A block of mass M is connected to a spring of mass m and oscillates in simple ha

Question

A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a horizontal, frictionless track (Figure P15.62). 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, vx = (x/ l )v. Also, note that the mass of a segment of the spring is dm = (m/ l )dx. Figure P15.62 (a) Find the kinetic energy of the system when the block has a speed v. (Use M, m, and v as necessary.) (b) Find the period of oscillation. (Use M, m, and k as necessary.)

Explanation / Answer

(a) Kinetic Energy,ke = 0.5*M*v² + 0.5*integral{(m/l)v²(x/l)²}dx from 0 to l

ke =0.5*M*v² + 0.5*(m/l)v²(1/l²) (1/3) l3 = 0.5*M*v² + (1/6) mv²

ke = [0.5*M + (m/6)]v²

(b)Time Period,T= 2/= 2/(k/[M+m/2])

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