nCv/T)dT + ∫ (nR/V)dV Ti Vi = nCvln(Tf/Ti) + nRln(Vf/Vi) Done! (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 計算自由膨脹過程計算”熵”的變化。 也請說明 你為什麼要透過這個方法求的原因。 Sol: Free expansion為不可逆的反應過程且反應前後內能不變；又△S僅與初狀態和末狀 態有關。 所以找一可逆反應(等溫膨脹)，計算其△S，其亦為Free expansion的△S Vf △S = ∫ (nR/V)dV = nRln2 Done ！ Vi (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%) 請問這題怎麼著手呢？ : 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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