A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a horizontal, frictionless track (Fig. P15.66). The force constant of the spring is k and the equilibrium length is l. Assume that all portions of the spring oscillate in phase and that the velocity of a segment dx is proportional to the distance x from the fixed end; that is, ??x = (x/l)??. Also, note that the mass of a segment of the spring is dm = (m/l)dx. Find (a) the kinetic energy of the system when the block has a speed v and (b) the period of oscillation.
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