Simple Pendulum Calculator

A simple pendulum is an idealized model consisting of a point mass (called a bob) suspended from a fixed pivot by a massless, inextensible string. When displaced from its equilibrium position and released, it oscillates back and forth in a regular harmonic motion. In practice, any small dense object on a string approximates a simple pendulum well.

The time period is calculated using the formula T = 2π√(L/g), where L is the length of the pendulum in meters and g is the acceleration due to gravity (9.807 m/s² on Earth). For example, a 1-meter pendulum on Earth has a period of approximately 2.006 seconds. This formula is valid for small oscillation angles (typically less than 15°).

You can rearrange the pendulum period formula to solve for g: g = 4π²L / T². Measure the length of the pendulum and time several complete oscillations, then divide by the number of swings to get the period T. Plugging L and T into the formula gives you a reliable experimental estimate of local gravitational acceleration.

Rearranging T = 2π√(L/g) gives L = g(T/2π)². Simply square the desired period, multiply by g, and divide by 4π². For instance, to achieve a period of exactly 2 seconds on Earth, the required length is about 0.993 meters — nearly 1 meter.

Using L = g(T/2π)² with T = 2 s and g = 9.807 m/s², the length works out to approximately 0.993 meters. This is the basis for the historical definition of the 'seconds pendulum', which was once proposed as a standard unit of length.

No — the mass of the bob has no effect on the period of a simple pendulum. The period depends only on the pendulum's length and the local gravitational acceleration. This is one of the key properties that makes the simple pendulum such a useful and elegant physical model.

The formula T = 2π√(L/g) is an approximation valid for small angles, generally below 15°. For larger amplitudes, the actual period becomes longer than this formula predicts, and a more complex series expansion or numerical integration is required. For everyday use and most physics problems, the small-angle approximation is perfectly sufficient.

Frequency is the number of complete oscillations per second and is the reciprocal of the period: f = 1/T = (1/2π)√(g/L), measured in Hertz (Hz). A pendulum with a 2-second period has a frequency of 0.5 Hz, meaning it completes one full swing every 2 seconds.