Chapter 5, Problem 71RE

To determine

Find an equation of the form $y=a\cos \omega t$ that describes the variation in water level.

Answer to Problem 71RE

  $y=4\cos \frac{\pi }{6}t$ .

Explanation of Solution

Calculation:

As we know that simple harmonic motion is a form of periodic motion in which object oscilate freely about a cantral point. The sine and cosine can be used to describe simple harmonic motion.

The equation,

  $y=a\sin \omega t,y=a\cos \omega t$

The object displacement from its equilibrium point at time $t$ , where,

  $\left|a\right|=$ the maximum displacement of the object.

  $2\pi /\omega =$ Period, time required to complete one cycle.

  $\omega /2\pi =1/period$ , the number of cycles per unit time.

Now the graph shows the variation in water level as a function to time,

  

  $y-axis$ represents the change in water level in feet, from the graph we identify,

  $\begin{array}{l}\left|a\right|=4feet\\ \omega /2\pi =1/12cycles/hour\\ \omega =\frac{\pi }{6}\end{array}$

Substitute all values in standard equation,

  $y=4\cos \frac{\pi }{6}t$ .

Hence, the equation is , $y=4\cos \frac{\pi }{6}t$ .