Frequency, Time Period and Angular Frequency

The motion of any system that occurs about the axes of rotation is called rotational motion, and the system has a moment of inertia that tries to oppose the change in its motion.

On this page, we will learn about the following:

• What is the time period?

• Time period formula

• What is the frequency?

• Frequency formula

• What is the angular frequency?

• Angular frequency formula

What is the Moment of Inertia?

It states that when a body rotates around its axis continues to rotate until or unless an external force is applied to it. Such type inertia is called the inertia of rotational motion or moment of inertia.

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You can observe in the above figure that the distance between the axis of rotation and the point at which the mass of the body is supposed to be concentrated at the center is called the radius of gyration.

So the formula of the moment of inertia is given by:

\[ P = \displaystyle\sum\limits_{i=1}^N m_{i} r_i^2\]

Moment of inertia  is directly proportional to the square of the radius of gyration .

\[D \alpha h^2\]

\[D= m\times h\]

We can see that the radius of gyration is varying with the different axis which we can see in the image above.

What is the Application of Inertia of Rotational Motion?

We can consider some real-life examples to explain this phenomenon.

Hands of the wall clock, Spinning the stone attached to the string, ceiling fan, rotating tires of the moving bicycle, the pendulum. These are having a rotational motion, and they continue to move or stay at rest. That’s why they have rotational inertia.

We can see that the hands of a wall clock after completing one revolution or a period repeats its motion and continues to do the motion periodically.

Oscillatory Motion: When an object makes a to and fro motion periodically then the object is said to be in oscillatory motion. Just like the pendulum oscillates from one end to the other one (to-and-fro) about its mean position and keeps on repeating its motion is having an oscillating motion.

Now, what is the basic difference between the motions of the wall clock and the pendulum?

Well, the wall clock has a periodic motion, which means after completing one cycle, and keeps on doing it continuously, but it doesn’t oscillate about its mean position whereas the pendulum oscillates about the mean position, and after completing its one period of to and fro motion repeats its motion. 

Here, via these two examples, we came to the conclusion that all the oscillatory motions are periodic by nature whereas all the periodic motions are not oscillatory by nature.  

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Figure 1 (above) of sine waves represents the periodic motion of the objects. The sine wave also represents the simple harmonic motion of any object.

The simple harmonic motion of any object indicates that the motion of an object is continuous by nature just like the sine wave, and it repeats its motion at a fixed duration or after completing its one cycle (T) as shown in figure 1.

Here, the positive half cycle of the wave is called the crest, and the negative half cycle or the lower part is called the trough.

What Attributes are Common among all these Objects besides being in Rotational Motion?

Well, all these have the following attributes: 

• Time Period 

• Frequency

• Angular Frequency

You might be new to all these terminologies, right?

As we all know, many forms of energy, such as light and sound, travel in waves. A wave's frequency, amplitude, and speed are all characteristics that describe it. In wave mechanics, parameters such as frequency, time period, wavelength, amplitude, and so on are assigned to each wave. This page goes into great detail about frequency, time period, and angular frequency.

Frequency is defined as the number of complete cycles of waves passing a point in a given amount of time. The time period refers to how long it takes for a whole wave cycle to pass through a given location. The angular displacement of any wave constituent per unit time is defined as the angular frequency.

Time Period

A time period (denoted by the letter 'T') is the amount of time it takes for one complete cycle of vibration to pass through a given point. The time period of a wave decreases as its frequency increases. The time period is measured in 'seconds.'

Frequency and time period have a mathematical relationship that can be expressed as T = 1/f or f = 1/T.

• The orbital period of a pendulum is the amount of time it takes for one object to revolve around another. The period is the amount of time it takes for a pendulum to swing from one side to the other and return. 

• T=2m/qB is the period of an electron in a spiral orbit with the axis parallel to the magnetic field, where m is the electron's mass, q is its charge, and B is the magnetic field in the region, where m is the electron's mass, q is its charge, and B is the magnetic field in the region, where m is the electron's mass, q is its charge, and B is the magnetic field in the region.

Let’s say, we have a wall clock with the second’s, minute’s and the hour’s hands all coinciding at 12.

The angle made after one revolution is 360° or 2 x π  (π  =180°)

We can represent one cycle of a seconds hand by a sine wave:

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The sine wave (figure.3)  represents one revolution of the second’s hand in a given period of time and continues to make revolutions further.

Now, you can see that the time taken to complete such a revolution is called the time period

The time period formula is given as:

\[T = \frac{2\pi}{\omega}\]

Dimensional Formula =  \[M^{0}L^{0}T^{1}\]

Frequency

In the chapter of statistics, we observe that the frequency of the data of any given estimation is the mode which means the data which have high frequency is called the mode of that data. 

So the data which is repeated in a fixed period of time is called the frequency.

Similarly, rotational motion frequency enumerates the number of complete cycles or oscillations in a given period of time.

Frequency is defined as the number of times a repetitive event occurs per unit of time. Ordinary frequency, rather than angular frequency, and temporal frequency, rather than spatial frequency, are other names for it. One hertz (Hz) unit of frequency equals one occurrence per second.

Because it is the duration of one cycle of a repeated occurrence, the period is the reciprocal of the frequency.

A heart beats 120 times per minute (2 hertz), for example, and its period is half a second (60 seconds divided by 120 beats). Frequency is the rate at which oscillatory and vibratory phenomena, such as mechanical vibrations, sound, radio waves, and light, occur in Science and Engineering.

The formula of frequency is:

\[f = \frac{1}{T} = \frac{\omega}{2\pi}\]   

Dimensional Formula =  \[M^{0}L^{0}T^{-1}\]

Angular Frequency

It is known as radial or circular frequency which measures the angular displacement per unit time. 

The scalar measure of rotation rate is the angular frequency (also called angular speed, radial frequency, circular frequency, orbital frequency, radian frequency, and pulsatance in physics). It is the rate at which the phase or angular displacement of a sinusoidal waveform changes per unit time (for example, in rotation). 

(For example, in oscillations and waves), or as the sine function's input rate of change. The vector quantity's magnitude the angular frequency is the angular velocity (or angular speed).

Because, one revolution is equal to 2π radians, therefore,

\[\omega = \frac{2\pi}{T} = 2\pi f\]

Dimensional Formula =  \[M^{0}L^{0}T^{-1}\]

Here, 

• ω is the angular frequency. It measured in radians per second

• T is the period. It is measured in seconds

• f is the ordinary frequency. It is measured in hertz. It is sometimes denoted as ν).

Even though it does not convey a rotational value, angular frequency is usually expressed in radians per second in SI units.

Angular Displacement

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In figure.5 we can see that the disc is making a revolution that goes from point A to point B by sweeping an angle Θ where Θ is the angular displacement made by the disc, the motion of the same is represented by a sine wave as shown in the figure.6

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We learned that frequency is defined as the time period at which a clock completes one cycle is its frequency. This was the basic term.

Summary

When a body or an object free is set free to rotate about a given axis can make angular oscillations. For example, a pendulum in the wall clock consists of a heavy particle suspended from fixed support via a light inextensible string when given an external force to it, it starts oscillating making a certain angle Θ.

Where Θ is the angle between the string and the vertical that describes the position of the pendulum which remains the same on either side of the mean position.

Moment of inertia D plays the same role in the rotational motion as mass p plays in the linear motion.