课程名称︰力学 课程性质︰系必修 课程教师︰张庆瑞 开课学院：理学院 开课系所︰物理学系 考试日期（年月日）︰2011/11/14 考试时限（分钟）：170分钟 是否需发放奖励金：是，感谢～ （如未明确表示，则不予发放） 试题 : 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)求的轨迹是指脱离大球之後的轨迹） 图：一个圆 x^2+(y-R)^2=R^2 θ是y轴与球心粒子连线的夹角 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)没有关系 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 图 一个第一卦限 封闭曲线是从原点出发沿x-轴到(1,0,0)--->x-y平面上单位圆的轨迹到(0,1,0) --->y-z平面上y=1那条线到(0,1,2)--->回到原点 如果看不太懂而手上有电磁学课本(Griffiths)的 Problem 1.56有一模一样的题目（更好的叙述XD） 完整的图，然後根据唯一性定理（误）答案也是一样的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. 题目可能有些不严谨，因为考试中助教也一直在修正orz；考力学期中考写到电磁学 的繁琐题，还真温馨（远望） --

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