Pendulum Calculator

Understanding Simple Pendulums

What is a Simple Pendulum?

A simple pendulum consists of a mass (bob) suspended from a fixed point by a string or rod of negligible mass. When displaced from its equilibrium position, it swings back and forth under the influence of gravity.

The Pendulum Equation

The period of a simple pendulum is given by:

Where:

• T = Period (time for one complete swing, in seconds)
• L = Length of the pendulum (from pivot to center of mass, in meters)
• g = Gravitational acceleration (9.81 m/s² on Earth at sea level)
• π = Pi (approximately 3.14159)

Key Observations

• Independent of mass: The period does not depend on the mass of the bob
• Length matters: Longer pendulums swing more slowly (longer period)
• Square root relationship: To double the period, you must quadruple the length
• Gravity dependence: Period changes with gravitational acceleration (varies by location and altitude)

Small-Angle Approximation

The pendulum equation above is accurate for small oscillations (typically less than 15 degrees from vertical). For larger angles, the motion becomes non-linear and more complex calculations are required. The small-angle approximation assumes that sin(θ) ≈ θ for small angles, which simplifies the differential equation of motion to produce the formula shown above.

Frequency vs Period

Frequency (f) and period (T) are reciprocals of each other:

• f = 1/T (frequency in Hz, or cycles per second)
• T = 1/f (period in seconds)

A pendulum with a period of 2 seconds has a frequency of 0.5 Hz (half a cycle per second).

Gravitational Variations

Gravitational acceleration varies slightly depending on location:

• Earth's equator: approximately 9.78 m/s²
• Earth's poles: approximately 9.83 m/s²
• Sea level average: 9.81 m/s² (standard value)
• Moon: approximately 1.62 m/s²
• Mars: approximately 3.71 m/s²

Real-World Applications

• Clocks: Pendulum clocks use the regular period for timekeeping
• Seismology: Understanding pendulum motion helps interpret seismograph data
• Engineering: Analyzing structural vibrations and resonance frequencies
• Education: Demonstrating harmonic motion and gravitational principles
• Measurement: Historical method for measuring gravitational acceleration

Limitations

This calculator assumes ideal conditions:

• No air resistance or friction
• Massless, inextensible string or rod
• Point mass concentrated at the bob
• Small oscillation angles (less than 15 degrees)
• No external forces

Real pendulums experience damping due to air resistance and friction, causing the amplitude to decrease over time.