Pendulum Calculator

Category: Chemistry
- April 02, 2025
| |

Calculate different properties of a pendulum, such as 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