Physical Pendulum
Calculate the period and frequency of a physical pendulum using its moment of inertia, mass, and pivot distance. Free online physics pendulum calculator with interactive charts and step-by-step breakdowns.
About This Calculator
The Physical Pendulum Calculator computes the period, frequency, and radius of oscillations for any rigid body swinging about a horizontal pivot. Whether you are a physics student studying rotational dynamics, an engineer designing a pendulum-based mechanism, or a hobbyist building a pendulum clock, this tool provides accurate results using the standard physical pendulum formula.
A physical pendulum (compound pendulum) differs from a simple pendulum because its mass is distributed throughout the body rather than concentrated at a single point. The period depends on the moment of inertia about the pivot point, the total mass, and the distance from the center of mass to the pivot. The formula used is T = 2π × √(I / (g × m × R)), where I is the moment of inertia, g is gravitational acceleration, m is the mass, and R is the distance from the center of mass to the pivot. The calculator also computes the frequency f = 1/T and the radius of oscillations L = I / (m × R).
How to Use
Enter the moment of inertia of the pendulum about its pivot point in kg·m², the total mass in kg, and the distance from the center of mass to the pivot in meters. The gravity field defaults to Earth's standard gravity (9.81 m/s²) but can be adjusted for other planets or custom scenarios. Click Calculate to instantly see the period, frequency, and radius of oscillations with interactive charts and a detailed breakdown.
Common Physical Pendulum Scenarios
Uniform rod pivoted at one end: For a rod of length L and mass m, I = (1/3)mL² and R = L/2. The period becomes T = 2π × √(2L/(3g)).
Uniform rod pivoted at center: For a rod pivoted at its center, I = (1/12)mL² and R = 0, making this an unstable equilibrium — the rod will not oscillate as a pendulum about its center.
Simple pendulum as special case: When all mass is concentrated at distance L from the pivot, I = mL² and R = L, giving the familiar simple pendulum formula T = 2π × √(L/g).
Applications
Physical pendulum calculations are used in clock design (determining pendulum length and bob shape for accurate timekeeping), sports science (analyzing swing dynamics of bats and clubs), engineering (vibration analysis of mechanical systems), and physics education (demonstrating rotational inertia and harmonic motion).
Frequently Asked Questions
What is a physical pendulum?
A physical pendulum (also called a compound pendulum) is any rigid body that is free to swing about a horizontal pivot point. Unlike a simple pendulum where the mass is concentrated at a point, a physical pendulum has its mass distributed throughout the body, making its period depend on the moment of inertia.
How is the period of a physical pendulum calculated?
The period T of a physical pendulum is calculated using the formula T = 2π × √(I / (g × m × R)), where I is the moment of inertia about the pivot, g is the acceleration due to gravity (9.81 m/s² on Earth), m is the mass, and R is the distance from the center of mass to the pivot point.
What is the radius of oscillations for a physical pendulum?
The radius of oscillations L = I / (m × R) is a derived length that combines the moment of inertia, mass, and pivot distance. Two different physical pendulums with the same radius of oscillations will have the same period, making it a useful parameter for comparing pendulums.
How does the moment of inertia affect the pendulum period?
A larger moment of inertia increases the pendulum period because it resists rotational motion more strongly. Objects with mass distributed farther from the pivot have higher moment of inertia and thus swing more slowly. For example, a uniform rod pivoted at its end swings slower than one pivoted near its center.
What is the small-angle approximation for physical pendulums?
The standard formula T = 2π × √(I / (g × m × R)) assumes small oscillations where the maximum angular displacement does not exceed about 15 degrees (0.26 radians). Beyond this angle, the period becomes amplitude-dependent and the simple formula no longer applies accurately.
How is a simple pendulum different from a physical pendulum?
A simple pendulum assumes all mass is concentrated at a point at the end of a massless string, with period T = 2π × √(L/g). A physical pendulum accounts for the actual mass distribution of a rigid body. The simple pendulum is a special case of the physical pendulum where I = m × L² and R = L, which gives T = 2π × √(L/g).
Can I use this calculator for pendulums on other planets?
Yes, the gravity field allows you to set any gravitational acceleration value. For Earth use 9.81 m/s², for the Moon use 1.62 m/s², and for Mars use 3.71 m/s². This lets you compute how a physical pendulum would behave on different celestial bodies.
What are common examples of physical pendulums?
Common examples include a playground swing (the chains and seat form a rigid system), a pendulum clock's swinging weight, a baseball bat pivoted at the handle, a door swinging on its hinges, and a metronome. Any rigid object swinging about a fixed horizontal axis is a physical pendulum.