Pendulum Calculator

Category: Chemistry
- March 25, 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 center 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 an interactive tool for exploring the motion of pendulums. It calculates the period, frequency, angular frequency, velocity, and energy for different types of pendulums: simple, physical, and spring-based. This calculator is ideal for students, educators, and anyone interested in learning about harmonic motion in physics.

Key Formulas Used:

Simple Pendulum:
T = 2π√(L/g)
f = 1/T
ω = √(g/L)

Physical Pendulum:
T = 2π√(I/(m·g·d))
f = 1/T
ω = √(m·g·d/I)

Spring Pendulum:
T = 2π√(m/k)
f = 1/T
ω = √(k/m)

Purpose of the Calculator

This calculator helps users analyze the motion of pendulums in various forms, including:

  • Simple Pendulum – A weight on a string swinging under gravity.
  • Physical Pendulum – A rigid body swinging about a pivot point.
  • Spring Pendulum – A mass oscillating on a spring.

It’s a helpful companion to educational tools like the Molarity Calculator and Molar Mass Calculator, which support chemistry and physics learning through visualization and computation.

How to Use the Pendulum Calculator

  1. Select the type of pendulum: Simple, Physical, or Spring.
  2. Enter the required values such as length, angle, mass, or spring constant.
  3. Optionally include damping to simulate energy loss over time.
  4. Choose output preferences such as number of decimal places and whether to show visualizations and calculation steps.
  5. Click the Calculate button to see the results.
  6. Review the visual animation, results, and step-by-step breakdown for better understanding.

What You’ll Learn from the Results

The calculator provides detailed insight into pendulum motion:

  • Period (T) – How long one full swing takes.
  • Frequency (f) – Number of swings per second.
  • Angular Frequency (ω) – How fast the angle changes over time.
  • Maximum Velocity – Speed at the lowest point of the swing.
  • Kinetic and Potential Energy – Energy transformations during motion.

These calculations mirror the kinds of insights provided by tools like the Entropy Calculator or Theoretical Yield Calculator, making it easier to analyze physical systems with confidence.

Why This Calculator Is Useful

The Pendulum Calculator is especially valuable in educational and practical contexts:

  • Students can visualize motion and check homework or lab results.
  • Teachers can use it as a classroom demonstration tool.
  • Engineers and hobbyists can explore dynamics in mechanical designs.

Just like the Percent Yield Calculator helps with chemical reactions, this tool simplifies analysis of motion and energy in pendulums.

Frequently Asked Questions (FAQ)

Can I use this calculator for real-world pendulums?

Yes. While it simplifies some scenarios (e.g., assuming small angles for simple pendulums), it can give reliable approximations, especially for educational purposes.

What’s the difference between simple and physical pendulums?

A simple pendulum uses a point mass and massless string, while a physical pendulum accounts for the mass distribution and pivot location.

Why include damping?

Damping simulates real-world friction or air resistance, helping you understand how energy is gradually lost during motion.

Is this tool useful outside of physics?

Yes. Pendulums are found in timekeeping, engineering systems, and seismology. This tool helps analyze and understand their behavior in those areas.

Are there similar tools for chemistry?

Absolutely. You might also explore:

Explore More

Whether you're studying motion or chemical reactions, calculators like this and the Atomic Mass Calculator or Electron Configuration Calculator can save time and support learning. Combine physics tools with chemistry calculators for a fuller scientific understanding.