Pendulum — Period, Frequency & Simple Harmonic Motion

A pendulum is a mass that swings about a pivot under gravity. If you are trying to find the pendulum period or frequency, the key result is this: for a simple pendulum swinging through a small angle, the period is

$$T = 2\pi\sqrt{\frac{L}{g}}$$

and the frequency is

$$f = \frac{1}{T}$$

Here $L$ is the length from the pivot to the bob's center of mass, and $g$ is gravitational acceleration. This formula works for the simple pendulum model at small angles, so the condition matters.

What a simple pendulum means in physics

In the standard model, a simple pendulum has a point-like bob, a light string or rod, and a fixed pivot. Air resistance and friction are small enough to ignore over the time you are modeling.

That idealization matters because real pendulums lose energy and can drift away from the simple formula. The model is still useful because it predicts the timing of many small oscillations well.

When a pendulum is simple harmonic motion

A pendulum is not exactly simple harmonic motion for every angle. It is approximately SHM when the angular displacement $\theta$ is small enough that

$$\sin \theta \approx \theta$$

with $\theta$ measured in radians.

Under that condition, the equation of motion becomes

$$\frac{d^2\theta}{dt^2} + \frac{g}{L}\theta = 0$$

which is the standard SHM form. That is why a pendulum behaves like SHM only as an approximation for small swings.

Pendulum period and frequency formula

For a simple pendulum in the small-angle limit,

$$T = 2\pi\sqrt{\frac{L}{g}}$$

and

$$f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$$

The period is the time for one full cycle. The frequency is the number of cycles per second.

Two quick takeaways:

• A longer pendulum has a longer period, so it swings more slowly.
• A larger local value of $g$ gives a shorter period, so it swings more quickly.

In the ideal small-angle model, the period does not depend on the bob's mass.

Worked example: 1 m pendulum period and frequency

Suppose a simple pendulum has length $L = 1.00\ \mathrm{m}$ and we use $g = 9.8\ \mathrm{m/s^2}$. Assume the swing angle is small.

Start with the period formula:

$$T = 2\pi\sqrt{\frac{L}{g}}$$

Substitute the values:

$$T = 2\pi\sqrt{\frac{1.00}{9.8}}$$ $$T \approx 2\pi\sqrt{0.102}$$ $$T \approx 2.01\ \mathrm{s}$$

So one full oscillation takes about $2.01$ seconds.

Now find the frequency:

$$f = \frac{1}{T} \approx \frac{1}{2.01} \approx 0.50\ \mathrm{Hz}$$

So the pendulum completes about half a cycle per second. This is a good reference value: a $1\ \mathrm{m}$ pendulum near Earth's surface takes about $2$ seconds per cycle.

Common pendulum mistakes

Using the formula for large swings

The standard period formula is accurate only when the small-angle approximation is good. If the swing is large, the real period is longer than the simple small-angle prediction.

Measuring the wrong length

For a simple pendulum, $L$ is measured from the pivot to the bob's center of mass, not just to the top of the bob or to the end of the string by itself.

Mixing up period and frequency

Period is time per cycle. Frequency is cycles per second. They are reciprocals, so a larger period means a smaller frequency.

Assuming every oscillation is SHM

Back-and-forth motion by itself is not enough. The pendulum behaves approximately like SHM only under the small-angle condition.

Where the pendulum model is used

Pendulums are used to introduce oscillations, restoring forces, and approximation methods in physics. They also appear in timekeeping history, seismometers, and classroom experiments that show how period depends on length.

They are especially useful in teaching because one system connects several ideas at once: gravity, periodic motion, angular displacement, and SHM as an approximation.

Try solving a similar pendulum problem

Change the example to $L = 0.25\ \mathrm{m}$ and compute the new period and frequency. That one change makes it clear how strongly the timing depends on length.

If you want to check your setup after trying it yourself, GPAI Solver can walk through the same pendulum model step by step with your own numbers.