# Physical Pendulum Calculator

Created by Miłosz Panfil, PhD
Reviewed by Małgorzata Koperska, MD and Steven Wooding
Last updated: Jun 05, 2023

With the physical pendulum calculator, you can compute the frequency and period of a physical pendulum. The text describes what exactly a physical pendulum is and how the moment of inertia affects its oscillations. You will also learn the equation describing the period of a physical pendulum and what the radius of its swings is.

On a side note, if you are looking to refresh basic concepts, we've got you covered with our frequency calculator and pendulum period calculator.

## Physical pendulum

A physical pendulum is any object performing small oscillations around its equilibrium position. An example of a physical pendulum is a swing in a kids' playground or a swinging weight of a pendulum clock.

Oscillations are small when the maximal angle doesn't exceed 15º. If it does, the physics of the pendulum is more complicated.

During this movement, kinetic energy is transformed into potential energy, and vice versa, but the sum of these energies is conserved (ignoring losses).

## The period of a physical pendulum

The period $T$ of a physical pendulum is:

$T = 2\pi \sqrt \frac{I}{g m R}$

In this equation:

• $I\ \rm [kg \cdot m^2]$ – Moment of inertia (see moment of inertia calculator);
• $g\ \rm [m/s^2]$ – Acceleration due of a gravity;
• $m\ \rm [kg]$ – Mass of the object; and
• $R\ \rm [m]$ – Distance from the center of mass to the pivot point.

The moment of inertia in the formula must be computed with respect to the pivot. On the Earth's surface, the acceleration due to gravity is $\small g = 9.81\ \rm m/s^2$. The combination,

$L = \frac{I}{mR},$

that appears in the equation for the period of a physical pendulum is called radius of oscillations $L$. It has a dimension of length. Two different pendulums with the same radius of oscillations have the same period.

## Moment of inertia

The period of oscillation depends on the mass of the object and its moment of inertia. Moment of inertia tells us how mass is distributed in the object.

Objects of the same shape and weight may have different moments of inertia. For example, if in one of them, the mass is spread uniformly and in the other, it is concentrated only in some areas.

The moment of inertia depends on the reference point. The one appearing in the equation for the period of oscillations is computed with respect to the pivot.

## Physical pendulum calculator

With our calculator, computations of the period of oscillation of a physical pendulum are simple. Just specify the moment of inertia of the object, its mass, and the distance from its center of mass to the pivot point.

You can also play with the acceleration due to gravity to see how the period would change if we were on a different planet. The calculator also computes the reciprocal of the period, that is, the frequency of oscillations.

## Radius of oscillations of a simple pendulum

Let us describe a pendulum made of a small ball of mass $m$ hanging at the end of a rope of length $l$. The pivot point is at the other end of the rope. We assume the cord is very light, so most of the mass is concentrated in the ball. The center of mass is then in the ball.

What is the frequency of this simple pendulum? The length measured from the pivot point to the center of mass is $R = l$. The moment of inertia is $ml^2$. The radius of oscillations $L$ equals then the length of the rope. In this way, we recover the period of oscillations of a simple pendulum:

$T = 2 \pi \sqrt \frac{l}{g}$
Miłosz Panfil, PhD
Moment of inertia
kg·m²
Mass
oz
Distance from center of mass to pivot
in
in
Period and frequency
Acceleration of gravity
m/s²
Period
sec
Frequency
Hz
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