Q)
(chatquestion)...A mass M oscillates at the end of a spring that has spring constant k and finite mass m. (a) Show that the total energy E of the system for oscillations of small amplitude is given by E = 12(M +m/3)v2 + 12kx2 where v and x are the velocity and displacement of the mass M, respectively. (Hint: To find the kinetic energy of the spring, consider it to be divided into infinitesimal elements of length dl and find the total kinetic energy of these elements, assuming that the mass of the spring is evenly distributed along its length. The total energy E of the system is the sum of the kinetic energies of the spring and the mass M and the potential energy of the extended spring.) (b) Hence show that the frequency of the oscillations is equal to √k/(M + m/3).
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