Simple Pendulum Calculator | Calculate Period, Frequency & More

What is a Simple Pendulum?

Core Concepts
The Small-Angle Approximation
Ideal vs. Real Pendulums

A simple pendulum is an idealized model consisting of a point mass (called a bob) suspended from a fixed point by a massless, inextensible string or rod. When displaced from its equilibrium position, the pendulum swings back and forth under the influence of gravity. This oscillatory motion is a classic example of Simple Harmonic Motion (SHM), provided the angle of swing is small (typically less than 15 degrees).

Key Components and Assumptions

The behavior of a simple pendulum is primarily determined by its length (L) and the local acceleration due to gravity (g). In the ideal model, we assume: 1. The bob is a point mass. 2. The string is massless and cannot stretch. 3. There is no friction or air resistance. 4. The swing (amplitude) is small.

The small-angle approximation is crucial because it simplifies the restoring force equation, making the motion sinusoidal and thus, simple harmonic. For larger angles, the motion is still periodic but no longer simple harmonic, and the period calculation becomes more complex.

Conceptual Examples

A playground swing can be approximated as a simple pendulum.
The swinging incense burner (Botafumeiro) in the Santiago de Compostela Cathedral is a famous large-scale pendulum.

The Mathematical Formula for a Simple Pendulum

Deriving the Period Formula
Calculating Frequency
The Role of Gravity

The period (T) of a simple pendulum is the time it takes to complete one full swing (back and forth). For small amplitudes, it is determined by the formula: T = 2π * sqrt(L / g)

Where: T is the period in seconds, L is the length of the pendulum in meters, g is the acceleration due to gravity in m/s², and π (pi) is approximately 3.14159.

Frequency and Angular Frequency

Frequency (f), the number of oscillations per second, is the reciprocal of the period (f = 1/T). Angular frequency (ω), measured in radians per second, is related by ω = 2πf. These values are essential for describing the oscillatory state of the system.

Notice that the mass of the bob and the amplitude of the swing do not appear in the basic formula. This 'isochronism' is a key feature of simple pendulums at small angles.

Calculation Examples

A pendulum with a length of 1 meter on Earth (g ≈ 9.81 m/s²) has a period of approximately 2.006 seconds.
To have a period of exactly 1 second, a pendulum on Earth would need a length of about 0.248 meters (24.8 cm).

Step-by-Step Guide to Using the Simple Pendulum Calculator

Entering Pendulum Length
Handling Gravity
Interpreting the Results

Inputting Pendulum Length

Start by entering the length of your pendulum in the 'Pendulum Length (L)' field. Be sure to select the correct unit of measurement (meters, centimeters, feet, or inches) from the dropdown menu. The calculator will automatically convert the value to meters for the calculation.

Setting Gravitational Acceleration

Next, define the gravitational acceleration (g). You can choose from presets like Earth, Moon, or Mars, which use standard gravity values. Alternatively, select 'Custom' to input your own value, for instance, if you are at a different altitude or performing a specific experiment. Select the appropriate units for your custom value.

Analyzing the Output

After clicking 'Calculate', the tool will display three key results: the Period (T) in seconds, the Frequency (f) in Hertz, and the Angular Frequency (ω) in radians per second. These results provide a complete picture of the pendulum's simple harmonic motion.

Real-World Applications of the Simple Pendulum

Timekeeping and Horology
Geophysical Measurements
Educational Demonstrations

Clocks and Metronomes

The most famous application is in timekeeping. Christiaan Huygens invented the pendulum clock in 1656, which used the regular period of a pendulum to keep time with unprecedented accuracy. The length of the pendulum was adjusted to set the clock's rate. Metronomes also use an adjustable pendulum to beat time for musicians.

Measuring Gravity and Surveying

Since the period depends on 'g', a pendulum can be used as a gravimeter to measure the local gravitational field. By precisely measuring the period of a pendulum of known length, geophysicists can detect variations in gravity, which can indicate underground mineral deposits or geological structures.

Foucault's pendulum is a famous experiment that uses a very long pendulum to demonstrate the rotation of the Earth.

Beyond the Basics: Factors Affecting Real Pendulums

The Effect of Amplitude
Physical Pendulums
Damping and Driving Forces

Large Amplitudes

The formula T = 2π sqrt(L/g) is an approximation that works well for small angles. As the initial angle (amplitude) increases, the period also increases. The exact calculation requires elliptic integrals and is significantly more complex. For an initial angle θ₀, the period is given by a series: T ≈ T_small (1 + (1/16)θ₀² + ...).

Physical vs. Simple Pendulums

A 'physical pendulum' (or compound pendulum) is any real swinging object, like a baseball bat or a swinging leg, where the mass is distributed. Its period depends on the moment of inertia and the distance from the pivot to the center of mass, not just its length.

Air Resistance and Friction

In the real world, forces like air resistance and friction at the pivot point cause the pendulum's oscillations to decrease over time. This effect is called 'damping'. To keep a pendulum swinging, like in a clock, a driving force must be applied to counteract the energy lost to damping.

Standard Earth Pendulum

Pendulum on the Moon

Grandfather Clock Pendulum