Simple Pendulum Calculator

Understanding Simple Pendulum Calculator

A simple pendulum consists of a mass on a string swinging under gravity. For small angles, the period depends only on length and gravity, not mass.

How to use this calculator

1. Enter the pendulum length, usually in meters, into the length field.
2. Enter the gravitational acceleration, usually in \(\mathrm{m/s^2}\), or use the default if the calculator provides one.
3. Click calculate to find the period \(T\).
4. Read the result and the step-by-step explanation to check how the value was found.

The formula explained

The formula \(T = 2\pi\sqrt{\frac{L}{g}}\) computes the period of a simple pendulum, which is the time for one full back-and-forth swing. It shows that longer pendulums take more time to swing, while stronger gravity makes the period shorter.

• T = period of the pendulum, in seconds
• L = length of the pendulum, in meters
• g = acceleration due to gravity, in \(\mathrm{m/s^2}\)
• \(\pi\) = the constant pi, about 3.14159

Step by step method

1. Start with the formula \(T = 2\pi\sqrt{\frac{L}{g}}\).
2. Divide the length \(L\) by gravity \(g\), then take the square root of the result.
3. Multiply by \(2\pi\) to get the period \(T\).

Worked example

Problem. Find the period of a simple pendulum with length \(L = 0.80\) m on Earth, where \(g = 9.8\,\mathrm{m/s^2}\).

1. Substitute the values into the formula: \(T = 2\pi\sqrt{\frac{0.80}{9.8}}\).
2. Compute the fraction: \(\frac{0.80}{9.8} \approx 0.08163\). Then take the square root: \(\sqrt{0.08163} \approx 0.2858\).
3. Multiply by \(2\pi\): \(T \approx 2\pi(0.2858) \approx 1.80\) s.

Tips and common mistakes

• Make sure the length is measured from the pivot point to the center of the bob, not just the string length if the bob has size.
• This formula works best for small swings, because very large swing angles can make the period slightly different.

Frequently asked questions