Physical Pendulum Calculator

Mass (m)

Distance pivot to centre of mass (d)

Moment of inertia about the pivot (I)

Acceleration due to gravity (g)

Angular frequency (omega)

Frequency (f)

Period (T)

Enter the mass, the pivot distance, and the moment of inertia, and the calculator finds the angular frequency, the frequency, and the period. Gravity is already set to Earth's value. Working the other way? Enter a measured period with any two of mass, distance, and inertia to solve for the third, or clear gravity to measure it.

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A physical pendulum is a rigid body that oscillates under the influence of gravity about a pivot point. It can be a swinging bar, a hanging sign or even a human foot. Unlike the simple pendulum described in textbooks, the mass of a physical pendulum is not concentrated at one end of the string but distributed throughout the body, which means that its motion depends on the distribution of mass. This calculator establishes relationships between mass, distance from pivot point, moment of inertia and angular velocity, frequency and period.

Enter known values and leave the unknown fields blank. As this calculator is bi-directional you can either calculate the period from the formula or enter a measured period to find the moment of inertia, distance from pivot point or even local gravity value.

What is a physical pendulum?

A physical pendulum is a rigid body that swings under the influence of gravity about a pivot point. It can be a swinging bar, a hanging shield or even a human leg. Unlike the simple pendulum described in textbooks, in a physical pendulum the mass is not concentrated at one end of the string but distributed throughout the body, which means that its oscillation behavior depends on the distribution of mass. This calculator establishes relationships between mass, distance from pivot point, moment of inertia and angular velocity, frequency and period.

The restoring moment is created by the gravity acting on the center of mass which is at a distance d from the pivot point.

\tau = -m g d \sin\theta

The minus sign indicates that the torque always acts in the direction of the rest position. As it depends on the sine of the angle, this is not a complete harmonic oscillation. However, for small amplitudes of oscillation, the sine of the angle very closely approximates to the angle itself and the motion becomes a harmonic oscillation with constant period. The following main formula describes this case for small angles.

Angular velocity, frequency and period equations

There are three quantities that describe the temporal properties of a vibration, and they all stem from the same physical relationship of a pendulum. The angular velocity describes the speed of oscillation and is measured in radians per second.

\omega = \sqrt{\dfrac{m g d}{I}}

The normal frequency is the number of complete oscillations that occur per second while the period is the time it takes to complete one full oscillation. Both can be directly calculated from angular velocity.

f = \dfrac{\omega}{2\pi} = \dfrac{1}{2\pi}\sqrt{\dfrac{m g d}{I}} \qquad T = \dfrac{2\pi}{\omega} = 2\pi\sqrt{\dfrac{I}{m g d}}

Each symbol represents a physical attribute of a pendulum.

Symbol	Meaning	Typical unit
m	Mass of the body	kilograms
d	Distance from pivot to centre of mass	metres
I	Moment of inertia about the pivot	$\mathrm{kg\,m^2}$
g	Acceleration due to gravity	$9.81\ \mathrm{m/s^2}$
omega	Angular frequency	radians per second
f	Frequency	hertz
T	Period	seconds

How to use this calculator:

Enter the known values and set the units. When enough information is entered, the remaining values will be shown in their respective fields with the period highlighted as a key result. It is not necessary to specify what quantity you are solving for; the blank fields indicate which quantities the calculator computes.

Normally mass, distance to pivot and moment of inertia are entered in order to calculate the period. The acceleration due to gravity is set by default to 9.81 meters per second squared and only needs to be changed if calculations for other planets are being done or a high degree of accuracy is required. Conversely you can enter the period and specify two of three values (mass, distance, moment of inertia) in order to calculate the third value. If the acceleration due to gravity value is cleared and the period as well as mass, distance and moment of inertia are entered then the local value for g will be displayed.

For example, a uniform bar.

A uniform rod of length one meter and mass two kilograms is fixed at one end. Since the center of gravity is in the middle, the distance to the pivot point is 0.5 meters. The moment of inertia of the rod about its end point is a third of the product of the mass and the square of the length.

I = \tfrac{1}{3} m L^2 = \tfrac{1}{3}(2.0)(1.0)^2 = 0.667 \ \text{kg m}^2

When these values are put into the formula for period, it gives the time required to complete one full oscillation.

T = 2\pi\sqrt{\dfrac{I}{m g d}} = 2\pi\sqrt{\dfrac{0.667}{(2.0)(9.81)(0.5)}} \approx 1.64 \ \text{s}

Thus it takes about 1.64 seconds for this pendulum to complete one full oscillation. The frequency is approximately 0.61 hertz while the angular velocity is about 3.84 radians per second.

Equivalent simple pendulum.

All physical pendulums have the same oscillation properties as a simple pendulum of an equivalent length. This length is equal to the ratio of the moment of inertia over mass divided by the distance from the pivot point.

L_{\text{eq}} = \dfrac{I}{m d} = \dfrac{g}{\omega^2}

The effective length of the upper bar is about 0.667 meters so its oscillation frequency corresponds to that of a simple pendulum with a length of two-thirds of a meter. For this reason, in addition to the period, the calculation tool also indicates the effective length. You can immediately feel how fast this pendulum swings compared to a normal string pendulum.

Large amplitude and small angular limit

The simple formula for the period assumes a very small amplitude since it replaces the sine of the angle by the angle itself. This approximation is very accurate within a few degrees and works well up to about ten degrees. Outside this range the actual period becomes longer because the restoring moment no longer matches the torque that an actual spring could provide.

A more accurate formula for the period includes correction terms that depend on the amplitude of the oscillation.

T \approx T_0\left(1 + \tfrac{1}{16}\theta_0^2 + \tfrac{11}{3072}\theta_0^4\right)

Here the amplitude is given in radians. At thirty degrees the actual period is about 1.7 points longer than the small angle value and this difference gets larger as the angle increases. If you turn on the large amplitude correction in the calculator, you can apply it to your own oscillation and see how the period changes with increasing angle.

Moments of Inertia for Common Shapes

The moment of inertia is a value often required. It describes the distribution of mass relative to an axis passing through a pivot point. The value must be used with respect to the pivot point, not the center of mass. If only the value for the center of mass is known, then the parallel axis theorem can be applied to transfer the value to the position of the pivot point.

I_{\text{pivot}} = I_{\text{cm}} + m d^2

The following are typical results for moment of inertia about a pivot point for shapes supported by the integrated utility tools.

Shape and pivot	Moment of inertia about the pivot
Uniform rod, pivot at one end	$I = \tfrac{1}{3} m L^2$
Uniform rod, pivot at centre	$I = \tfrac{1}{12} m L^2$ (d = 0, so it does not swing)
Solid disk, pivot at the rim	$I = \tfrac{3}{2} m R^2$
Solid sphere, pivot at the surface	$I = \tfrac{7}{5} m R^2$
Thin hoop, pivot on the rim	$I = 2 m R^2$

Open the Moment of Inertia calculator tool, select the appropriate shape and enter length or radius. The calculation tool will then calculate the values and populate them into the respective fields.

A physical pendulum for measuring gravity

Since the period is a measurable quantity that depends on gravity, a physical pendulum is an amazingly practical device for measuring gravity. If we rearrange the formula for the period, we get:

g = \dfrac{4\pi^2 I}{T^2 m d}

By carefully measuring the swing period of a pendulum, one can determine the strength of local gravity. Long before electronic gravimeters were developed, the inverted Kater's pendulum was used to measure g with great precision, based on the same principle. A calculation tool allows you to check your calculated values for g by entering the measured time period as well as mass, distance and moment of inertia.

Examples of physical pendulums in use:

The most familiar application is the pendulum clock. A heavy mass on a pendulum rod marks time with a pendulum rod of about one meter length making a known swing per second. A metronome is an inverted physical pendulum where a sliding mass changes the effective length and moment of inertia, thus adjusting the speed.

These concepts are not limited to clocks. In biomechanics, leg movements while walking are modeled as physical pendulums in order to predict natural stride lengths and design prosthetics accordingly. Engineers consider skyscrapers, bridges, and mass dampers as slow moving physical pendulums in order to decouple the natural oscillations from earthquake or wind frequencies.

Tips for accurate results:

Measure the distance from the pivot point to the actual center of mass. An inaccuracy in d is the most common source of error. For objects with multiple parts, you will need to determine the center of mass based on each part and its position.

Use the moment of inertia about the pivot point and not the moment of inertia about the center of mass. If a conversion is needed use Steiner's theorem. Keep oscillations small so that the formulas for small angles are valid or enable large amplitude correction. Also, use local gravity values to measure time accurately since g varies by a few tenths of a percent at different locations on Earth's surface.

Frequently asked questions

What is the period of a physical pendulum?

The period is the time taken for one complete oscillation and is calculated as T = 2 pi multiplied by the square root of the value obtained by dividing the moment of inertia I by m g d. Here, I is the moment of inertia about the pivot point, m is the mass, d is the distance from the pivot to the center of gravity, and g is the acceleration due to gravity. The longer the length and the larger the moment of inertia, the longer the period will be.

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

In the simple pendulum, all of the mass is considered to be a point at the end of a weightless string so that the period depends only on length and gravity. A physical pendulum is an object with a particular shape, so its period depends on the distribution of mass which is reflected in the moment of inertia. The oscillation of any physical pendulum is equivalent to that of a simple pendulum with length L = I / (m d).

Does period depend on amplitude?

For small amplitudes the period does not depend on the amplitude. The period is determined by mass, distance, moment of inertia and gravity and not by the amplitude. When the oscillation exceeds about ten degrees the actual period lengthens slightly, at thirty degrees it is about 1.7 points longer than the value for small angles. The correction for large amplitudes in this calculator compensates for that.

Is it possible to measure gravity using a physical pendulum?

Yes. Rearranging the formula for period gives that gravity is equal to the product of moment of inertia and four pi squared divided by the product of period squared and mass times distance. So if you know the mass, distance, and moment of inertia of a pendulum and measure its period, then you can determine the local value of gravity. Historically, this was how gravity was measured very accurately using the Kater's Pendulum. If you enter in a measured period value without knowing the value for gravity, then you can try it out here.

What are some examples of physical pendulums?

A uniform bar with one end pivoted as a fulcrum, a swinging disc, a hanging shield, the pendulum arm of a metronome and even a person's leg can act as physical pendulums. Pendulum clocks and the tuned mass dampers in skyscrapers are examples of engineering applications where natural frequency plays an important role.

Related calculators

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Angular Momentum CalculatorFind the angular momentum of a spinning body from its moment of inertia and angular velocity (L = I omega), or of a point mass from its mass, speed and radius (L = m v r). Solve for any variable, work the moment of inertia out from a shape, apply the conservation of angular momentum, and read the rotational kinetic energy.

Angular Velocity CalculatorCalculate angular velocity (ω) from an angle and time, a linear speed and radius, a period or a frequency, and read it in rad/s, RPM, rev/s and deg/s. Solves every method backwards too.

Angular Frequency CalculatorFind angular frequency (omega) from frequency, period, rpm, or an oscillating system, and get the matching frequency, period, wavelength, and peak motion.

Angle of Twist CalculatorCalculate the angle of twist of a shaft from torque, length, shear modulus, and the polar moment of inertia, or solve backward for any of them. Computes the polar moment for solid and hollow shafts plus the maximum shear stress and torsional stiffness.

Frequency CalculatorFind the frequency of a wave from its period, or from its speed and wavelength. Enter any two of frequency, period, wave speed, or wavelength and the rest fill in, with angular frequency and photon energy.

Disclaimer: This calculator is provided for general informational and educational purposes only. Our calculators are under active development, and results may be inaccurate, incomplete, or unsuitable for your situation. Always verify the figures independently and seek advice from a qualified professional before relying on them. We make no warranties and accept no liability for any loss or decision arising from use of this tool.

References

Wikipedia: Pendulum (mechanics)
The physical pendulum, its equation of motion, and the small-angle period.
HyperPhysics: Physical Pendulum
Derivation of the physical-pendulum period and the equivalent simple-pendulum length.
Harvard Natural Sciences Lecture Demonstrations: Physical Pendulum
A demonstration of a rigid-body pendulum and how its inertia sets the period.