What is a Periodic Function?
One can understand periodic function meaning as the motion that occurs repetitively over the course of fixed time intervals. Periodic function examples include rocking a chair, which is a circular motion. In other words, one can also define a periodic function as the motion that returns to its initial position after a fixed duration of time.
Understanding the Difference Between Periodic and Oscillatory Motion
After going through the periodic function definition, one can easily get confused with oscillatory motion at first glance. But, not all periodic functions are oscillatory at the same time. One of the biggest differences between the two is that, while periodic motions can be repetitive at times, oscillatory motion is only constrained at an equilibrium point.
For better understanding, one can take the example of a bob of a pendulum. It oscillates along its equilibrium position in a periodic manner. During its movement, the displacement takes place from zero to positive to negative passing through its initial position. Such a motion is periodic and oscillatory at the same time. Another aspect of the oscillating motion is Simple Harmonic Motion (SHM), where the restoring force of the periodic motion is directly proportional to that of its displacement.
The Formula for Periodic Function
One can define the periodic function f, along with a non-zero constant in the same case:
f (x+P) = f (x)
The function is applicable for all the values of x in the same domain. While the constant P is termed as the period of a function.
Derivation of Periodic Function Equation
For an oscillating object, its periodic function can be defined as:
f(t) = Acosωt
With the cosine part repeating itself after a certain point of time, it can defined as:
cosθ = cos(θ+2π)
⇒ cos(ωt) = cos(ωt+2π) ——(1)
Considering the time period to be T:
f(t) = f(t+T)
⇒ Acosωt = Acosω(t+T)
⇒ Acosωt = Acos(ωt + ωT) ——(2)
So from equation 1 and 2, we can derive:
ωT = 2π
Thus, T = 2πω
Time Period of Periodic Function
The time period of the periodic function is given by;
T = 2πω, ω is the angular frequency of the oscillating object.
Frequency of Periodic Function
We all know that the frequency is given by the total number of oscillations per unit time. For periodic motion, frequency is given by;
F = 1/T;
F = 1/ (2πω);
Solved Questions
Example 1
State whether a motion can be periodic but not oscillatory or not?
a) True
b) False
Solution
The answer is option A. You can often find motions that are periodic but not oscillatory. For example, a uniform circular motion is a periodic motion, but there is no restoring force being applied on it. So, it is not an oscillatory motion.
Example 2
For a given pendulum, if l is the length of the bob, while its mass is m, and it is moving along the circular arc with angle θ. So if a spherical mass M is placed at the end of the circle, what is the momentum of the sphere gained by the moving bob?
a) Infinity
b) Zero
c) Constant
d) Unity
Solution
The answer is B. The sphere will not attain any momentum through the bob at the end of the circle. This is because, at the end of the circle, the velocity of the bob becomes zero.
Example 3
Let us assume that a 2 kg body is suspended from a stretchable spring. So, if someone pulls down the spring, it is released with an oscillating motion vertically. What is the name of the force that is applied to the body, when the spring passes through its mean position?
a) Force equal to the pull
b) Force equal to the weight of the body
c) Force equal to gravity
d) Conservative force
Solution
The answer is B. It is imperative to understand that at the mean position, the total acceleration of the body is zero. So, the resultant force that is applied by the spring is the same as that of the weight of the body.
FAQs on Periodic Function
1. What is a periodic function in the context of Physics?
A function in Physics is described as a periodic function if it repeats its values at regular intervals or periods. Any motion that repeats itself over a fixed duration of time is considered periodic. Mathematically, a function f(t) is periodic if there exists a non-zero constant T (the time period) such that f(t + T) = f(t) for all values of t. The smallest positive value of T for which this is true is called the fundamental period.
2. What are some common real-life examples of periodic functions or motion?
Many phenomena in our daily lives can be described by periodic functions. Some common examples include:
- The motion of the hands on a clock, which repeat their positions every 12 hours, 60 minutes, or 60 seconds.
- The revolution of the Earth around the Sun, which completes its path in approximately 365.25 days.
- The swinging motion of a child on a swing, returning to the same point periodically.
- The vibration of a guitar string after being plucked, which produces a sound wave with a repeating pattern.
- A uniform circular motion, like a point on a spinning fan blade.
3. What is the fundamental difference between periodic motion and oscillatory motion?
While all oscillatory motions are periodic, not all periodic motions are oscillatory. The key difference lies in the presence of an equilibrium position. Oscillatory motion is the to-and-fro movement of an object about a stable equilibrium point, driven by a restoring force (e.g., a pendulum). In contrast, periodic motion simply needs to repeat over time; it does not require an equilibrium position or a restoring force. For instance, the uniform circular motion of a planet is periodic but not oscillatory.
4. How is Simple Harmonic Motion (SHM) related to a periodic function?
Simple Harmonic Motion (SHM) is a special and very important type of oscillatory motion, and therefore, it is also periodic. What makes SHM unique is that the restoring force acting on the object is directly proportional to its displacement from the mean position and is always directed towards it. This specific condition (F = -kx) leads to motion described by sinusoidal periodic functions, such as sine and cosine. So, SHM is a subset of periodic motion.
5. What is the general formula for the time period of a periodic function like Acos(ωt)?
For a periodic function describing simple harmonic motion, such as f(t) = Acos(ωt), the time period (T) is the time taken to complete one full oscillation. It is calculated using the formula: T = 2π / ω. Here, ω (omega) represents the angular frequency of the oscillation, which measures how rapidly the oscillations occur in radians per second.
6. How can you determine if a given mathematical function is periodic?
To determine if a function, say f(x), is periodic, you must check if there is a non-zero constant 'P' (the period) that you can add to the variable 'x' without changing the function's value. You need to test if the condition f(x + P) = f(x) holds true for all x in the function's domain. For trigonometric functions, we use known periods; for example, sin(x) and cos(x) have a period of 2π, while tan(x) has a period of π. For combinations of functions, you often need to find the Least Common Multiple (LCM) of the individual periods.
7. What does the graph of a simple periodic function typically look like?
The graph of a simple periodic function, like a sine or cosine wave, has a characteristic wave-like shape that repeats itself continuously along the horizontal axis. This repeating segment is called a cycle. Key features of the graph include the amplitude (the maximum displacement or height from the centreline) and the period (the horizontal length of one complete cycle). The graph oscillates smoothly between its maximum and minimum values.
8. Why is angular frequency (ω) an important concept for describing periodic functions?
Angular frequency (ω) is crucial because it provides a direct link between circular motion and oscillatory motion, which are fundamental types of periodic phenomena. It represents the rate of change of the phase of the waveform, measured in radians per second. It simplifies the mathematical representation of periodic functions (e.g., y = A sin(ωt)) and directly relates the time period (T) and frequency (f) of the motion through the simple equations ω = 2π/T and ω = 2πf. This makes it a highly efficient parameter for calculations in waves and oscillations.
9. Can complex repeating sounds or signals be described using simple periodic functions?
Yes, even very complex periodic waveforms, such as the sound from a musical instrument or a human voice, can be described using simple periodic functions. According to a principle known as the Fourier theorem, any complex periodic function can be broken down into a sum of simple sine and cosine functions, each with different amplitudes and frequencies. This powerful concept is fundamental to signal processing and acoustics, allowing us to analyse and reconstruct complex waves from basic components.
