Free Simple Pendulum Calculator

Understanding the Simple Pendulum

A simple pendulum consists of a point mass (the bob) attached to a massless, inextensible string or rod. When displaced from its equilibrium position, it swings back and forth under the influence of gravity, exhibiting simple harmonic motion for small displacements. During each cycle, kinetic energy converts to potential energy and vice versa.

The simple pendulum period formula gives the time required for one complete oscillation:

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

where:

• $T$ is the period (in seconds),
• $L$ is the pendulum length (in meters),
• $g$ is the acceleration due to gravity (default $9.80665\ \text{m/s}^2$ on Earth).

This relationship holds for small angular amplitudes, typically between $-15^\circ$ and $15^\circ$. Notably, the period does not depend on the mass of the bob or on the amplitude under this condition.

The pendulum frequency—the number of oscillations per second—is simply the reciprocal of the period:

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

How to Use a Pendulum Period Calculator

With this pendulum oscillation calculator, you can quickly find the period and frequency for any given length. Here's the process:

1. Measure the length of the pendulum from the suspension point to the center of the bob.
2. Choose the gravity value. For Earth, use $9.80665\ \text{m/s}^2$; you can adjust it for other planets.
3. Apply the formula: compute $T = 2\pi\sqrt{L/g}$ and $f = 1/T$.
4. Alternatively, let the calculator do the work—enter the length, and it returns both period and frequency instantly.

For example, a pendulum of length $2\ \text{m}$ on Earth yields:

$$T = 2\pi\sqrt{\frac{2}{9.80665}} \approx 2.837\ \text{s},\qquad f \approx 0.352\ \text{Hz}.$$

The same tool can act as a pendulum length calculator if you need to find the length that produces a desired period. Rearranging the formula gives:

$$L = \left(\frac{T}{2\pi}\right)^2 g$$

Beyond the Simple Pendulum: Large‑Angle Oscillations

When the angular amplitude exceeds about $15^\circ$, the small‑angle approximation breaks down, and the period becomes dependent on the amplitude and the mass distribution. For such a physical pendulum, the period equation involves the moment of inertia $I$ and the distance $D$ from the pivot to the center of mass:

$$T = 2\pi\sqrt{\frac{I}{mgD}}$$

where $m$ is the mass of the pendulum. This more complex behavior is handled by dedicated physical‑pendulum models.