Chapter 14.3, Problem 52SR

(a)

To determine

To Find: The period of the function.

Answer to Problem 52SR

The period of the string is $0.0023\text{sec}$ .

Explanation of Solution

Given:

When a guitar string is plucked, it is displaced from a fixed point in the middle of the string and vibrates back and forth, producing a musical tone. The exact tone depends on the frequency, or number of cycles per second, that the string vibrates. To produce an $A$ , the frequency is 440 cycles per second, or 440 hertz $(Hz).$

Calculation:

Consider that the guitar string vibrate at 440 Hz then,

  $\begin{array}{c}P=\frac{1}{f}\\ =\frac{1}{440}\sec \\ =0.0023\text{sec}\end{array}$

The, the period of the string is $0.0023\text{sec}$ .

(b)

To determine

To Find: The graph for the height of the fixed point on the string from its resting position as a function of time.

Answer to Problem 52SR

The required graph is shown in Figure 1

Explanation of Solution

Given:

Let the maximum distance above the resting position have a value of 1 unit, and let the minimum distance below this position have a value of 1 unit.

Calculation:

Consider the general equation of the vibrating string is,

  $y=a\sin \left(\frac{2\pi }{b}t\right)$

Here, $a$ is the amplitude and $b$ is the period then,

  $y=\sin \left(\frac{2\pi }{0.0023}\right)t$

The required graph is shown in Figure 1

  

Figure 1