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Problems & Solutions >Tutorial-3 Coupled Oscillator_ PHYSICS Q&A

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Tutorial-3 Coupled Oscillator Problem 1 There are two small massive beeds, each of mass ?, on a taut massless string of length 8?, as shown. The tension in the string is ?. 1. Derive expressions ... for the normal mode angular frequencies of the system, for small displacements of the system in a plane. Assume that ? is suffieciently large that you may ignore gravity. 2. If at ? = 0 both masses are stationary, one is at equilibrium and the other displaced from equilibrium by a distance ?, write an expression for displacement ?1(?) as a function of time of the mass initially in the equilibrium p Problem 2 The figure above shows a system of masses in the x-y plane. The mass of 2? is connected to an immobile wall with a spring of constant 2?, while the mass of ? is connected to an immobile wall with a spring of constant ?. The masses are then coupled to each other with an elastic band of length ?, under tension ? = 2??. The masses are constrained to move in the x direction only. At equilibrium the masses have the same x position and the springs are uncompressed. There is no friction orgravity. The displacements from equilibrium are small enough (?1, ?2 ≪ ?), so that the tension in the band stays constant. 1. Find the normal mode frequencies of the system. 2. What are the ratios of the displacements from equilibrium for two different normal modes? (Hint: Basically, you have to find out the eigen functions for corresponding normal mode frequencies) Problem 3 Two identical spring-mass systems ? and ? are coupled by a weak middle spring having a spring constant smaller by a factor of 100 (i.e. 100??????? = ?? = ??).Mass ? is pulled by a small distance and released from rest, while mass ? is released from rest at its equilibrium position, at ? = 0. Calculate the approximate number of oscillations completed by mass ? before its oscillations die down first. Problem 4 Three identical masses ? are connected in series to four identical springs each of force constant ? (each mass coupled to two strings, and the end springs to walls): a) Calculate the eigenfrequencies of small oscillations (normal modes) of this coupled oscillator system. b) What sets of initial displacements would you give to the three masses so that the system oscillates in each of these normal modes. [Show More]

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