Solutions_HW7

# Solutions_HW7 - Rachel 4o l

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Rachel Lee Physics 601 Homework 7---Due Friday October 29 4o l<o You may find Mathematica very helpful for some of these problems!! 1, Three beads are free to move along a wire. They are connected two immovable walls by 4 springs. The masses and spring constants are as indicated on the figure. . a. Find the normal modes and their associated frequencies by finding the eigenvectors and eigenvalues of the appropriate matrix. . b. Verifu that the modes are orthogonal witi respect to the mass matrix. "c. Suppose at t=0 all three particles are in their equilibrium positions with the two particle on the end at rest and the one in the middle moving with velocity v. [This can happens as a result of an impulse acting on the middle particleJ. Find the motion ofthe three particles. ' 2. Consider the system described in the previous problem. Suppose that all the particles begin at rest in equilibrium a force given by/(r) = f sinh(r/7")0(l) acts on the middle particle. Use Green's functions to determine the motion of the three particles. . 3. Co4q4!e13n infinite ch4in qf q4qsqe_q and springs as shown below: 2mm2mm2mm 2m

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The masses alternate between two types, one twice as heavy as the other. The springs connecting the masses are all equal and in equilibrium the masses are all a distance a apart. The masses are constrained to move longitudinally. Find the dispersion relation relating frequency to wave number. Three beads are free to move along a wire. They are connected two immovable walls by 4 springs. The masses and spring constants are as indicated on the figure. The third particle.ia**end has a mass q. The first particle- is attached to a shock absorber with a force given by F : -maoi where arj : a. Using the . la\ - (o (i)-' \ phase-space formalism derived in class: ri = | '-l W :l -- .: '- , I with the ' \n) \-Y -F(n-'l Suppose the system described in the problem above has the middle particle charged fwith charge qJ and is place in an electric field, oriented to the right with a magnitude of E = E,tcos(2a.t)

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Solutions_HW7 - Rachel 4o l

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