Chapter 2.6, Problem 65ES

If an object suspended from a spring is displaced vertically from its equilibrium position by a small amount and released, and if the air resistance and the mass of the spring are ignored, then the resulting oscillation of the object is called simple harmonic motion. Under appropriate conditions the displacement $y$ from equilibrium in terms of time $t$ is given by

$y=A\cos \omega t$

where $A$ is the initial displacement at time $t=0$ , and $\omega$ is a constant that depends on the mass of the object and the stiffness of the spring (see the accompanying figure). The constant $\left|A\right|$ is called the amplitude of the motion and $\omega$ the angular frequency.

(a) Show that

$\frac{d^{2}y}{d{t}^2}=-{\omega }^{2}y$

(b) The period $T$ is the time required to make one complete oscillation. Show that $T=2\pi /\omega .$

(c) The frequency $f$ of the vibration is the number of oscillations per unit time. Find f in terms of the period $T$ .

(d) Find the amplitude, period, and frequency of an object that is executing simple harmonic motion given by $y=0.6\cos 15t$ , where $t$ is in seconds and $y$ is in centimeters.