Pendulum Calculator

A simple pendulum consists of a mass (the “bob”) suspended from a fixed pivot by a string or rod of negligible mass. When displaced from its resting position and released, gravity causes it to swing back and forth in a regular, predictable rhythm. This regularity made pendulums the basis of clocks for over 300 years.

Period of a simple pendulum formula: T = 2π × √(L / g)

Frequency formula: f = 1 / T

What each variable means:

• T: period: the time for one complete oscillation (one full swing back and forth), in seconds
• L: length of the pendulum from pivot to center of mass of the bob, in meters
• g: gravitational acceleration = 9.81 m/s² on Earth’s surface
• f: frequency: oscillations per second (Hertz, Hz)
• π ≈ 3.14159

Worked example: A grandfather clock pendulum has a length of 0.9940 meters.

T = 2π × √(0.9940 / 9.81) = 2π × √(0.10133) = 2π × 0.31833 = 2.0000 seconds

This is why the classic grandfather clock pendulum is almost exactly 1 meter long — it produces a 2-second period (1 second each way), giving a satisfying “tick… tock” at exactly 1-second intervals.

Frequency = 1 / 2.0 = 0.5 Hz (half an oscillation per second)

Worked example 2: A child’s playground swing has a rope length of 3 meters. T = 2π × √(3 / 9.81) = 2π × 0.5528 = 3.47 seconds per full cycle

Key physics insight: The period depends only on length and gravity — not on the mass of the bob, and not on the amplitude (for small angles up to about 20°). Doubling the mass has zero effect on the period. Doubling the length increases the period by √2 ≈ 1.414×.

Gravity variation: On the Moon (g = 1.62 m/s²), the same 1-meter pendulum would have a period of T = 2π × √(1/1.62) = 4.94 seconds — nearly 2.5× slower.