1. (a) The vibrations of an undamped free mass-spring system is governed by the initial value...

 

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1. (a) The vibrations of an undamped free mass-spring system is governed by the initial value problem mu" + ku = 0, u(0) = a, u'(0) = 6, where u denotes position and u' denotes velocity. Write the solution to this i.v.p. by finding the coefficients from the i.c. 1 1 (b) Note that the total initial energy of the system is E(0) = mb? + ka" , where the first term is its initial kinetic energy and the second term is its potential energy. Show that 1 the total energy E(t) =m(u'(t))? +k(u(t))? of the system at any time t20 is equal to E(0) thereby confirming the principle of conservation of energy for this system. (c) Now take a = 0. If you plot u' on the vertical axis and u on the horizontal axis using t as a parameter, what curve do you get? Prove your claim. 1. (a) The vibrations of an undamped free mass-spring system is governed by the initial value problem mu" + ku = 0, u(0) = a, u'(0) = 6, where u denotes position and u' denotes velocity. Write the solution to this i.v.p. by finding the coefficients from the i.c. 1 1 (b) Note that the total initial energy of the system is E(0) = mb? + ka" , where the first term is its initial kinetic energy and the second term is its potential energy. Show that 1 the total energy E(t) =m(u'(t))? +k(u(t))? of the system at any time t20 is equal to E(0) thereby confirming the principle of conservation of energy for this system. (c) Now take a = 0. If you plot u' on the vertical axis and u on the horizontal axis using t as a parameter, what curve do you get? Prove your claim.

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