What Is a Pendulum Calculator?

A pendulum calculator helps you quickly determine the period, frequency, and length of a simple pendulum using the fundamental physics formula T = 2π√(L/g).

A simple pendulum consists of a mass (bob) suspended from a fixed point by a string or rod. When displaced from its equilibrium position, it swings back and forth in a periodic motion. The time it takes to complete one full swing is called the period (T).

Key Variables

Period (T)

Time for one complete oscillation, measured in seconds

Frequency (f)

Number of oscillations per second, measured in Hertz (Hz). f = 1/T

Length (L)

Distance from the pivot point to the center of the bob, measured in meters

Gravitational Acceleration (g)

Varies by planet. Earth: 9.807 m/s²

The Formula

The period of a simple pendulum depends only on its length and the local gravitational acceleration:

Calculate Period

T = 2π√(L/g)

Input: Length (L)
Output: Period in seconds

Calculate Frequency

f = 1/(2π√(L/g))

Input: Length (L)
Output: Frequency in Hz

Calculate Length

L = gT²/(4π²)

Input: Period (T)
Output: Length in meters

Important Note: These formulas assume small-angle oscillations (less than about 15°) where the simple pendulum approximation is valid.

How to Use the Pendulum Calculator

Choose What to Calculate

Select one of three solve modes at the top:

Period (T) — Enter the pendulum length to find the period and frequency
Frequency (f) — Enter the pendulum length to find the frequency and period
Length (L) — Enter the period to find the required pendulum length

Select a Planet

Choose a planet or celestial body from the dropdown to set the gravitational acceleration (g). Available options include Earth, Moon, Mars, Venus, Jupiter, Saturn, Mercury, and Sun. Select Custom to enter any value for g.

Enter Your Value

Type a number in the input field and choose your preferred unit. The calculator supports multiple units:

Length: m, cm, ft, in, km, yd
Time: s, ms, min
Frequency: Hz, mHz, kHz, rpm

View Results

Results appear instantly as you type:

Formula display — Shows the equation with your actual values
Result cards — Primary result highlighted, with secondary values below
Pendulum animation — Watch the pendulum swing at the calculated period
Unit conversions — All values shown in multiple units
Planet comparison — See how the period changes across all planets

Features

Three Calculation Modes

Switch between solving for Period (T), Frequency (f), or Length (L). Each mode adjusts the input field and formula display accordingly.

Multi-Planet Support

Compare pendulum behavior across 8 celestial bodies with preset gravitational values:

Earth (9.807 m/s²)
Moon (1.62 m/s²)
Mars (3.721 m/s²)
Venus (8.87 m/s²)
Jupiter (24.79 m/s²)
Saturn (10.44 m/s²)
Mercury (3.70 m/s²)
Sun (274.0 m/s²)
Custom gravity value

Animated Pendulum

A real-time canvas animation shows the pendulum swinging at the exact period calculated. You can pause and resume the animation at any time using the toggle button.

Unit Conversion Table

All calculated values are displayed in multiple units simultaneously:

Length: m, cm, ft, in, km, yd
Period: s, ms, min
Frequency: Hz, mHz, kHz, RPM

Planet Comparison Table

See how the period of your pendulum changes across all available planets at a glance. The currently selected planet is highlighted for easy comparison.

Real-Time Calculation

Results update instantly as you type or change any parameter — no need to press a calculate button.

Frequently Asked Questions

What is the formula for a simple pendulum?

The period of a simple pendulum is given by T = 2π√(L/g), where T is the period in seconds, L is the length in meters, and g is the gravitational acceleration in m/s².

The frequency is f = 1/T, and the length can be found using L = gT²/(4π²).

Does the mass of the bob affect the period?

No. For a simple pendulum with small oscillations, the period depends only on the length of the pendulum and the gravitational acceleration. The mass of the bob does not affect the period.

Key Insight: This is one of the most important properties of simple pendulums and is why they were historically used in clocks for accurate timekeeping.

Why does the pendulum swing faster on Jupiter?

Jupiter has a much stronger gravitational pull (24.79 m/s²) compared to Earth (9.807 m/s²). Since the period is inversely related to the square root of g, a higher gravitational acceleration results in a shorter period and faster oscillation.

Earth

Standard Gravity

g = 9.807 m/s²
Slower oscillation
Longer period

Jupiter

Strong Gravity

g = 24.79 m/s²
Faster oscillation
Shorter period

What is the small-angle approximation?

The formula T = 2π√(L/g) is accurate when the pendulum swings at small angles (typically less than about 15°). At larger angles, the actual period becomes longer than what this formula predicts. This calculator uses the small-angle approximation.

Important Limitation: For angles greater than 15°, the simple pendulum formula becomes less accurate. At 30°, the error is approximately 1.7%, and it increases significantly at larger angles.

How accurate are the results?

The calculator provides results with up to 6 significant digits for standard values and uses exponential notation for very large or very small numbers. The accuracy is limited only by the small-angle approximation assumption.

High precision calculations (6 significant digits)
Accurate for angles less than 15°
Supports exponential notation for extreme values

Can I use this for a real pendulum clock?

Yes. For example, a pendulum clock with a 1-second period (ticking once per second) requires a pendulum length of approximately 0.248 m (24.8 cm) on Earth. Use the Length mode and enter T = 1 s to find the exact length.

Practical Example: Traditional grandfather clocks typically use a pendulum with a 2-second period (1 second each way), requiring a length of approximately 0.994 m (99.4 cm).