Frequency Formula: Equations and Solved Examples

Frequency is the number of complete cycles or oscillations that occur in one second. The frequency formula is used to calculate how frequently a repeating event occurs over a given time interval. It can be determined using the time period, wave speed and wavelength, or angular frequency.

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Frequency Formula

The frequency of a periodic event is calculated by dividing the total number of cycles by the time taken to complete them.

The general frequency formula is:

\boxed{f=\frac{N}{t}}

Where:

$f$ = Frequency (Hz)
$N$ = Number of complete cycles
$t$ = Total time (s)

For a single complete cycle, the frequency is the reciprocal of the time period.

1. Frequency Formula Using Time Period

The relationship between frequency and time period is expressed as:

\boxed{f=\frac{1}{T}}

Where:

$f$ = Frequency (Hz)
$T$ = Time period (s)

The time period represents the time required to complete one full cycle or oscillation.

Frequency and time period are inversely proportional. Therefore, an increase in the time period results in a decrease in frequency, while a shorter time period produces a higher frequency.

The time period can also be calculated using:

\boxed{T=\frac{1}{f}}

2. Frequency Formula Using Wave Speed and Wavelength

The frequency of a travelling wave can be calculated from its propagation speed and wavelength.

The relationship is given by:

\boxed{f=\frac{v}{\lambda}}

Where:

$f$ = Frequency (Hz)
$v$ = Wave propagation speed (m/s)
$\lambda$ = Wavelength (m)

This equation is derived from the wave velocity relationship:

v=f\lambda

For a constant wave speed, frequency is inversely proportional to wavelength. A shorter wavelength corresponds to a higher frequency, whereas a longer wavelength corresponds to a lower frequency.

3. Frequency Formula Using Angular Frequency

Frequency and angular frequency are related through the following equation:

\boxed{f=\frac{\omega}{2\pi}}

Where:

$f$ = Frequency (Hz)
$\omega$ = Angular frequency (rad/s)
$\pi$ = Mathematical constant (approximately 3.1416)

Angular frequency represents the rate of change of phase angle with respect to time.

The angular frequency can also be calculated as:

\boxed{\omega=2\pi f}

One complete cycle corresponds to an angular displacement of $2\pi$ radians. Therefore, angular frequency is $2\pi$ times the frequency.

Solved Examples Using Frequency Formula

The following examples explain how to calculate frequency using different given quantities.

Example 1: Calculate Frequency Using Time Period

Problem: A vibrating object completes one full cycle in 2 seconds. Calculate its frequency.

Given:

Time period, (T=2,s)

Formula:

f=\frac{1}{T}

Substituting the values:

f=\frac{1}{2}

\boxed{f=0.5,Hz}

Answer: The frequency of the vibrating object is 0.5 Hz.

Example 2: Calculate the Frequency of a Light Wave

Problem: A light wave has a wavelength of 500 nm. Calculate its frequency. Take the speed of light as $(3\times10^8,m/s).$

Given:

Wavelength, $(\lambda=500,nm)$

Speed of light, $(v=3\times10^8,m/s)$

First, convert the wavelength into metres:

\lambda=500\times10^{-9}

\lambda=5\times10^{-7},m

Formula:

f=\frac{v}{\lambda}

Substituting the values:

f=\frac{3\times10^8}{5\times10^{-7}}

\boxed{f=6\times10^{14},Hz}

Answer: The frequency of the light wave is $6\times10^{14} Hz.$

Example 3: Calculate the Frequency of a Pendulum

Problem: A pendulum takes 8 seconds to complete one full oscillation. Determine its frequency.

Given:

Time period, (T=8,s)

Formula:

f=\frac{1}{T}

Substituting:

f=\frac{1}{8}

\boxed{f=0.125,Hz}

Answer: The frequency of the pendulum is 0.125 Hz.

Example 4: Calculate Angular Frequency from Frequency

Problem: An alternating signal has a frequency of 10 Hz. Calculate its angular frequency.

Given:

Frequency, (f=10,Hz)

Formula:

\omega=2\pi f

Substituting:

\omega=2\times\pi\times10

\omega=20\pi

\boxed{\omega=62.83,rad/s}

Answer: The angular frequency is approximately 62.83 rad/s.

Example 5: Calculate Time Period from Frequency

Problem: A pendulum completes 20 oscillations per second. Calculate the time required to complete one oscillation.

Given:

Frequency, (f=20,Hz)

Formula:

T=\frac{1}{f}

Substituting:

T=\frac{1}{20}

\boxed{T=0.05,s}

Answer: The pendulum takes 0.05 seconds to complete one oscillation.

Conclusion

The frequency formula is used to determine the number of complete cycles occurring per second. Depending on the available quantities, frequency can be calculated using the time period, wave speed and wavelength, or angular frequency.

The three important equations are:

f=\frac{1}{T}

f=\frac{v}{\lambda}

f=\frac{\omega}{2\pi}

These relationships are widely used in wave motion, oscillations, and periodic signal calculations.

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