Introduction
You have learned about the frequency velocity of an object and its different motions. We do have an idea of the frequency. Whenever an object rotates along its axis, it makes an angle. For example, take a string and tie a rubber ball at its one end and make the ball swing in a circle on its axis. (just imagine that the string doesn’t exist and the ball is rotating in a circle, making an angle along its imaginary axis) You will notice that it moves at a certain speed completing 360°. This is the situation when we need angular frequency for computation. Let us do a detailed analysis of Angular Frequency.
What is Angular Frequency?
Before understanding angular frequency, let us go through the meaning of time period and angular displacement.
The time period is the time taken by an object to complete one oscillation.
Angular displacement is the shortest angle made by an object from the first position to the second position of stop while rotating in a circle.
Definition of Angular Frequency – It is the angular displacement of an element of a wave per unit of time or in other words, you can say that it is the rate at which the change in rotation takes place or the rate at which change in the sinusoidal waves occurs. For example, if you say that the object has a high angular frequency, it turns very speedily. Angular frequency is the magnitude of the angular velocity. Thus, it is the scalar quantity, i.e. it does not have direction. Angular frequency helps find the rate of rotation of a body in periodic motion.
Different names- Angular speed, radial frequency, circular frequency, orbital frequency, radian frequency and pulsatance.
Derivation of Formula
Imagine that a rubber attached to a string makes a complete circle, i.e. 2π in a time T. We know that angular frequency is the radian of the angle through which the body moves per unit time.
Thus we can write angular frequency as,
ω =2π /T, where ω is the angular frequency.
Also, we know that frequency is 1/T. Thus, angular frequency in terms of frequency (f) can be written as,
ω =2πf
Unit
The SI unit of angular frequency in radian per second.
Examples of Angular Frequency
- Circular Motion
For an orbiting object, there exists a relation between the distance from the axis ‘r’, tangential velocity ‘v’ and angular frequency of rotation. The rotating body travels a distance of ‘vT’ in a time period ‘T’. The distance travelled by the body is equal to the circumference of a circle which we know is 2πr. Thus we get,
ω= v/r.
- Electricity Supply.
In a power supply station, generators rotate at a particular rate. This rotation rate is termed angular frequency, which decides the frequency at which the electricity is produced. If the power supply has to be slowed down, the angular frequency of the generators has to be reduced.
- Satellites
If the satellite wants to escape the earth’s gravity, then the force exerted by the satellite has to be equal to the gravitational force of the earth, i.e.
GMm /r2 =mω2r, where M is the mass of the larger body and G is the gravitational constant.
To make a circle, the angular frequency must be equal to,
ω circle=√GM/√r2
Solved Examples
- Calculate the angular frequency of an object rotating with a time period of 1 min.
Solution:
Given- T= 1min= 60s
Formula- ω =2π/T
By applying the formula, ω=2*180/60 = 360/60= 60rad/s or 600 /sec
Therefore, angular frequency is 600 /sec.
- Calculate the time taken by an object to complete the rotation if the angular frequency is 900/sec.
Solution:
Given: ω=900/sec
Formula – T =2π/ ω
Thus, by applying the formula, we get
T= 2*180/90= 3600/900= 40sec.
Thus, the time taken by an object to complete the rotation is 40 seconds.
Conclusion
Thus, we can conclude that angular frequency exists when an object is following a circular path or if it is oscillating. Angular frequency is the rate of angular displacement change, so its unit is radians per second. Its unit depends on the problem taken. Suppose the problem is related to the merry-go-round. In that case, we will take the angular frequency in radians per second, while if we are talking about the rotation of the moon around the earth, then the angular frequency will be in radians per day. It is useful to communicate the time period of the rotating object.