課程名稱︰普通物理學甲上 課程性質︰機械系必修 課程教師︰蔡爾成 開課學院：理學院 開課系所︰物理系開給機械系 考試日期（年月日）︰100/11/11 考試時限（分鐘）：120分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 1.[10%]A centripetal addict rides in uniform circular motion with period → 2 T = 2.0 s and radius r. At t1 his acceleration is a = (6.00m/s )i 2 + (-4.00m/s )j. At that instant, what are the values of → → → → (a) v ． a and (b) r ╳ a ? 2. The potential energy of a diatomic molecule (a two atom system like H2 or O2) is given by A B U= ── - ── r^12 r^6 where r is the seperation of the two atoms of the molecule and A and B are positive constants. This potential energy is associated with the force that binds the two atoms together.(a)[5%]Find the equilibrium seperation - that is, the distance between the atoms at which the force on each atom is zero. (b)[5%]Is this equilibrium configuration stable? 3.[10%]Consider a rocket that is in deep space and at rest relative to an inertial reference frame. The rocket's engine is to be fired for a certain interval. What must be the rocket's mass ratio (ratio of initial to final mass) over that interval if the rocket's finalspeed relative to the inertial frame is ti be equal to the exhaust speed (speed of the exhaust products relative to the rocket) ? 4.[10%]An object rotates about a fixed axis, and the angular position of a 2t reference line on the object is given by θ(t) = e , where θis in radius and t is in seconds. Consider a point on the object that is at a distance of r⊥ from the axis of rotation. At t = 0, what is the → linear acceleration a of the point? 5.[10%]In Fig. 11-54, a bullet of mass m is fired into a block of mass M attached to the end of a nonuniform rod of length L and mass Mrod. The block-rod-bullet system then rotates in the plane of the figure, about a fixed axis at A.The rotational inertia of the rod alone about that A is I. Treat the block as a particle. (a) What then is the rotational inertia of the block-rod-bullet system about point A? (b) If the angular speed of the system about A just after impact is ω, what is the bullet?'s speed just before impact? Fig. 11-54 A ┌┐ ││ ││ ││Rod ││ ││ ││ ││ ││ ┌───┐ │ │Block ▆ → └───┘ Bullet → → → 6.(a)[5%]Is the uniform force field F (x,y,z) = Fo, where Fo is a constant → vector, conservative? Why? (b)[5%]Suppose F(x,y,z) = f(z)k points only in the z direction and its z component f(z) is a function of z only. Is this force conservative? 7.[10%]The total kinetic energy for a system of particles depends on the inertial reference frame. Prove that the total kinetic energy observed in the center of mass frame is minimal. 8. For a system of particles, if the strong version of Newton's third law holds, show that (a)[5%]only external forces will contribute to the total net torque: → → → → → τnet = Σxi ╳ Fi = Σxi ╳ Fi(ext) → → → → → where Fi = Fi(int) + Fi(ext) with Fi(int) the internal force and Fi(ext) the external force acting on partical i. (b)[5%]The total angular momentum → → → → → dL L = Σxi ╳ Li where Li = xi ╳ pi is conserved, i.e., ── = 0, for an i dt isolated system. 9.[10%]To get a billard ball to roll without slipping from the start, the cue 2 must hit the ball at a height ─R above the center. Prove this result. 5 2 2 The rotational inertia for a homogeneous sphere is ─MR . 5 10.[10%]For a fast spinning heavy symmetrical top rotates around its symmetry axis ─in the coordinate system whose origin is the fixed point on the → symmetry axis, the angular momentum L may be assumed to be parellel to → → → the angular velocity ω, L = Iω, where I is the rotational inertia. Ml → → (a)[3%]Show that the total torque exerted on the top is τ = ──g ╳ w dω Ml → → ω (b)[3%]Prove that ── = - ── g ╳ w dt Iω → (c)[4%]What is the angular velocity with which ω rotates about the vertical axis? What is the precession rate of the spinning top? P.S.第八題以及第十題中的dL/dt和dω/dt 上面的L和ω皆有向量符號，不過因排版問題 與以省略。 --

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