Pendulum Calculator

- May 10, 2025
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Calculate various properties of a pendulum including period, frequency, and maximum velocity. This calculator helps with physics problems related to simple harmonic motion.

Pendulum Type

Simple Pendulum Parameters
°

Angle from vertical (less than 10° for small-angle approximation)

m/s²

Default: Earth's gravity (9.81 m/s²)

Damping Options

Output Settings

About Pendulums

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced from its resting equilibrium position, it oscillates due to the restoring force of gravity.

Simple Pendulum

A simple pendulum consists of a mass (called a bob) attached to a massless string or rod. When the angle of displacement is small, the period of oscillation is independent of the amplitude.

  • Period: T = 2π√(L/g)
  • Frequency: f = 1/T = 1/(2π)·√(g/L)
  • Angular Frequency: ω = 2πf = √(g/L)
Physical Pendulum

A physical pendulum has an extended mass distribution rather than a point mass. The period depends on the moment of inertia and the distance to the centre of mass.

  • Period: T = 2π√(I/(m·g·d))
  • Frequency: f = 1/T = 1/(2π)·√(m·g·d/I)
  • Angular Frequency: ω = 2πf = √(m·g·d/I)
Spring Pendulum

A spring pendulum consists of a mass attached to a spring. Its motion follows simple harmonic motion with the restoring force provided by the spring.

  • Period: T = 2π√(m/k)
  • Frequency: f = 1/T = 1/(2π)·√(k/m)
  • Angular Frequency: ω = 2πf = √(k/m)
Damped Pendulum

In real-world pendulums, damping forces like air resistance and friction cause the amplitude to decrease over time.

  • Underdamped: The pendulum oscillates with decreasing amplitude
  • Critically Damped: The pendulum returns to equilibrium in minimum time without oscillating
  • Overdamped: The pendulum returns to equilibrium without oscillating, but slower than critical damping
Energy in Pendulums

In an ideal pendulum, energy is conserved and continuously transforms between potential and kinetic energy.

  • Potential Energy: Highest at maximum displacement (maximum amplitude)
  • Kinetic Energy: Highest when passing through the equilibrium position
  • Total Energy: Remains constant in an ideal system without damping
Applications of Pendulums
  • Timekeeping: Pendulum clocks were historically very accurate timepieces
  • Seismometers: Used to detect and measure earthquakes
  • Demonstration of Earth's Rotation: Foucault pendulum
  • Engineering: Analysis of oscillatory systems and structural vibrations
  • Physics Education: Demonstrating principles of mechanics, harmonic motion, and energy conservation

Understanding the Pendulum Calculator

The Pendulum Calculator is a helpful tool for anyone interested in the Physics of pendulums. It allows you to calculate important properties like the period, frequency, and maximum velocity of different types of pendulums. Whether you're tackling homework problems or just curious about pendulum motion, this calculator simplifies the process by giving you quick and accurate results.

Types of Pendulums You Can Explore

This calculator supports multiple pendulum types, making it versatile for various physics applications. You can choose between:

  • Simple Pendulum: A mass on a string swinging freely.
  • Physical Pendulum: An object with mass distributed along its length.
  • Spring Pendulum: A mass attached to a spring that oscillates.

Key Parameters for Simple Pendulums

When using the Pendulum Calculator for a simple pendulum, you'll need to input some key parameters. Important inputs include the length of the pendulum, amplitude, initial angle, and gravitational acceleration. These values help the calculator determine how the pendulum will behave during its motion.

Advanced Options for Physical Pendulums

For those using the calculator for physical pendulums, there are advanced parameters to consider. These include the moment of inertia, mass, distance to the centre of mass, and the initial angle. Each of these factors influences the pendulum's period and behaviour, allowing for a more detailed analysis of its motion.

Spring Pendulum Input Details

If you're interested in spring pendulums, the calculator asks for specific inputs like spring constant, mass, amplitude, and rest length. These parameters determine how the spring oscillates. By adjusting them, you can see how the spring's characteristics affect its motion.

Damping Options for Real-World Applications

In real-life situations, damping plays a significant role in pendulum motion. The calculator offers an option to include damping effects. You can specify a damping coefficient, which simulates real-world forces like air resistance or friction. This adds another layer of depth to your calculations.

Visualising Pendulum Motion

The Pendulum Calculator doesn’t just provide numbers; it also visualises pendulum motion. You can view a diagram showing how the pendulum moves over time. This visual representation helps you understand the principles of oscillation and the effects of different parameters on motion.

Learning More About Pendulum Theory

Along with calculations, the calculator offers insights into pendulum theory. You can learn about the energy transformations between kinetic and potential energy, the role of damping, and various types of pendulums. This additional information deepens your understanding of pendulum mechanics.