Frequency, Time period and Angular frequency

What Is Rotational Motion?

The motion of any system that occurs about the axes of rotation is called the 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.


Moment of inertia is a Scalar Quantity.

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 (h).


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


Moment of inertia (D) is directly proportional to the square of the radius of gyration (h).


D  ∝ h ^ 2  


D =  m x h ….(1)


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 with the conclusion that all the oscillatory motions are periodic by nature whereas all the periodic motions are not oscillatory by nature.  


The figure1 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 is that attribute 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?

Time Period

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:


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 (K)

Time Period Formula

        Time period (K): 2 x  π   /  e

         Dimensional formula = 

         [M ^ 0  L ^0 T ^ 1]


Here, e is the angular frequency.


Where q is the radius of revolution, K is the time period and n is the frequency.

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 Formula


      Frequency (n) =  1 /  K  (measured in cycle per second or hertz)

      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. 


Let’s observe the rotational motion of the disc:


Angular Displacement

In figure.5 we can see that the disc is making a revolution goes from point A to the 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 figure.6


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

Angular Frequency Formula


      Angular frequency (af) = 1 / K =  

      e / 2 x π  = radian/second 


     Which is the inverse of the time period.


     Dimensional formula


     = [M ^0 L ^0 T ^- 1]


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.

SAQs

Q1: If the bob of a pendulum of length k is negatively charged and a positively charged metal plate is placed just below it's bob then the bob is made to oscillate. Will there be any effect on the time period of the pendulum? If yes, explain it.

Ans: Yes, because there are two opposite charges and due to the force of attraction, the effective value of ‘g’ increases, by the relationship T α √k/ g, T α  √1/ g, the time period will decrease.


Q2: A boy sitting on the swing suddenly stands up. Will there be any effect on the time period of the swing? If yes, explain it.

Ans:  Yes, because while standing up, the location of the center of the mass of that boy shifts upwards and by the relationship T α √k/ g, T α √k, which means the effective length of the pendulum increases, the time period decreases.


FAQ (Frequently Asked Questions)

Suppose you are taking a ride on ABC’s wheel, and when you are sitting at the top of the ABC’s wheel. You observed that the wheel moved 1 /5 of rotation in 12 sec. Calculate the angular frequency.

Solution: Since the formula for angular frequency is given by  f  = e / 2 x  π

  1/ 5 of revolution is made in 12 seconds


One complete cycle would take …..5 x 12 = 60 seconds.


Frequency would be 1/60 or 60  sec ^ -1.


e = 2 x  π x f = 2 x  π x 1/ 60  


e  =  π / 30 radian/second


2. The flywheel of an engine has a diameter of 60 cm and completes 6000 rotations each minute. Point A is on the rim of the wheel and point B is a point on the wheel which is 20cm from the center. Calculate the period of rotation.

Solution: Since the body rotates 6000 times each minute, which means it must rotate

6000/ 60 or 100 times per second. The time period is the reciprocal of the frequency.


We know that K = 1/ n 


= 1 / 100 


Hence, time period (K) = 0.01 seconds