Chapter 5, Problem 14T

An object is moving up and down in damped harmonic motion. Its displacement at time t = 0 is 16 in.: this is its maximum displacement. The damping constant is c = 0.1, and the frequency is 12 Hz.

1. (a) Find a function that models this motion.
2. (b) Graph the function.

To determine

To find: The function for the models of this motion.

Answer to Problem 14T

The function is $y=16e^{-0.1t}\cos 24\pi t$ of this model.

Explanation of Solution

Given:

The maximum displacement is 16 in.

The damping constant is,

$c=0.1$

Frequency of the object is 12 Hz.

Formula Used:

The formula of the number of cycles per unit time is,

$\text{frequency}=\frac{\omega }{2\pi }$

Calculation:

The general equation of damping harmonic motion is,

$y=k{e}^{-ct}\cos \omega t$. (1)

Where, c is damping constant and k is initial amplitude.

The formula of the number of cycles per unit time is,

$\text{frequency}=\frac{\omega }{2\pi }$

Substitute 12 for frequency in the above formula.

$\begin{array}{l}12=\frac{\omega }{2\pi }\\ \omega =12\times 2\pi \\ \omega =24\pi \end{array}$

Substitute $24\pi$ for $\omega$, 0.1 for c and 16 for k in the equation (1).

$y=16e^{-0.1t}\cos 24\pi t$

Hence, the function is $y=16e^{-0.1t}\cos 24\pi t$ of this model.

To determine

To sketch: The graph for $y=16e^{-0.1t}\cos 24\pi t$.

Explanation of Solution

An object oscillates in ups and down in damped harmonic motion.

The graph of the cosine curve for the given amplitude is,

Figure (1)

The Figure (1) shows the graph of the function $y=16e^{-0.1t}\cos 24\pi t$.