Pendulum Calculator

- May 10, 2025
| |

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 handy tool designed for anyone interested in learning about pendulums and their movements. Whether you're a student studying Physics or a hobbyist exploring mechanics, this calculator can help you find various properties. With it, you can calculate the period, frequency, and maximum velocity of different types of pendulums. This way, you can easily solve physics problems related to simple harmonic motion.

Types of Pendulums Available

This calculator allows you to choose from three types of pendulums, each with its own unique characteristics:

  • Simple Pendulum: Consists of a mass on a string, ideal for basic calculations.
  • Physical Pendulum: Takes into account mass distribution and is used for more complex scenarios.
  • Spring Pendulum: Incorporates a spring, showing how elasticity affects motion.

Input Parameters Made Easy

Using the Pendulum Calculator is straightforward. You enter specific parameters based on the type of pendulum you select. For a simple pendulum, you will input the length, amplitude, initial angle, and gravitational acceleration. For physical and spring pendulums, you'll include additional details like moment of inertia or spring constant. This clarity helps ensure accurate calculations.

Visualising Pendulum Motion

A fascinating feature of this calculator is its ability to visualize pendulum motion. When enabled, it displays a diagram of the pendulum swinging back and forth. This adds an engaging visual element to understand how the pendulum behaves over time, enhancing your learning experience.

Advanced Options for Enthusiasts

If you want to delve deeper, the calculator offers advanced options. You can include damping effects, which are real-world factors that affect pendulum motion like air resistance. Entering a damping coefficient enables you to see how it impacts the pendulum's performance and energy loss, making your calculations more realistic.

Output Settings Tailored to You

The Pendulum Calculator lets you customize the output. You can select the number of decimal places for your results. You also have the option to view calculation steps and even adjust the visibility of the motion diagram. This flexibility ensures the results meet your specific needs and preferences.

Learning Through Energy Concepts

Alongside calculating motion, the calculator provides insights into energy concepts. It shows maximum kinetic energy, maximum potential energy, and total energy within the pendulum system. Understanding these energy transitions is essential for grasping the principles of physics and how energy conservation works in pendulum motion.

The Importance of Pendulums in Science

Pendulums are not just theoretical constructs; they have practical applications in various fields. They were historically important for timekeeping, serving as the backbone of clocks. Today, they help in seismometers to detect earthquakes and demonstrate concepts like Earth’s rotation with devices like the Foucault pendulum. Their study enriches both engineering and physics education, making them a vital subject in learning mechanics.