※ 引述《hydest ()》之铭言： : 90年林敏聪 大一普通物理学 考古题 : total 150 points : 1. Please write down the three laws of thermodynamics with brief : interpretation. : (15%) : 请写出热力学三大定律并作简单的解释。 r 2. Please prove the relation, pV = a(常数) constant,during an adiabatic process of an ideal gas,where r=Cp/Cv(p,v下标),the ratio of the molar specific heats for the gas. (5%) r 请证明此关系式 pV =a(常数)，在一个理想气体的绝热过程，r=Cp/Cv(p,v下标)， 是气体莫耳比热的比率。 Sol: First,We have to know the following conditions : (i) equation of state: PV = nRT → PdV + VdP = nRdT (ii) first law : dQ = dU + dW (iii) Constraint condition: dQ = 0 0 = nRdT + PdV = nCv*dT + PdV From (i) , dT=(PdV+VdP)/nR So nCv*(PdV+VdP)/nR + PdV = 0 Find the relation between V and P , We can get PV^r = const. Done! : 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,(原卷有附图)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%) : 一个实心均匀的圆柱体系在一个水平没什麽质量的弹簧上,如此以至於它(圆柱体)可以滚而不会沿着水平面滑动。又这个弹黄的k (弹性系数)是 3N/m。 : 这个系统由静止释放，而绳子的原长是0.25m，则: : (a) 平移动能? : (b) 当圆柱体通过平衡位置时，圆柱体的转动动能是多少? : (c) 当这个圆柱体的质心在做简谐运动时，它的周期为何? : 4. (a)Please derive the entropy change:DS=S-S=nRln(Vf/Vi)+nCvln(Tf/Ti) (D为 : delta的大写,打不出来;f,i,v均为下标) for all reversible processes that : take the gas from state i to state f. : 请推论”熵”变化: DS=S-S=nRln(Vf/Vi)+nCvln(Tf/Ti) (D为delta的大写,打不出来;f,i,v均为下标)对所有可逆反应过程，且气体由状态i变成状态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. : 请用上面的关系求体积V 到 2V ，此这个自由膨胀过程计算”熵”的变化。 也请说明你为什麽要透过这个方法求的原因。 : (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.) : 用统计力学推论”熵”的增加 (使用 Boltsmann's 的”熵等式”: S=klnW，k是Boltzmann常数，W架构的多样性) : (你可以使用:the Stirling's Approximation:lnN!=NlnN-N.) : (each10%) : 5. Please express (in string tension T(tou打不出来) and mass length density : u(miew打不出来)) and derive the equation for the wave speed v on stretched : string from Newton's second law. : 请表达(在绳子张力 T 和线密度u中和从牛顿的第二定律得到伸开的绳子上波速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%) : 一个装置保持在300K的低温环境之下保持氦(He)的液化。如果氦(He)所处的环境变成了4.0K,把氦(He)由300K的环境移到4.0K的环境的热的最小比率是多少? : 7. Please construct the plots of P versus V,T versus S,and S versus Einternal : (internal下标)for the isothermal expansion and isobaric expansion : thermodynamic process. : (15%) : 请建构一个恒温膨胀和等压膨胀热力学过程之中:P相对於V的图表,T相对於S ,和S相对於Einternal(下标)。 : 8.One mole of an ideal gas are expanded from V1 to V2 =3V1 (1,2下标). 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%) : 一个1mole的理想气体从V1膨胀到V2 =3V1( 1,2 个下标)。如果扩张在温度为300K的恒温过程,则: : (a) 膨胀气体所对外做的功? 和 : (b)”熵”的变化? : (c)如果膨胀是可逆绝热而不恒温过程, 气体”熵”的变化是多少? : 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”的变化是多少在下列所述之环境: : (a) 任何一个可逆的封闭循环 和 : (b)不可逆转过程, 最後的T和V和一开始的相同。 : 10. Consider a damped simple harmonic motion with the total force EF=-kx-bv : (E表Sigma打不出来),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%) : 用总力把减弱的简谐运动看作 EF=- kx - bv , k是弹簧中的"力常数", x是位移, v是速度, b减弱的常数。请写出和解出牛顿第二定律运动的微分方程式。 : 11.Good luck!!!and Happy NEW YEAR! : 祝好运!!!新年快乐! --

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