Chapter 7, Problem 85GP

(III) We usually neglect the mass of a spring if it is small compared to the truss attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length $l$and mass $M_S$ uniformly distributed along the length of the spring. A mass $m$ is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7–30). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed ${\vartheta }_0$, the midpoint of the spring moves with speed ${\vartheta }_0/2$ Show that the kinetic energy of the mass plus spring when the mass is moving with velocity $\vartheta$ is

$K=\frac{1}{2}M{\vartheta }^2$

where $M=m\frac{1}{3}{M}_{\text{S}}$ is the “effective mass” of the system. [Hint: Let D be the total length of the stretched spring. Then the velocity of a mass $dm$ of a spring of length $dx$ located at $x$ is $\vartheta (x)-{\vartheta }_0(x/D)$. Note also $dm=dx(M_S/D)$.]

FIGURE 7-30 Problem 68.