A spring with Mass the preceding problems in this chapter have assumed that the springs had negligible mass. But of course no spring is completely mass less. To find the effect of the spring's mass, consider a spring with mass M, equilibrium length Lo, and spring constant k. When stretched or compressed to a length L, the potential energy is kx2 , where x = L - Lo.
(a) 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 I 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 tens of M and v. (Hint: Divide the spring into pieces of length dl; find the speed of each piece in tens of I, v, and L; find the mass of each piece in tens of dl, M, and L; and integrate from 0 to L. The result is not Mv2, since not all of the spring moves with the same speed.)
(b) Take the time derivative of the conservation of energy equation, Eq. (13.21), for a mass m moving on the end of a mass less spring. By comparing your results to Eq. (13.8), which defines cu, show that the angular frequency of oscillation is cu = Vkj;
(c) Apply the procedure of part (b) to obtain the angular frequency of oscillation cu of the spring considered in part (a). If the effective mass M' of the spring is defined by", = ~, what is M' in terms of M?