課程名稱︰力學上 課程性質︰物理系必帶 課程教師︰龐寧寧 開課學院：理學院 開課系所︰物理系 考試日期（年月日）︰2019/01/07 考試時限（分鐘）：180分鐘 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : Mechanics 2019/01/07 1. Show that the geodesic on the surface of a right cylinder is a segment of a helix. (16%) 2. In an Atwood (the left graph below) machine mass m_1 is hung on the left side of a fixed massless pulley which rotates without slipping on a massle- ss string. A mobile pulley of mass m_2 and radius R can move vertically and also rotate without slipping on the string. Use the Lagrange undertermined multipliers to calculate the accelerations of m_1 and m_2 and tensions t_1 and t_2. (20%) ---------------- | | - | |o| | | |t_1 |t_2 | \ o / | - ↓ ↓ m_1g m_2g 3. A particle of mass m is constrained to move without friction on a circular loop of radius R. The loop (the right graph above) is rotating around its vertical diameter with constant angular velocity ω. (a) Find the equilibr- ium position. (b) Calculate the frequency of small oscillation about the equilibrium position. (c) There is a critical angular frequency which is a threshold of two very different particle behaviors. Find it. (d) Write down the Hamiltonian. Is it the total energy? Is it conserved? Why? (24%) 4. In field theory people usually use Lagrage density L in place of Lagrangian because they consider 4-dimension space-time: J=∫Ldxdydzdt where J is the action. Here let's consider only 1 space and 1 time dimension: J=∫L(Ψ*,Ψ,∂Ψ*/∂t,∂Ψ/∂t,∂Ψ*/∂x,∂Ψ/∂x;x,t)dxdt. Now we have two independent variables, x and t. Hence, the ordinary derivative d/dt in textbook should be replaced by ∂/∂x and ∂/∂t. The Euler-Lagrange equat- ion is the Schrodinger equation:-(hbar^2/2m)∂^2Ψ/∂x^2+V(x)Ψ(x,t) = i hbar ∂Ψ/∂t where Ψ is the wave function. Its complex conjugate Ψ* is treated as another independent function and we obtain another equation as -(hbar^2/2m)∂^2Ψ*/∂x^2+V(x)Ψ*(x,t) =-i hbar ∂Ψ*/∂t. Find the form of the Lagrangian. (16%) 5. In a sun-earth system there are five Lagrange points (denoted by L1~L5 in graph below) where a massive particle can be at "equilibrium". At these po- ints the net gravitational force is equaal to the needed centripetal force for the particle to perform uniform circular motion around the sun with the same period as that of earth. (a) Let the masses of the sun and earth be M_S and M_E respectively and the distance between the sun and earth be R, find the positions of L1 and L2. (b) Are the orbits of L1 and L2 stable? If a particle is at L1(L2) moving with earth, will a little deviation in position change the orbit drast- ically? (c) As for L4, we have to consider the fact that the sun and earth are orb- iting about their common center of mass, denoted as CoM in the graph below. We can again approximate the orbit by a circle. Show that if sun , earth and L4 form a equilateral triangle (正三角形), then the combin- ed gravitational force by the sun and earth on a particle at L4 point points at CoM. Furthermore, show that the combined force is equal to the centriputal force of a particle at L4 point within the accuracy of M_E/M_S. (24%) ‧L4 / \ / \ / \ / \ / \ / \ L2 L3 ‧ sun‧-x------------‧ ‧ ‧ CoM L1 earth ‧L5 6. A system consists of two particles with masses m_1 and m_2 and position ve- ctors r_1, and r_2. There is no external force. But two particles interact with each other via the potential V(|r_1-r_2|). That is, V depends only on the distance between two particles. Use the Noether's theorem to find out what physical quantities are conserved. (20%) Hint: Consider the center-of-mass frame and rewrite the kinetic energy. --

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