Physical Pendulum Calculator – Period, Frequency & Compound Pendulum

  • Adam
  • November 27, 2025

Calculate physical pendulum period, frequency, and equivalent length for rods, disks, and custom shapes. Free calculator with moment of inertia analysis, parallel axis theorem, and step-by-step solutions for compound pendulums.

Physical Pendulum Calculator - Period, Frequency & Oscillation

A physical pendulum calculator determines the period of oscillation for extended rigid bodies pivoting about a fixed point, unlike simple pendulums with point masses. Physical pendulums, also called compound pendulums, account for the distribution of mass and moment of inertia, making them essential for analyzing real-world oscillating systems including meter sticks, doors, and human limbs.

Calculate Physical Pendulum Properties

Default: 9.8 m/s² (Earth's surface)

Result:

What is a Physical Pendulum?

A physical pendulum, also known as a compound pendulum, is any rigid body that oscillates about a fixed horizontal axis under the influence of gravity. Unlike a simple pendulum where all mass is concentrated at a single point, a physical pendulum has its mass distributed throughout its body. The period of oscillation depends on the body's moment of inertia, mass distribution, and the distance between the pivot point and the center of mass.

Key Characteristics of Physical Pendulums:

  • Extended body: Mass is distributed throughout the object, not concentrated at a point
  • Moment of inertia: Rotational inertia about the pivot point determines oscillation behavior
  • Real-world applicability: More accurately models actual pendulums like swinging doors, legs, and measuring instruments
  • Pivot location matters: Period depends critically on where the object is suspended
  • Reduces to simple pendulum: For small oscillations and specific geometries, behaves like simple pendulum with equivalent length

Physical Pendulum Formula

The period of oscillation for a physical pendulum is derived from the equation of rotational motion and torque analysis. The fundamental formula accounts for the object's moment of inertia and mass distribution.

Period Formula

T = 2π√(I / mgd)

Where:

  • T = period of oscillation (seconds)
  • I = moment of inertia about the pivot point (kg·m²)
  • m = mass of the pendulum (kg)
  • g = gravitational acceleration (9.8 m/s² on Earth)
  • d = distance from pivot point to center of mass (m)

Frequency and Angular Frequency

Frequency:

f = 1/T = (1/2π)√(mgd/I)

Angular Frequency:

ω = 2πf = √(mgd/I)

Equivalent Length (Radius of Oscillation)

A physical pendulum can be characterized by an equivalent length that produces the same period as a simple pendulum:

Leq = I / (md)

The period becomes T = 2π√(Leq/g), identical to simple pendulum form

Common Physical Pendulum Configurations

Different geometric shapes and pivot locations have specific formulas for calculating their oscillation periods:

Uniform Rod Pivoted at End

Moment of Inertia about end: I = (1/3)mL²

Distance to center of mass: d = L/2

Period:

T = 2π√[(1/3)mL² / (mg·L/2)] = 2π√(2L/3g)

Equivalent length: Leq = 2L/3

Uniform Rod Pivoted Off-Center

For pivot at distance x from one end:

Moment of inertia: I = (1/12)mL² + m(L/2 - x)²

Distance to center: d = |L/2 - x|

Period:

T = 2π√{[(1/12)mL² + m(L/2 - x)²] / [mg|L/2 - x|]}

Disk or Cylinder Pivoted at Edge

Moment of inertia about edge: I = (1/2)mR² + mR² = (3/2)mR²

Distance to center: d = R

Period:

T = 2π√[(3/2)mR² / (mgR)] = 2π√(3R/2g)

Equivalent length: Leq = 3R/2

Physical Pendulum vs Simple Pendulum

Understanding the differences between physical and simple pendulums is crucial for accurate modeling of real oscillating systems:

Characteristic Simple Pendulum Physical Pendulum
Mass Distribution Point mass at end of massless string Mass distributed throughout body
Period Formula T = 2π√(L/g) T = 2π√(I/mgd)
Key Parameter Length L Moment of inertia I and distance d
Real-World Accuracy Idealized approximation Accurately models real objects
Moment of Inertia I = mL² Depends on shape and pivot location
Examples Theoretical model, approximate for small bob Meter stick, door, human limb, real pendulum

How to Calculate Physical Pendulum Period

Calculating the period of a physical pendulum requires determining the moment of inertia and center of mass location. Follow these systematic steps:

  1. Identify the pendulum geometry: Determine the shape (rod, disk, irregular body) and dimensions.
  2. Locate the pivot point: Identify where the pendulum is suspended or pivots.
  3. Find the center of mass: Calculate or determine the location of the center of mass relative to the pivot.
  4. Calculate moment of inertia: Use standard formulas for common shapes or parallel axis theorem: I = Icm + md².
  5. Measure the distance d: Determine the perpendicular distance from pivot to center of mass.
  6. Apply the period formula: Use T = 2π√(I/mgd) with all values in SI units.
  7. Verify reasonableness: Check that the period makes physical sense for the system size and mass.

Calculation Example 1: Rod Pivoted at End

Problem: A uniform rod of length 1.0 m and mass 0.5 kg is pivoted at one end. Calculate the period of small oscillations.

Solution:

Step 1: Calculate moment of inertia about end

I = (1/3)mL² = (1/3) × 0.5 × 1.0² = 0.1667 kg·m²

Step 2: Find distance to center of mass

d = L/2 = 1.0/2 = 0.5 m

Step 3: Calculate period

T = 2π√(I/mgd) = 2π√(0.1667/(0.5 × 9.8 × 0.5))

T = 2π√(0.1667/2.45) = 2π√(0.068)

T = 1.64 seconds

Equivalent simple pendulum length:

Leq = 2L/3 = 2(1.0)/3 = 0.667 m

Calculation Example 2: Disk Pivoted at Edge

Problem: A uniform disk of radius 0.3 m and mass 2 kg is suspended from a point on its edge. Find the period.

Solution:

Step 1: Calculate Icm for disk

Icm = (1/2)mR² = (1/2) × 2 × 0.3² = 0.09 kg·m²

Step 2: Apply parallel axis theorem

Ipivot = Icm + md² = 0.09 + 2 × 0.3²

Ipivot = 0.09 + 0.18 = 0.27 kg·m²

Step 3: Calculate period

d = R = 0.3 m

T = 2π√(0.27/(2 × 9.8 × 0.3)) = 2π√(0.27/5.88)

T = 2π√(0.0459)

T = 1.34 seconds

Calculation Example 3: Finding Equivalent Length

Problem: A physical pendulum has I = 0.5 kg·m², m = 3 kg, and d = 0.4 m. What is its equivalent simple pendulum length?

Solution:

Leq = I/(md) = 0.5/(3 × 0.4) = 0.5/1.2

Leq = 0.417 m

This physical pendulum oscillates with the same period as a simple pendulum of length 0.417 m.

Period: T = 2π√(0.417/9.8) = 1.30 seconds

Applications of Physical Pendulums

Physical pendulum analysis is essential in numerous scientific, engineering, and practical applications:

  • Biomechanics: Modeling human limb motion during walking, running, and sports activities as compound pendulums
  • Seismology: Designing seismometers and pendulum-based instruments for detecting ground motion and earthquakes
  • Metrology: Precision measurement of gravitational acceleration using reversible pendulums and Kater's pendulum
  • Engineering Design: Analyzing oscillations in hanging structures, crane loads, and suspended equipment
  • Timekeeping: Historical pendulum clocks using compound pendulums for accurate time measurement
  • Robotics: Controlling robotic arm swinging motions and understanding oscillatory dynamics
  • Architecture: Analyzing building components like doors, shutters, and suspended elements
  • Physics Education: Demonstrating rotational dynamics, energy conservation, and harmonic motion principles

Frequently Asked Questions

What is the formula for a physical pendulum?
The period formula for a physical pendulum is T = 2π√(I/mgd), where T is the period, I is the moment of inertia about the pivot point, m is mass, g is gravitational acceleration, and d is the distance from the pivot to the center of mass. This differs from the simple pendulum formula T = 2π√(L/g) by explicitly accounting for mass distribution through the moment of inertia. The formula can also be written as T = 2π√(Leq/g) where Leq = I/(md) is the equivalent length.
What is the difference between a simple pendulum and a physical pendulum?
A simple pendulum consists of a point mass suspended by a massless string, while a physical pendulum is an extended rigid body with mass distributed throughout. Simple pendulums have period T = 2π√(L/g) depending only on length, while physical pendulums have period T = 2π√(I/mgd) depending on moment of inertia and mass distribution. Physical pendulums more accurately model real-world oscillating objects like meter sticks, doors, and human limbs, whereas simple pendulums are idealized theoretical models.
How do you calculate the period of a physical pendulum?
To calculate the period: (1) Determine the moment of inertia I about the pivot point using standard formulas or parallel axis theorem, (2) Find the distance d from pivot to center of mass, (3) Use the formula T = 2π√(I/mgd) with mass m and gravity g = 9.8 m/s². For example, a uniform rod of length L pivoted at its end has I = (1/3)mL², d = L/2, giving T = 2π√(2L/3g). Always use SI units for consistent results.
What is the moment of inertia of a physical pendulum?
The moment of inertia of a physical pendulum depends on its shape and pivot location. Common examples: uniform rod about end I = (1/3)mL², uniform rod about center I = (1/12)mL², disk about edge I = (3/2)mR², solid sphere about diameter I = (2/5)mR². Use the parallel axis theorem I = Icm + md² to calculate I about any pivot point, where Icm is moment of inertia about center of mass and d is the distance between axes. Units are kg·m².
What is equivalent length of a physical pendulum?
The equivalent length (also called radius of oscillation) is Leq = I/(md), representing the length of a simple pendulum with the same period as the physical pendulum. Using this, the period formula becomes T = 2π√(Leq/g), identical in form to a simple pendulum. For a rod pivoted at its end, Leq = 2L/3. For a disk at its edge, Leq = 3R/2. The equivalent length is always measured from the pivot point.
How does pivot location affect physical pendulum period?
Pivot location dramatically affects the period through both the moment of inertia I and the distance d to the center of mass. As the pivot moves away from the center of mass, d increases (increasing period), but I also increases (increasing period further). There exists an optimal pivot location that minimizes the period. For a uniform rod, the minimum period occurs when pivoted at a distance L/√12 ≈ 0.289L from the center. At the center of mass, d = 0 and the period becomes infinite (no restoring torque).
Can a physical pendulum have the same period as a simple pendulum?
Yes, a physical pendulum has the same period as a simple pendulum of length Leq = I/(md). This equivalent length depends on the physical pendulum's geometry and pivot location. For example, a meter stick (L = 1 m) pivoted at its end has Leq = 2L/3 = 0.667 m, so it oscillates with the same period as a 0.667 m simple pendulum. Any physical pendulum can be matched to some simple pendulum length.
What is a compound pendulum?
A compound pendulum is another name for a physical pendulum—any rigid body that oscillates about a fixed horizontal axis under gravity. The term "compound" emphasizes that the pendulum is composed of distributed mass rather than a single point mass. Examples include meter sticks, baseball bats, and doors. The compound pendulum formula T = 2π√(I/mgd) accounts for the complexity of mass distribution through the moment of inertia I.
How do you use the parallel axis theorem for physical pendulums?
The parallel axis theorem calculates moment of inertia about the pivot: Ipivot = Icm + md², where Icm is the moment about the center of mass and d is the distance between the pivot and center of mass. For example, a disk with Icm = (1/2)mR² pivoted at its edge (d = R) has Ipivot = (1/2)mR² + mR² = (3/2)mR². This Ipivot value is then used in the period formula T = 2π√(Ipivot/mgd).
What factors affect the period of a physical pendulum?
The period depends on: (1) Moment of inertia I—larger I increases period, (2) Mass m—cancels out in the ratio I/m but affects I calculation, (3) Distance d from pivot to center of mass—larger d generally decreases period, (4) Gravitational acceleration g—stronger gravity decreases period, (5) Shape and mass distribution—affects I for given mass, (6) Pivot location—changes both I and d. Temperature can also affect period by changing dimensions through thermal expansion.

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