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 $\frac{1}{2} k x^{2},$ where $x=L-L_{0}$ . (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 $l$ 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 $v$ . (Hint: Divide the spring into pieces of length $d l$ ; find the speed of each piece in terms of $l$ , $v,$ and $L ;$ find the mass of each piece in terms of $d l, M,$ and $L ;$ and integrate from 0 to $L$ . The result is $n o t_{2}^{1} M v^{2}$ , 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 massless spring. By comparing your resultsto Eq. $(13.8),$ which defines $\omega$ , show that the angular frequency of oscillation is $\omega=\sqrt{k} / m .$ (c) 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 $M^{\prime}$ of the spring is defined by $\omega=\sqrt{k} / M^{\prime},$ what is $M^{\prime}$ in terms of $M ?$