KAHIBARO
Table of Contents
Measuring Repetition in Oscillations
When motion repeats itself over and over in the same way, two of the most important quantities are the period and the frequency. These tell us how quickly the motion repeats.
The period is the time required for one complete cycle of motion. A cycle means one full repeat of the motion, returning to the same position and moving in the same direction as before. The symbol for period is usually $T$.
If a mass on a spring moves from the center to the right, back through the center to the left, and then returns to the center moving right again, that is one complete cycle. The time for that cycle is the period.
The frequency is the number of complete cycles that occur in one second. The symbol for frequency is $f$.
If an oscillator completes 3 cycles every second, then its frequency is $3 \, \text{Hz}$. The unit hertz, written as Hz, means cycles per second.
Relationship Between Period and Frequency
Period and frequency describe the same repetition in two different ways. If the motion takes a long time for one cycle, then few cycles happen each second. If the motion has a short period, then many cycles happen each second.
Their relationship is
$$
f = \frac{1}{T}
$$
and equivalently,
$$
T = \frac{1}{f}
$$
Important relationship:
$$
f = \frac{1}{T}, \qquad T = \frac{1}{f}
$$
where $T$ is measured in seconds and $f$ is measured in hertz.
Units
The period is a time quantity, so its SI unit is the second, $\text{s}$.
The frequency is measured in hertz:
$$
1 \, \text{Hz} = 1 \, \text{s}^{-1}
$$
This means that frequency has the dimensions of inverse time.
| Quantity | Symbol | Meaning | SI Unit |
|---|---|---|---|
| Period | $T$ | Time for one cycle | $\text{s}$ |
| Frequency | $f$ | Cycles per second | $\text{Hz}$ |
| Hertz | Hz | $1/\text{second}$ | $\text{s}^{-1}$ |
Counting Cycles
A practical way to find the frequency or period is to count how many oscillations happen in a certain time.
If $N$ cycles occur in a total time $\Delta t$, then
$$
f = \frac{N}{\Delta t}
$$
and therefore
$$
T = \frac{\Delta t}{N}
$$
This is very useful in experiments, especially when one cycle is too short to measure accurately by itself.
If $N$ complete oscillations take time $\Delta t$:
$$
f = \frac{N}{\Delta t}, \qquad T = \frac{\Delta t}{N}
$$
Always count complete cycles.
Simple Examples
Suppose one oscillation takes $2 \, \text{s}$. Then
$$
T = 2 \, \text{s}
$$
and
$$
f = \frac{1}{2} = 0.5 \, \text{Hz}
$$
Now suppose an oscillator has frequency $5 \, \text{Hz}$. This means it completes 5 cycles each second, so its period is
$$
T = \frac{1}{5} = 0.20 \, \text{s}
$$
If 20 oscillations take 10 seconds, then
$$
f = \frac{20}{10} = 2 \, \text{Hz}
$$
and
$$
T = \frac{10}{20} = 0.5 \, \text{s}
$$
Reading Period from Motion
The period is not just the time to go from one side to the other. In oscillatory motion, going from one extreme position to the opposite extreme is only half a cycle. Returning back to the original extreme completes the full cycle.
For simple harmonic motion, some useful fractions of a cycle are:
| Part of motion | Time taken |
|---|---|
| One full cycle | $T$ |
| From center to one extreme | $T/4$ |
| From one extreme to the opposite extreme | $T/2$ |
| From one extreme back to the same extreme | $T$ |
This helps when interpreting graphs or watching motion directly.
Visualizing One Cycle
In this sketch, the motion at time $t=0$ repeats at time $t=T$. That is why $T$ is called the period.
Common Mistakes
A common mistake is to confuse frequency with angular frequency. Angular frequency belongs to a separate topic, so here it is enough to note that frequency $f$ counts cycles per second, not radians per second.
Another common mistake is to call half a cycle the period. The period must always refer to one complete repetition of the motion.
The period is the time for one complete cycle, not half a cycle.
The frequency is the number of complete cycles per second.
Physical Meaning
Period and frequency tell us how fast an oscillating system repeats its motion. A large frequency means rapid oscillation. A large period means slow oscillation.
For example, a vibrating guitar string can have a high frequency, meaning many oscillations each second. A slowly swinging pendulum has a lower frequency and a longer period.
These two quantities are basic tools for describing all periodic motion.