課程名稱︰力學上 課程性質︰系必修 課程教師︰高英哲 開課學院：理學院 開課系所︰物理學系 考試日期（年月日）︰2008/11/17 考試時限（分鐘）：160 是否需發放獎勵金：是。 試題 : Do all the problems. Please put down the details of your calculation/derivation as well as your reasoning. Good luck!! Total Score: 100 1.Using the solutions of a damped oscillator, explain why it is better to design a door closing mechanism such that it is critically damped. (10%) 2.Given a force F = (xy, cx^2, z^3) on a unit mass, where c is a constant. (a) Find the value of c such that the force is conservative. (5%) (b) For the c above, find the work required to move the particle from (0, 0, 0) to (1, 1, 1). (5%) (c) If the point (1, 1, 1) is a turning point, what is the velocity of the particle at the origin? (5%) 3.A block of mass m is attached to a massless spring with a force constant k. The system is placed horizontally on a table and the friction coefficient between the block and the table surface is μs = μk = μ. The block is originally placed at a position x0 (greater than μmg/k and the equilibrium position is at x = 0) and released at rest, . (a) find x(t) and x(t) after the block is released and plot the curve x(t) vs. t, at least for the first two cycles of oscillation; (10%) (b) plot the corresponding phase diagram. (10%) 4.Consider a uniform ring of radius a with total mass M(Fig. 1), and a point particle of mass m is placed in the plane of the ring, (a) find the gravitational potential at r' from the center (10%); (b) find the equilibrium point and discuss the stability of this equilibrium point. (10%) 5.Given two fixed points (x1, y1) and (x2, y2), find the curve connecting these two points such that the surface area generated by revolving the curve about the y-axis is minimum. (15%) (Fig. 2) 6.Taipei and Tokyo are connected by a subway tunnel (Fig. 3). A train of mass m travels between the two cities powered only by the gravitational force of the earth. Assume the length of the tunnel is L, and the earth is a homogeneous sphere of radius R with total mass M. The train starts form rest at the earth's surface. Calculate the maximum speed of the train, and the time taken to travel from Taipei to Tokyo. (20%) 1 dt ∫ ───── = π 0 √(t(1-t)) --

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