Chapter 6.6, Problem 10E

To determine

To find: the equation for the cosine function that represents the movement of leaf in relationship to the equilibrium point.

Answer to Problem 10E

  $y=2\cos \left(0.2\pi \right)t$

Explanation of Solution

Given information:

Distance between the highest and lowest point = 4 cm

Time period for 1 complete cycle = 10 seconds

Calculation:

The general equation for cosine function is as follows:

  $y=A\cos k\theta$

Here, $A$ is coefficient of absolute value of amplitude and $k$ is absolute value of coefficient of $\theta$ .

The value of $A$ can be evaluated as follows:

  $\begin{array}{c}\text{Amplitude}=\frac{1}{2}\left(\text{Highest point}-\text{Lowest point}\right)\\ \left|\pm A\right|=\frac{4}{2}\\ A=\pm 2\end{array}$

The value of $k$ can be evaluated as follows:

  $\begin{array}{c}\text{Period}=\frac{2\pi }{k}\\ 10=\frac{2\pi }{k}\\ k=0.2\pi \end{array}$

Substitute the value of $A\text{ and }k$ in the general equation of cosine as shown below.

  $\begin{array}{l}y=A\cos \theta \\ y=2\cos \left(0.2\pi \right)t\left[\theta =t\right]\end{array}$

Thus, the equation is $y=2\cos \left(0.2\pi \right)t$ .