Find the small-angle motion of a rigid pendulum by entering either pivot inertia directly or a common shape preset.
Advanced options
How to use our Physical Pendulum Calculator
- Choose Solve for to pick the main result: period, frequency, angular frequency, gravity, or equivalent simple-pendulum length.
- Choose Input method. Use Generic: enter inertia and center-of-mass distance if you already know Moment of inertia about pivot (kg*m^2), Center-of-mass distance from pivot (m), and Mass (kg). Use Shape preset: derive inertia from geometry for a rod, disk, or plate.
- If you use a preset, select Shape preset and then fill in only the size inputs that shape needs, such as Length (m), Radius (m), Width (m), or Height (m), plus Mass (kg).
- Enter Gravity (m/s^2) unless you are solving for gravity. Enter Period (s) only when you are solving for gravity or equivalent length from measured period. Geometry-based equivalent length uses inertia, mass and pivot distance instead.
- Optional: open Advanced options and enter Release angle (degrees) to get a small-angle warning, then choose Display format and Significant figures for cleaner output.
- Click Calculate to see the main result along with Frequency, Angular frequency, Equivalent simple-pendulum length, and the actual Moment of inertia used about pivot and Center-of-mass distance used.
- Sanity-check the result: the Center-of-mass distance used must be positive, and a larger Moment of inertia used about pivot usually makes the period longer while a larger Gravity (m/s^2) usually makes it shorter.
Definitions
Physical pendulum: A rigid body that swings about a fixed axis under gravity, unlike an ideal simple pendulum that puts all mass at one point [1].
Moment of inertia about pivot: A measure of how hard it is to rotate an object about the chosen axis; it depends on how the mass is spread out and on which axis you use [2].
Center-of-mass distance from pivot: The straight-line distance from the pivot to the object's center of mass. In this calculator it must be positive for gravity to create a restoring torque.
Period: The time for one full back-and-forth oscillation [1].
Frequency: The number of oscillations per second, equal to 1 divided by period [1].
Angular frequency: The oscillation rate in radians per second, equal to 2 pi divided by period [1].
Equivalent simple-pendulum length: The simple-pendulum length that would give the same small-angle period under the same gravity.
Small-angle approximation: The standard model used here assumes the starting angle is not too large, so the real period can be a little longer at bigger release angles.
Common mistakes and quick fixes
Mistake: Entering Moment of inertia about pivot (kg*m^2) from the center of mass instead of the actual pivot axis in Generic mode.
Fix: Use the pivot-axis value directly, or switch to Input method = shape preset so the calculator builds the pivot value for you.
Mistake: Using Center-of-mass distance from pivot (m) as 0 or a negative number.
Fix: Enter a positive distance from the pivot to the center of mass. If the geometry really gives zero, that setup does not work as a gravity-driven physical pendulum.
Mistake: Filling in Period (s) when solving for period, then expecting that number to control the answer.
Fix: Period (s) is only used when Solve for is gravity or equivalent length from measured period. In other modes, check the geometry and Gravity (m/s^2) instead.
Mistake: Choosing a rod or disk preset but forgetting the matching size field such as Length (m) or Radius (m) .
Fix: After selecting Shape preset , fill in every visible required size input for that shape and keep all lengths positive.
Mistake: Treating Release angle (degrees) as part of the main equation.
Fix: Use Release angle (degrees) only as an interpretation check. It helps judge whether the small-angle result is likely close, but it does not replace the main small-angle model.
Mistake: Ignoring the transparency outputs Moment of inertia used about pivot and Center-of-mass distance used .
Fix: Compare those outputs to your intended setup. If they do not match your object or pivot location, revise Input method or Shape preset .
Limitations & Key Assumptions / Boundary Conditions
- This calculator uses the standard small-angle physical-pendulum model, so accuracy drops as the release angle gets larger.
- It assumes a rigid body swinging about a fixed horizontal axis with negligible friction, air drag, and pivot slop.
- The setup must be a gravity-driven hanging pendulum, which means the Center-of-mass distance used must be greater than 0.
- Center-pivot presets such as a rod about its center, a disk about its center, or a plate about its center have d = 0, so they do not produce gravity-driven oscillation in this model.
- Shape presets use ideal textbook formulas for uniform objects; real objects with holes, attachments, nonuniform density, or off-axis pivots can differ.
- The optional release-angle note is only an interpretation aid. It does not replace the main result with a full large-angle solution.
- If you are solving for gravity from a measured Period (s), measurement error in period, pivot location, or geometry can strongly affect the result.
Methodology
Core model
This calculator uses the small-angle physical-pendulum relation for a rigid body swinging about a fixed pivot [1].
T = 2*pi*sqrt(I_pivot/(m*g*d))
Here, T is period, I_pivot is moment of inertia about the pivot, m is mass, g is gravitational acceleration, and d is the pivot-to-center-of-mass distance.
Derived outputs
Once period is known, the calculator also reports frequency, angular frequency, and equivalent simple-pendulum length.
f = 1/T
omega = 2*pi/T
L_eq = I_pivot/(m*d)
For Equivalent length from measured period, enter only gravity and the measured period. This uses L_eq = g*(T/(2*pi))^2 and does not require object geometry or mass.
If you solve for gravity from a measured period, the equation is rearranged as follows.
g = 4*pi^2*I_pivot/(m*d*T^2)
How presets get the pivot inertia
In Shape preset mode, the calculator finds the needed pivot-axis inertia from standard rigid-body formulas and, when needed, the parallel-axis theorem [2].
I_pivot = I_cm + m*r^2
I_cm_rod = (1/12)*m*L^2
I_end_rod = (1/3)*m*L^2
I_cm_disk = (1/2)*m*R^2
I_rim_disk = (3/2)*m*R^2
I_cm_plate = (1/12)*m*(w^2 + h^2)
For example, a uniform rod of length 1 m and mass 2 kg pivoted at one end has I_pivot = (1/3)*2*1^2 = 0.6667 kg*m^2 and d = 0.5 m. With g = 9.80665 m/s^2, the period is about 1.638 s, the frequency is about 0.6104 Hz, and the angular frequency is about 3.835 rad/s.
Release-angle interpretation
If you enter a release angle, the calculator gives a practical note about whether the small-angle answer is likely close. For larger starting angles, the true period is typically a bit longer than the small-angle prediction [3].
T_large/T_small ~= 1 + theta0^2/16 + 11*theta0^4/3072
The angle in this estimate is in radians, after converting from degrees using pi/180.
Assumptions used by the math
The formulas assume a rigid object, a fixed pivot axis, small oscillations for the main result, and a positive center-of-mass distance from the pivot. If d = 0, gravity does not create the restoring torque needed for this physical-pendulum model.