How to Use the Calculator?
To use the calculator, simply enter a value in either the Frequency or Period field, and the other field will update automatically. You can choose the unit you want from the dropdown next to each field, and if you change the unit, the value will adjust to match.
Introduction to Frequency and Period
When we look around in daily life, many things repeat again and again in a regular way. A swinging pendulum moves to and fro. A fan keeps rotating. The Earth revolves around the Sun every year.
Whenever something repeats its motion or action after equal intervals of time, we say it is showing a Periodic Motion.
To understand such repeating events properly, two very important ideas are used in physics: Frequency and Period. These two ideas help us describe how fast something repeats and how much time one complete repetition takes.
Let us begin with a simple thought. Imagine a swing in a park. Suppose one child moves from one side to the other side and then comes back to the starting point. That full back-and-forth motion is called One Complete Oscillation.
Now suppose this complete oscillation takes 2 seconds. That 2 seconds is the time taken for one full cycle. In physics, the time taken for one complete cycle or one complete oscillation is called the Period. So period tells us the duration of one repetition. If the swing is moving slowly, the period will be larger. If the swing is moving faster, the period will be smaller.
Now think in another way. Instead of asking, “How much time does one oscillation take?”, we may ask, “How many oscillations happen in one second?” This gives us the idea of Frequency.
Frequency tells us the number of complete cycles, vibrations, oscillations, or rotations completed in one second. So if a source vibrates 5 times in one second, its frequency is 5. If it vibrates 100 times in one second, its frequency is 100.
Relationship Between Frequency and Period
So, period and frequency describe the same repeating motion, but from two different viewpoints. Period focuses on the time for one cycle. Frequency focuses on the number of cycles in one second. Because of this, they are closely related. In fact, one is the reciprocal of the other.
$$T=\frac{1}{f}$$
$$f=\frac{1}{T}$$
Here, $T$ stands for period and $f$ stands for frequency.
These formulas are extremely important. They show that when the frequency increases, the period decreases. This makes sense. If more cycles are happening every second, then each cycle must be taking less time. Similarly, if each cycle takes more time, then fewer cycles can happen in one second.
Units of Frequency and Period
Let us understand the units now. The unit of period is second, written as $s$, because period is a measure of time. The unit of frequency is hertz, written as $Hz$. One hertz means one complete cycle per second.
$$1\ \text{Hz} = 1\ \text{cycle per second}$$
For example, if a tuning fork vibrates 256 times every second, its frequency is $256\ \text{Hz}$. If a pendulum completes one oscillation in 2 seconds, its period is $2\ \text{s}$.
Examples
Let us try several examples.
Example 1: Finding frequency from period
Suppose the period of a pendulum is $4\ \text{s}$. This means one complete oscillation takes 4 seconds. To find frequency, we use the relation:
$$f=\frac{1}{T}$$
Substituting $T=4\ \text{s}$:
$$f=\frac{1}{4}=0.25\ \text{Hz}$$
So the frequency is $0.25\ \text{Hz}$. This means the pendulum completes only one-fourth of a cycle in one second, or one full cycle in 4 seconds. This is a slow oscillation, so it makes sense that the frequency is small.
Example 2: Finding period from frequency
Suppose a machine vibrates with a frequency of $50\ \text{Hz}$. This means it completes 50 vibrations every second. To find the period, we use:
$$T=\frac{1}{f}$$
Substituting $f=50\ \text{Hz}$:
$$T=\frac{1}{50}=0.02\ \text{s}$$
So the period is $0.02\ \text{s}$. This means one vibration takes only 0.02 second. Since the machine is vibrating many times in one second, the time for one vibration must be very small.
Example 3: Using number of oscillations and time
Sometimes questions do not directly give the period or the frequency. Instead, they tell you how many oscillations happen in a certain time. In that case, frequency can be found by dividing the number of oscillations by the total time.
$$f=\frac{\text{number of oscillations}}{\text{total time}}$$
Suppose a pendulum completes 20 oscillations in 40 seconds. Then:
$$f=\frac{20}{40}=0.5\ \text{Hz}$$
Now we can find the period:
$$T=\frac{1}{f}=\frac{1}{0.5}=2\ \text{s}$$
So the pendulum has frequency $0.5\ \text{Hz}$ and period $2\ \text{s}$. This means it completes half an oscillation in one second, or one full oscillation in 2 seconds.
This method is very useful in practical observations. In real experiments, it is often hard to measure the time for just one oscillation accurately, especially if the motion is very quick. So you can measure the time for many oscillations and then divide. For example, if 50 oscillations take 100 seconds, the period is:
$$T=\frac{\text{total time}}{\text{number of oscillations}}$$
$$T=\frac{100}{50}=2\ \text{s}$$
Then the frequency becomes:
$$f=\frac{1}{2}=0.5\ \text{Hz}$$
