0.5791
s
1.7268
Hz
10.8499
rad/s
0.0833
m
0.5791
s
1.7268
Hz
10.8499
rad/s
0.0833
m
The Physical Pendulum Calculator determines the oscillation period of a rigid body swinging about a fixed pivot point. Unlike a simple pendulum that assumes all mass is concentrated at a single point, a physical (or compound) pendulum accounts for the actual mass distribution through the moment of inertia.
Any rigid body that is free to swing about a horizontal pivot will oscillate when displaced from equilibrium. For small angular displacements, the restoring torque is proportional to the angle, producing simple harmonic motion with period:
$$T = 2\pi\sqrt{\frac{I}{mgd}}$$
where I is the moment of inertia about the pivot axis, m is the total mass, g is the gravitational acceleration, and d is the distance from the pivot to the center of mass. This formula applies to any shape — rods, discs, irregular bodies — as long as the oscillation amplitude remains small (typically less than 15°).
The physical pendulum concept is foundational in engineering and physics. It governs the swing of clock pendulums, the rocking motion of ships, and the dynamics of robotic limbs. Geophysicists use physical pendulum measurements to determine local gravitational acceleration with high precision.
A key insight is the equivalent simple pendulum length, defined as $$L_{eq} = \frac{I}{md}$$. This is the length of a simple pendulum that would have the same period. Because I includes contributions from mass elements at varying distances from the pivot, Leq is always greater than d, meaning the physical pendulum swings slower than a simple pendulum of length d.
The parallel axis theorem connects the pivot-axis moment of inertia to the center-of-mass moment: $$I = I_{cm} + md^2$$. By varying the pivot location along the body, one can find a minimum period, which occurs when d = √(Icm/m). This principle underpins Kater's reversible pendulum, a classic instrument for precise gravity measurement.
This calculator accepts the moment of inertia, mass, pivot distance, and gravitational acceleration, then computes the period, frequency, angular frequency, and equivalent simple pendulum length. It is useful for physics courses, engineering design of oscillating mechanisms, and experimental verification of rotational inertia.
The calculator uses the small-angle approximation for a rigid body oscillating about a fixed pivot:
Period:
$$T = 2\pi\sqrt{\frac{I}{mgd}}$$
Frequency and Angular Frequency:
$$f = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{mgd}{I}}, \quad \omega = \sqrt{\frac{mgd}{I}}$$
Equivalent Simple Pendulum Length:
$$L_{eq} = \frac{I}{md}$$
A simple pendulum of this length produces the same period as the physical pendulum. Since the moment of inertia about the pivot satisfies I = Icm + md², we always have Leq = Icm/(md) + d > d.
A larger moment of inertia increases the period — the body swings more slowly because angular inertia resists rotational acceleration. Increasing the mass or pivot distance reduces the period by strengthening the gravitational restoring torque. The equivalent length lets you intuitively compare any physical pendulum to the familiar simple pendulum. If you know Icm from tables (e.g., mL²/12 for a uniform rod), you can quickly find the total I using the parallel axis theorem and compute the period for any pivot location.
Inputs
Results
A 2 kg body with I = 0.05 kg·m² pivoted 0.3 m from its center of mass oscillates with a period of about 0.82 s. The equivalent simple pendulum length is only 8.3 cm — much shorter than d because the mass distribution is compact.
Inputs
Results
An 8 kg disc with I = 1.2 kg·m² pivoted 0.25 m from its center oscillates with T ≈ 1.56 s. The large moment of inertia produces a slower swing and an equivalent length of 0.60 m.
A simple pendulum models all mass as a point at the end of a massless string, giving T = 2π√(L/g). A physical pendulum accounts for the actual mass distribution of a rigid body through its moment of inertia I, giving T = 2π√(I/(mgd)). The simple pendulum is a special case where I = mL² and d = L.
It is the length Leq = I/(md) of a simple pendulum that has the same period as the physical pendulum. This quantity is always greater than d because the distributed mass increases the effective inertia. It provides an intuitive way to compare physical pendulums of different shapes.
The formula T = 2π√(I/(mgd)) is accurate for small angles (typically below 15°). For larger amplitudes, the period increases and requires elliptic integral corrections, just as with a simple pendulum. At 30° amplitude, the true period is about 1.7% longer than the small-angle prediction.
For standard shapes, use known formulas: Icm = mL²/12 for a uniform rod, mR²/2 for a solid disc, 2mR²/5 for a solid sphere. Then apply the parallel axis theorem: I = Icm + md² to get the moment about the actual pivot. For irregular objects, measure the period experimentally and solve I = T²mgd/(4π²).
The period is minimized when the pivot distance equals d = √(Icm/m), which is the radius of gyration about the center of mass. Moving the pivot closer or farther from the center of mass increases the period. This minimum-period condition is exploited in Kater's pendulum for precision gravity measurements.
Yes. Pendulum clocks use physical pendulums, and the period determines the clock's timekeeping rate. By adjusting the pivot distance or adding a movable weight (bob), clockmakers tune the period to exactly 1 or 2 seconds. Temperature compensation (using materials with different thermal expansion) ensures long-term accuracy.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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