Chapter 17, Problem 1E

Determine the fundamental frequency, fundamental radian frequency, and period of the following: (a) 5 sin 9t; (b) 200 cos 70t; (c) 4 sin(4t − 10°); (d) 4 sin(4t + 10°).

To determine

The fundamental frequency, fundamental angular frequency and period of the given function.

Answer to Problem 1E

The fundamental angular frequency of the given function is $9\text{rad}/\text{s}$, fundamental frequency is $\frac{9}{2\pi }\text{Hz}$ and the time period is $\frac{2\pi }{9}\text{s}$.

Explanation of Solution

Given data:

The given function $f\left(t\right)$ is $5\sin 9t$.

Calculation:

The general form of the function is given as,

$f\left(t\right)=A\sin \left({\omega }_{0}t+\varphi \right)$        (1)

Here,

$A$ is the constant.

${\omega }_0$ is the fundamental radian frequency.

$\varphi$ is the phase angle of the function.

Compare the given function with equation (1).

The fundamental radian frequency is,

${\omega }_0=9\text{rad}/\text{s}$

The fundamental frequency is given as,

$f_0=\frac{{\omega }_0}{2\pi }$        (2)

Substitute $9\text{rad}/\text{s}$ for ${\omega }_0$ in the above equation.

$\begin{array}{c}{f}_0=\frac{9\text{rad}/\text{s}}{2\pi }\\ =\frac{9}{2\pi }\text{Hz}\end{array}$

The time period is given as,

$T=\frac{1}{f_0}$        (3)

Substitute $\frac{9}{2\pi }\text{Hz}$ for $f_0$ in the above equation.

$\begin{array}{c}T=\frac{1}{\frac{9}{2\pi }\text{Hz}}\\ =\frac{2\pi }{9}\text{s}\end{array}$

Conclusion:

Therefore, the fundamental angular frequency of the given function is $9\text{rad}/\text{s}$, fundamental frequency is $\frac{9}{2\pi }\text{Hz}$ and the time period is $\frac{2\pi }{9}\text{s}$.

To determine

The fundamental frequency, fundamental angular frequency and period of the given function

Answer to Problem 1E

The fundamental angular frequency of the given function is $70\text{rad}/\text{s}$, fundamental frequency is $\frac{35}{\pi }\text{Hz}$ and the time period is $\frac{\pi }{35}\text{s}$.

Explanation of Solution

Given data:

The given function $f\left(t\right)$ is $200\cos 70t$.

Calculation:

Compare the given function with equation (1).

The fundamental radian frequency is,

${\omega }_0=70\text{rad}/\text{s}$

Substitute $70\text{rad}/\text{s}$ for ${\omega }_0$ in equation (2).

$\begin{array}{c}{f}_0=\frac{70\text{rad}/\text{s}}{2\pi }\\ =\frac{35}{\pi }\text{Hz}\end{array}$

Substitute $\frac{35}{\pi }\text{Hz}$ for $f_0$ in equation (3).

$\begin{array}{c}T=\frac{1}{\frac{35}{\pi }\text{Hz}}\\ =\frac{\pi }{35}\text{s}\end{array}$

Conclusion:

Therefore, the fundamental angular frequency of the given function is $70\text{rad}/\text{s}$, fundamental frequency is $\frac{35}{\pi }\text{Hz}$ and the time period is $\frac{\pi }{35}\text{s}$.

To determine

The fundamental frequency, fundamental angular frequency and period of the given function

Answer to Problem 1E

The fundamental angular frequency of the given function is $4\text{rad}/\text{s}$, fundamental frequency is $\frac{2}{\pi }\text{Hz}$ and the time period is $\frac{\pi }{2}\text{s}$.

Explanation of Solution

Given data:

The given function $f\left(t\right)$ is $4\sin \left(4t-10°\right)$.

Calculation:

Compare the given function with equation (1).

The fundamental radian frequency is,

${\omega }_0=4\text{rad}/\text{s}$

Substitute $4\text{rad}/\text{s}$ for ${\omega }_0$ in equation (2).

$\begin{array}{c}{f}_0=\frac{4\text{rad}/\text{s}}{2\pi }\\ =\frac{2}{\pi }\text{Hz}\end{array}$

Substitute $\frac{2}{\pi }\text{Hz}$ for $f_0$ in equation (3).

$\begin{array}{c}T=\frac{1}{\frac{2}{\pi }\text{Hz}}\\ =\frac{\pi }{2}\text{s}\end{array}$

Conclusion:

Therefore, the fundamental angular frequency of the given function is $4\text{rad}/\text{s}$, fundamental frequency is $\frac{2}{\pi }\text{Hz}$ and the time period is $\frac{\pi }{2}\text{s}$.

To determine

The fundamental frequency, fundamental angular frequency and period of the given function

Answer to Problem 1E

The fundamental angular frequency of the given function is $4\text{rad}/\text{s}$, fundamental frequency is $\frac{2}{\pi }\text{Hz}$ and the time period is $\frac{\pi }{2}\text{s}$.

Explanation of Solution

Given data:

The given function $f\left(t\right)$ is $4\sin \left(4t+10°\right)$.

Calculation:

Compare the given function with equation (1).

The fundamental radian frequency is,

${\omega }_0=4\text{rad}/\text{s}$

Substitute $4\text{rad}/\text{s}$ for ${\omega }_0$ in equation (2).

$\begin{array}{c}{f}_0=\frac{4\text{rad}/\text{s}}{2\pi }\\ =\frac{2}{\pi }\text{Hz}\end{array}$

Substitute $\frac{2}{\pi }\text{Hz}$ for $f_0$ in equation (3).

$\begin{array}{c}T=\frac{1}{\frac{2}{\pi }\text{Hz}}\\ =\frac{\pi }{2}\text{s}\end{array}$

Conclusion:

Therefore, the fundamental angular frequency of the given function is $4\text{rad}/\text{s}$, fundamental frequency is $\frac{2}{\pi }\text{Hz}$ and the time period is $\frac{\pi }{2}\text{s}$.