Physical Pendulum Calculator

How to use this physical pendulum calculator

What this calculator does. This tool estimates three linked small-oscillation quantities for a rigid body swinging about a fixed pivot: Radius of oscillations (L), Period (T), and Frequency (f). It uses the standard physical-pendulum model in which gravity supplies the restoring torque and the body's mass distribution is represented by the moment of inertia about the pivot. The result is an idealized mechanics estimate; it does not model large swing angles, air drag, bearing friction, flexing supports, or other energy losses. OpenStax's discussion of the physical pendulum and the small-angle approximation explains the underlying equation and why mass distribution matters.

When to use it. Use the calculator when comparing how two rigid objects swing from different pivot locations, checking a laboratory setup before measuring a period, estimating the natural timing of a suspended component, or exploring how a different gravitational acceleration changes oscillation timing. It is particularly useful when a simple point-mass pendulum is not a good model because the object's shape and mass distribution affect its inertia.

How to calculate. The calculator opens with a complete demonstration set so the results and example XLSX are available immediately.

1. Replace the demonstration values with your measured or calculated Moment of inertia (I), Mass (m), Distance from center of mass to pivot (R), and Acceleration due to gravity (g). Values use SI units and a decimal point; commas, scientific notation, unit text, zero, and negative values are rejected.
2. Read Period (T) first if you need the time for one complete oscillation. Use Frequency (f) for cycles per second, and Radius of oscillations (L) to compare the distributed-mass pendulum with an equivalent simple pendulum.
3. Select Download Excel to export the current validated inputs and results to an XLSX workbook. Reset clears the demonstration and all calculated state; after Reset, the Excel action is disabled until all four required inputs form a valid model again.

Input guide. Moment of inertia (I) is a required positive decimal in kg·m². The sample is 0.75 kg·m². It must be taken about the actual pivot axis; entering a center-of-mass inertia without applying the required axis shift is a common mistake. At fixed mass, pivot distance, and gravity, a larger inertia increases the radius of oscillations and therefore lengthens the period. The parallel-axis theorem is useful when your tabulated inertia is about a parallel axis through the center of mass.

Mass (m) is the total positive mass in kilograms; the sample is 2.5 kg. A larger mass lowers L and T only when I is held numerically fixed. For geometrically similar bodies, I often scales with mass too, so changing mass alone while leaving inertia unchanged may not represent a real object. Distance from center of mass to pivot (R) is a required positive distance in meters; the sample is 0.3 m. A larger R increases the gravitational torque lever arm and, with I fixed, decreases both L and T. Do not substitute total object length unless the center of mass really is that far from the pivot. Acceleration due to gravity (g) is a required positive acceleration in m/s²; the sample is 9.81. Larger g shortens the period and raises the frequency. For comparison, NIST gives the defined standard acceleration of gravity as 9.80665 m/s².

Output guide. Radius of oscillations (L) is the exact derived identity I/(mR), reported in meters. It is the length that would give an ideal simple pendulum the same small-angle period under the same gravity. Period (T), reported in seconds, is an estimate from the small-angle physical-pendulum model; high values mean slower oscillations, while a smaller positive value means faster oscillations. Frequency (f), reported in hertz, is exactly the reciprocal of the computed period, so high frequency corresponds to short period. All three live summary pills repeat these same canonical outputs rather than using separate calculations.

Worked example. With I = 0.75 kg·m², m = 2.5 kg, R = 0.3 m, and g = 9.81 m/s², the radius of oscillations is L = 0.75/(2.5×0.3) = 1.000000 m. The period is T = 2π√(1/9.81) = 2.006067 s, and the frequency is f = 1/T = 0.498488 Hz. Those are the first-open values shown above and the values written into the initial workbook.