Say goodbye to tedious manual calculations of a physical pendulum's period, frequency, and moment of inertia with our physical pendulum calculator! Instead, focus on learning how we perform these calculations and why there are significant! In this article, we shall discuss the fundamentals of a physical pendulum, including:

  • Frequency and period of a physical pendulum.
  • Moment of inertia of a pendulum.
  • Equation of a physical pendulum's radius of oscillations.

Being familiar with simple pendulums will help you understand physical pendulums, so we recommend you first go through our simple pendulum calculator.

What is a physical pendulum?

We know that a simple pendulum is a point mass suspended by a massless string that undergoes simple harmonic motion for small oscillations. On the other hand, a physical pendulum or a compound pendulum is more realistic. Instead of assuming a massless string or a point mass, a physical pendulum is a physical object with mass, like the swinging pendulum in a grandfather's clock or a swing set in a playground.

Like a simple pendulum, a physical pendulum also undergoes simple harmonic motion for small angular displacements (around 15°).

Equations of a physical pendulum for period and frequency

The period of a physical pendulum is given by:

T=2πImgRT = 2 \pi \sqrt{\frac{I}{mgR}}

Where:

  • TT - Period of the physical pendulum;
  • II - Moment of inertia of the physical pendulum;
  • mm - Mass of the physical pendulum;
  • gg - Acceleration due to gravity; and
  • RR - Distance between the center of mass of oscillations to the pivot point.

The formula for the physical pendulum's frequency ff is the reciprocal of its period:

f=1T=12πmgRIf = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{mgR}{I}}

Moment of inertia and radius of oscillation

The moment of inertia of the pendulum measures how much torque is required to rotate it about its pivot point. It tells us about the mass distribution in the pendulum.

The radius of oscillation is a term we use to express the distance between the pivot point and the center of oscillation. This depends on the mass distribution in the body - if the mass is concentrated toward the center of mass, then the center of oscillation will be close to the center of mass. Otherwise, it is commonly below the center of mass.

The radius of oscillation, also called equivalent length, is given by:

L=ImRL = \frac{I}{mR}

Where LL is the radius of oscillation.

Hence we can rewrite the equations for the period and frequency of a physical pendulum:

T=2πLgf=12πgL\begin{align*} T &= 2 \pi \sqrt{\frac{L}{g}} \\[2em] f &= \frac{1}{2\pi}\sqrt{\frac{g}{L}} \end{align*}

Notice how these physical pendulum formulae look similar to simple pendulum equations!

How to use this physical pendulum calculator

There are two methods to determine the period and frequency of a physical pendulum using this calculator:

  1. Using the moment of inertia, mass, and distance from the center of mass to the pivot point: If you know these three values, enter them into their corresponding fields. The physical pendulum calculator will determine:
    • Radius of oscillations;
    • Period; and
    • Frequency of the physical pendulum.
  2. Using the radius of oscillations: If the radius of oscillations is known, provide its value, and the calculator will find the following unknowns:
    • Period; and
    • Frequency of the physical pendulum.

Calculating moment of inertia: To find the moment of inertia, you need to enter the values of the following variables:

  • Mass of the physical pendulum;
  • Distance from the center of mass to the pivot point; and
  • Radius of oscillations.
Krishna Nelaturu
Radius of oscillations
Moment of inertia
kg*m²
Mass
oz
Distance from center of mass to pivot
in
Radius of oscillations
in
Period and frequency
Acceleration of gravity
m/s²
Period
sec
Frequency
Hz
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