Physical Pendulum Calculator

Physical Pendulum Calculator finds the swing period and frequency of a rigid body from moment of inertia, mass and pivot distance, with its equivalent length and wide-swing period.

Physical Pendulum Calculator: Period From I, m and R

The Physical Pendulum Calculator finds how long a real, rigid body takes to swing once about a fixed pivot. It needs the moment of inertia, the mass, the distance from the pivot to the center of mass, and gravity.

A physical pendulum is any solid object that hangs from a pivot and swings under its own weight. A baseball bat held at the knob, a swinging door sign and a clock pendulum with a heavy rod are all examples. Another name for it is the compound pendulum.

The period formula has a square root over four quantities. OpenStax University Physics derives it from the torque of gravity on the body.

$$T = 2\pi \sqrt{\frac{I}{m g R}}$$

Here T is the period in seconds, I is the moment of inertia about the pivot in kg·m², m is the mass in kg, g is gravity in m/s², and R is the distance from the pivot to the center of mass in meters. The frequency is the reciprocal of the period.

$$f = \frac{1}{2\pi} \sqrt{\frac{m g R}{I}}$$

The calculator opens with I = 0.5 kg·m², a mass of 1.5 kg and R = 0.4 m at standard gravity. The period is 1.832 s. The first card gives the frequency as 0.546 Hz, the angular frequency as 3.43 rad/s and the pace as 32.76 swings per minute.

Moment of Inertia and Pivot Distance

The formula wants the moment of inertia about the pivot, not about the center of mass. Tables usually list the second kind. The “Moment of Inertia Is About” setting in the Physical Pendulum Calculator accepts either, and it adds the parallel axis term m × R² when the center of mass value is entered.

The third card shows both sides of that step. For the default body the 0.5 kg·m² about the pivot is 0.26 kg·m² about the center of mass plus a parallel axis term of 0.24 kg·m².

For a regular shape, both I and R follow from the dimensions. These are the common cases, with L_eq as the length of the simple pendulum that keeps the same time.

A 1 m rod pivoted at one end swings in 1.638 s. A hoop of 0.3 m radius on a nail takes 1.554 s, and a disk of that radius takes 1.346 s. A plate 0.4 m wide and 0.6 m tall, hung from its top edge, takes 1.338 s. The mass drops out of each answer, because it sits in both I and the bottom of the formula.

The Body list does this work. Choose Uniform rod, Thin hoop, Solid disk, Solid sphere or Rectangular plate, enter its size, mass and pivot distance, and the Physical Pendulum Calculator finds the moment of inertia itself. Each choice loads a common pivot, such as the end of the rod or the rim of the hoop, and the distance can be changed.

A solid sphere of 0.1 m radius hung with its center 0.5 m below the pivot swings in 1.43 s. That is how a real pendulum bob differs from the ideal point mass, which would take 1.419 s at that distance.

For other shapes and axes, the Mass Moment of Inertia Calculator gives I for 23 shapes. Leave Body on “Any body: enter I and R”, take the value about the center of mass, and add the pivot distance.

Radius of Oscillation and Pendulum Length

Every physical pendulum keeps time like some simple pendulum. The length of that simple pendulum is called the radius of oscillation, or the equivalent length.

$$L = \frac{I}{m R}$$

For the default body the Physical Pendulum Calculator gives 83.33 cm, more than twice the 40 cm pivot distance. The point at that distance from the pivot is the center of oscillation, and the second card places it 43.33 cm below the center of mass.

That point has a useful property, which Huygens proved in 1673. Hang the body from its center of oscillation instead, and the period stays the same. The old pivot becomes the new center of oscillation.

With the length in hand, the simple pendulum period formula applies. The pendulum frequency formula is its reciprocal, the square root of g ÷ L divided by 2π.

$$T = 2\pi \sqrt{\frac{L}{g}}$$

The same formula answers the usual length questions. The period of a 1 m simple pendulum is 2.006 s. A pendulum with a period of 2 seconds, the seconds pendulum that ticks once each way, is 99.36 cm long. A pendulum with a 1 second period is 24.84 cm long.

To use the Physical Pendulum Calculator as a simple physical pendulum calculator, set the moment of inertia to 0 about the center of mass. That describes a point mass, and R is then the string length.

School books for class 8 give the time period of a simple pendulum another way. It is the total time divided by the number of oscillations. Twenty swings timed at 36.6 s give a period of 1.83 s, and that measured value can go into the reverse modes below.

Solving for I, R, Mass or Gravity

The Calculate list in the Physical Pendulum Calculator turns the formula around. Each option takes a measured period, or a frequency in hertz, and solves for one of the other quantities.

The first row is a standard lab method. Hang the part, time it, and the moment of inertia follows. With the default 1.5 kg, 0.4 m and a period of 1.8316 s, the answer is 0.5 kg·m² about the pivot.

The gravity option reproduces a textbook measurement. OpenStax times a 75 cm simple pendulum at 1.7357 s and gets 9.8281 m/s². Enter a point mass with R = 0.75 m and that period, and the calculator returns 9.828 m/s².

When the moment of inertia is given about the center of mass, a period can match two pivot distances. The calculator shows the larger one as the headline and names the second under it. A period that is too short for any pivot is refused, with the shortest possible period stated.

Best Pivot Point and Shortest Period

Moving the pivot does not change the period in one direction only. Close to the center of mass the body swings slowly, because gravity has little leverage. Far from it the body also swings slowly, because the moment of inertia grows.

In between there is one distance that gives the shortest period. It equals the radius of gyration about the center of mass. The “Period at Other Pivot Distances” panel of the Physical Pendulum Calculator lists it with five other distances for the default body.

The rows pair up. Pivots at 20.82 cm and 83.27 cm both give 2.047 s, and the two distances add up to the 1.041 m equivalent length. That is Huygens’ rule seen in numbers.

Wide Swings and the Small-Angle Limit

The period formula assumes a small swing. OpenStax treats it as good below about 15°, where the sine of the angle and the angle itself agree within 1%. Bigger swings take longer.

The fourth card of the Physical Pendulum Calculator uses the exact solution for the Swing Angle entered. At the default 20° the period is 1.846 s, which is 0.77% above the small-angle 1.832 s. The same card lists the peak restoring torque and the swing energy for that angle.

Friction and air drag are left out. A swing that dies away over time is a damped oscillation, and the Damping Ratio Calculator works from the shrinking peaks.

Pendulum Force and Energy

The force that drives a physical pendulum acts as a torque about the pivot. It is the weight times the sideways offset of the center of mass, m × g × R × sin θ. For the default body at 20° that is 2.012 N·m, from a weight of 14.71 N.

The energy of a pendulum is set by how high its center of mass is lifted. At a swing angle θ the stored energy is m × g × R × (1 − cos θ), which is 0.3548 J for the default body at 20°. All of it is kinetic at the bottom of the swing.

Both figures appear on the fourth card of the Physical Pendulum Calculator, next to the period at that angle. For a bob on a string, the Kinetic Energy of a Pendulum Calculator gives the energy, speed and string tension at any angle.

Physical vs Simple Pendulum and Other Types

Most pendulum questions come down to which kind of pendulum is meant. The answers below sort the types and say which ones the Physical Pendulum Calculator covers.

What is the difference between a physical pendulum and a simple pendulum?

A simple pendulum is an ideal one, with all its mass at a point on a weightless string. A physical pendulum has its mass spread out, so its moment of inertia matters. A 1 m rod pivoted at its end swings in 1.638 s, while a 1 m simple pendulum takes 2.006 s.

What are the types of pendulums?

The two main types are the simple and the physical, or compound, pendulum. OpenStax adds the torsional pendulum, a body that twists on a wire, as a third. Common physical pendulums include a bar, a hoop, a disk and Kater’s reversible pendulum, which was built to measure gravity.

What are the four laws of a pendulum?

They describe the simple pendulum at small angles. The period does not depend on the size of the swing. It does not depend on the mass. It rises with the square root of the length. It falls with the square root of gravity.

What are double, triple and inverted pendulums?

A double pendulum is one pendulum hung from the end of another, and a triple pendulum adds a third. A double pendulum is chaotic, so a tiny change in the starting position leads to very different motion. An inverted pendulum balances with its center of mass above the pivot and falls over unless it is actively steadied.

What is the most famous pendulum?

It is the Foucault pendulum that Léon Foucault hung in the Panthéon in Paris in 1851 on a 67 m wire. The slow turning of its swing plane showed that the Earth rotates. A length of 67 m gives a period of 16.42 s.

Oscillation, Period and Hertz

One oscillation is one full trip there and back, and the time it takes is one period. A count of oscillations divided by the time gives the frequency. One hertz is one cycle per second.

That makes 60 beats per minute equal to 1 Hz, and 50 Hz equal to 3,000 cycles per minute, or 3,000 RPM for a shaft. A 300 Hz tone means 300 cycles every second. The Frequency Calculator converts between period, hertz and RPM.

A wave is an oscillation that travels. One wavelength, written λ, passes a fixed point in one period, so one wave is one oscillation at that point. The Harmonic Wave Equation Calculator works with wavelength, and its page lists the main types of waves.

For ω alone, the Angular Frequency Calculator has pendulum and spring inputs. More tools for swinging and spinning bodies are on the Rotational Motion Calculators hub.