1. Understanding the Basic Quantities of Oscillatory Motion
Whenever an object oscillates, three quantities help describe its motion clearly: amplitude, time period, and frequency. These are the basic building blocks of understanding any repeated back-and-forth motion.
I like to think of these as the motion’s ‘size’, ‘duration’, and ‘speed of repetition’. Once these three are clear, almost every part of oscillatory motion becomes easier to visualize.
2. Amplitude
Definition: Amplitude is the maximum displacement of the oscillating object from its mean (equilibrium) position.
You can imagine amplitude as the ‘extent’ or ‘reach’ of the oscillation. A larger amplitude simply means the object swings or moves farther from the center.
2.1. How Amplitude Appears in Real Situations
When a swing moves higher, its amplitude increases. When you stretch a spring further before releasing it, the mass oscillates with a greater amplitude.
Amplitude does not tell you how fast the object moves — it only tells you how far it goes from the center position during the motion.
2.2. Mathematical Representation
In the equation of simple harmonic motion, the amplitude is represented by A:
\( x(t) = A \sin(\omega t + \phi) \)
Here, \( A \) determines how ‘wide’ the oscillation is.
3. Time Period
Definition: The time period is the time taken by the object to complete one full oscillation.
This means from one extreme position to the other extreme and back again. I like to think of it simply as ‘how long one cycle takes’.
3.1. Understanding Through an Example
If a pendulum swings from leftmost to rightmost and back to the leftmost position in 2 seconds, then its time period is:
\( T = 2~\text{seconds} \)
Every cycle takes exactly the same time as long as the motion stays regular.
3.2. Importance of Time Period
The time period tells us how fast or slow an oscillation is. A larger time period means the object takes more time to finish a cycle, indicating slower motion.
4. Frequency
Definition: Frequency is the number of complete oscillations made in one second.
It tells us how rapidly the cycles are repeating. A high frequency means the motion repeats more times per second.
4.1. Unit of Frequency
The unit of frequency is hertz (Hz). One hertz simply means one oscillation per second.
4.2. Relation Between Time Period and Frequency
Time period and frequency are closely connected. They are inversely proportional to each other:
\( f = \dfrac{1}{T} \)
This means if the time period increases, the frequency decreases and vice versa.
4.3. Quick Example to Visualize Frequency
If an object completes 5 oscillations in one second, its frequency is:
\( f = 5~\text{Hz} \)
This simply means it repeats the motion 5 times every second.
5. Putting Amplitude, Time Period and Frequency Together
These three quantities describe an oscillation completely:
- Amplitude tells how far it moves.
- Time period tells how long one cycle takes.
- Frequency tells how many cycles occur in one second.
Whenever I look at an oscillating system, I mentally check these three values. They help me predict how the motion will behave — whether it’s slow or fast, wide or narrow, or repeating quickly or slowly.