Chapter 10.1, Problem 30PS

Solution

To find: The period and frequency of given function.

As $z^{\text{ο}}$ equals $-{7.76}^{\text{ο}}$ , therefore, the option ( C ) is correct.

Given data: The function representing pressure wave produced by tuning fork is modelled as, $P\left(t\right)=0.001\sin 880t$ .

  

Method/Formula used:

A periodic function $P\left(t\right)$ with time period T , frequency f , and amplitude A is given by,

  $P\left(t\right)=A\sin \frac{2\pi t}{T}$ … (1)

Here, $f=1/T$ .

Calculations:

The function representing pressure wave produced by tuning fork is modelled is

  $P\left(t\right)=0.001\sin 880t$ … (2)

Comparing results (1) and (2),

  $\begin{array}{c}\frac{2\pi t}{T}=880t\\ T=\frac{2\pi }{880}\\ =\frac{2\cdot 22}{880\cdot 7}\\ =\frac{1}{140}\end{array}$

Thus, the period of the wave is $\frac{1}{140}$ seconds. Therefore, by the definition of frequency f is

  $\begin{array}{c}f=1/\left(1/140\right)\\ =140\end{array}$

Thus, the frequency of the wave produced by the tuning fork is 140 per second.

The graph of the function (2) is shown in Fig.1:

  

Therefore, the time period and frequency of the wave are $1/140$ and 140 /s respectively.