The preceding problems in this chapter have assumed that the springs had negligi

Question

The preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely massless. To find the effect of the spring's mass, consider a spring with mass M. equilibrium length L_0, and spring constant k. When stretched or compressed to a length L. the potential energy is 1/2kx^2, where x = L - L-0 Consider a spring, as described above, that has one end fixed and the other end moving with speed v. Assume that the speed of points along the length of the spring varies linearly with distance l from the fixed end. Assume also that the mass M of the spring is distributed uniformly along the length of the spring. Calculate the kinetic energy of the spring in terms of M and v. Take the time derivative of the conservation of energy equation. E = 1/2mv_x^ 2 + 1/2kx^2, for a mass m moving on the end of a massless spring. By comparing your results to a_x = -omega^2x, which defines omega, find the angular frequency of oscillation. Apply the procedure of part B to obtain the angular frequency of oscillation omega of the spring considered in part A. If the effective mass M' of the spring is defined by omega = squarerootk/M', what is M' in terms of M?

Explanation / Answer

Part C )

M' = M/3 + m

since KE - 1/6 Mv^2 = 0.5 ( M/3) v^2

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