課程名稱︰力學 課程性質︰系必修 課程教師︰張慶瑞 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰2011/11/14 考試時限（分鐘）：170分鐘 是否需發放獎勵金：是，感謝～ （如未明確表示，則不予發放） 試題 : Prob. 1: Please find the response of the damped oscillator to a forcing function of the form F(t)=f*(δ(t-a)-δ(t-2a)), where {=∞ if t=a; δ(t-a)={ {=0 if t≠a Prob. 2: A pendulum is suspended from the cusp of cycloid cut in a rigid support. The path described by the pendulum bob is cycloidal and is giiven by x=a(φ-sinφ) y=a(cosφ-1) where the length of the pendulum is L=4a, and where φ is the angle of the rotation of the circle generating the cycloids. Show that the oscillation are exactly isochronous with a frequency ω=(g/L)^0.5 Prob. 3: Consider the surface generated by revolving a line connecting two fixed points (x_1,y_1) and (x_2,y_2) about an axis coplanar with the two points. Find the equation of the line connecting the points such that the surface area generated by the revolution is a minimum. Obtain the solution using second form of the Euler equation. Prob. 4: A smooth ball(radius R) fixed in the plane, a particle starting form rest at the highest point slide down along the spherical surface. Find (a) the angle θ, velocity v when the particle just leaves the ball (b) and the path of the particle y(x). （(b)求的軌跡是指脫離大球之後的軌跡） 圖：一個圓 x^2+(y-R)^2=R^2 θ是y軸與球心粒子連線的夾角 Prob. 5: (a) A particle of mass m is constrained to move on the surface of a sphere of radius R by an applied force F(θ,φ). Find a_θ and a_φ. (b) Evaluate the integral ∫A╳A"dt. Note. 助教：(a)跟(b)沒有關係 Prob. 6 (a) Explain Stoke's Theorem. (b) Compute the line integral ^ ^ ^ v=(rcosθ)r-(rcosθsinθ)θ+3rφ around the path shown in the Figure. (c) Check your answer, using Stoke's Theorem 圖 一個第一卦限 封閉曲線是從原點出發沿x-軸到(1,0,0)--->x-y平面上單位圓的軌跡到(0,1,0) --->y-z平面上y=1那條線到(0,1,2)--->回到原點 如果看不太懂而手上有電磁學課本(Griffiths)的 Problem 1.56有一模一樣的題目（更好的敘述XD） 完整的圖，然後根據唯一性定理（誤）答案也是一樣的1.5π。 Note. The formula of spherical curl is NOT GIVEN in the exam. Note. The constants of the catenoid (the curve wanted in Prob.3) have to be determined. Note. 題目可能有些不嚴謹，因為考試中助教也一直在修正orz；考力學期中考寫到電磁學 的繁瑣題，還真溫馨（遠望） --

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