Exercise $1.$ An $8-$pound weight is placed upon the lower end of a coil spring suspended from the ceiling. The weight comes to rest in its equilibrium position, thereby stretching the spring $6$ inches. The weight is then pulled down $3$ inches below its equilibrium position and released at $t=0$ with an initial velocity of $1$ foot per second, directed downward. Neglecting the resistance of the medium and assuming that no external forces are present, determine the amplitude, period, and frequency of the resulting motion.

Ans. The spring constant is $k=16$ pounds per foot, $m=\frac{8}{32}=\frac{1}{4}$ slug. The equation or motion is $\frac{d^{2}x}{(d)t^2}+64x=0$ with the initial conditions $(0)=\frac{1}{4},x^{\text{'}}(0)=1.$ Hence the vertical displacement is $x=x(t)=\frac{1}{8}sin8t+\frac{1}{4}sin8t$

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