Pendulum Calculator

Simple pendulum with length L, mass m, and angle θ
Period equals 2 pi times the square root of length divided by gravity

Solution

—
Share:

Simple Pendulum Period

The period of a simple pendulum depends only on the string length and local gravity. A longer string swings more slowly; stronger gravity speeds the swing.

T = 2π√(L/g)

Simple Pendulum Frequency

Frequency is the reciprocal of period. It tells you how many complete swings per second the pendulum makes under the small-angle approximation.

f = 1/(2π) × √(g/L)

Physical Pendulum Period

A physical pendulum is any rigid body swinging about a pivot. Its period depends on the moment of inertia, mass, gravity, and distance from the pivot to the center of mass.

T = 2π√(I/(Mgd))

How It Works

A simple pendulum’s period (T = 2π√(L/g)) depends only on the string length and local gravity — not on the mass of the bob or the swing angle (for small angles). A longer string swings more slowly; stronger gravity makes it swing faster. The frequency is simply f = 1/T.

Example Problem

A grandfather clock has a pendulum 1 m long. What is its period on Earth (g = 9.81 m/s²)?

  1. Identify the knowns. Pendulum length L = 1 m (measured from the pivot to the center of mass of the bob) and local gravitational acceleration g = 9.81 m/s².
  2. Identify what we're solving for. We want the period T — the time in seconds for one complete back-and-forth swing under the small-angle approximation.
  3. Write the simple pendulum formula: T = 2π × √(L / g). Note that mass and swing amplitude do not appear.
  4. Substitute the known values: T = 2π × √(1 m / 9.81 m/s²).
  5. Simplify the ratio inside the square root: 1 / 9.81 = 0.10193 s². Then take the root: √0.10193 = 0.31928 s.
  6. Multiply by 2π: T = 2π × 0.31928 ≈ 6.2832 × 0.31928 ≈ **2.006 s** — almost exactly a 2-second period, which is by design for tall-case clocks where each tick (half-period) takes about 1 second.

When to Use Each Variable

  • Solve for Period (Simple) — when you know the string length and gravity, e.g., designing a clock pendulum with a specific tick rate.
  • Solve for Length (Simple) — when you know the desired period and gravity, e.g., determining the string length for a 1-second pendulum.
  • Solve for Gravity (Simple) — when you know the period and length, e.g., using a pendulum to measure local gravitational acceleration.
  • Solve for Frequency — when you need the oscillation rate in Hz instead of the period, e.g., matching a pendulum to a timing signal.
  • Solve for Period (Physical) — when the pendulum is a rigid body and you know its moment of inertia, mass, and pivot-to-CM distance.

Key Concepts

A simple pendulum's period depends only on string length and gravitational acceleration — not on mass or amplitude (for small angles). The small-angle approximation (sin(theta) ~ theta) makes the motion simple harmonic. A physical (compound) pendulum replaces the point mass with a rigid body, so its period also depends on the moment of inertia and the distance from the pivot to the center of mass.

Applications

  • Timekeeping: grandfather clocks use a ~1 m pendulum to produce a precise 2-second period
  • Geophysics: measuring local gravitational acceleration using a calibrated pendulum
  • Seismology: early seismometers used long pendulums to detect ground motion from earthquakes
  • Education: demonstrating simple harmonic motion and the independence of period from mass

Common Mistakes

  • Using the simple pendulum formula for large swing angles — the formula T = 2pi*sqrt(L/g) is only accurate for angles below about 15 degrees; larger angles increase the period
  • Measuring length to the bottom of the bob instead of to its center of mass — the effective length is from the pivot to the center of mass of the bob
  • Applying the simple pendulum formula to a physical pendulum — a rod or plate swinging about an end requires the physical pendulum equation that includes moment of inertia

Frequently Asked Questions

Does the mass of a pendulum affect its period?

No. For a simple pendulum, the period depends only on length and gravity. A heavier bob swings at the same rate as a lighter one (assuming the same string length).

Why does the small-angle approximation matter?

The formula T = 2π√(L/g) is exact only for infinitesimally small swings. For angles up to about 15° the error is less than 0.5%, but for large swings the period increases and the formula becomes less accurate.

Can you use a pendulum to measure gravity?

Yes. Measure the period T and length L, then solve for g = 4π²L/T². This method was historically used to map variations in gravity across the Earth.

What equation relates period, length, and gravity for a simple pendulum?

T = 2π√(L/g) connects all three. Period is in seconds, length in meters, and g in meters per second squared. Rearrange to solve for L (= g(T/2π)²) or g (= 4π²L/T²) when those are unknown.

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

A simple pendulum is an idealized point mass on a massless string; only L and g matter. A physical (compound) pendulum is any rigid body — a rod, a plate, a swinging leg — whose period depends on its moment of inertia I, total mass M, and the pivot-to-center-of-mass distance d via T = 2π√(I/(Mgd)).

How does pendulum length affect the period?

Period scales with the square root of length, so quadrupling L only doubles T. To halve the period of a clock pendulum you must shorten the rod to one quarter of its original length — which is why clockmakers use small adjustment nuts on the bob to make fine timing changes.

Why are pendulum clocks less accurate at high altitude or near the equator?

Local gravity g is slightly lower at the equator (Earth's rotation and equatorial bulge) and decreases with altitude. Lower g lengthens the period, so an unadjusted clock runs slow. 19th-century chronometers shipped with a regulator nut to retune the pendulum length when moved to a new latitude.

Reference:

Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.

Worked Examples

Civil Engineering

How long does a 25 m Foucault pendulum take to complete one swing?

A Foucault pendulum hangs from a 25 m cable in a museum atrium. On Earth (g = 9.81 m/s²), what is its period T = 2π√(L/g)? This is also the slow heartbeat that times its rotation against the floor markings.

  • Knowns: L = 25 m, g = 9.81 m/s²
  • T = 2π × √(L / g)
  • T = 2π × √(25 / 9.81)
  • T = 2π × √(2.548) ≈ 2π × 1.5964

T ≈ 10.03 seconds

Big Foucault pendulums use long cables to slow the swing — the longer T, the easier it is to see Earth's rotation deflect the swing plane (about 9.4°/h at 40° latitude). The Smithsonian's original 1851 installation used a 20 m cable for the same reason.

Biomechanics

What is the natural swing frequency of a human leg modeled as a simple pendulum?

Biomechanics models a relaxed walking leg as a simple pendulum of length L ≈ 0.9 m hanging from the hip joint. On Earth, what frequency does this give from f = (1/2π)√(g/L)? This is the cadence the body naturally falls into when walking.

  • Knowns: L = 0.9 m, g = 9.81 m/s²
  • f = (1 / 2π) × √(g / L)
  • f = (1 / 2π) × √(9.81 / 0.9)
  • f = (1 / 2π) × √(10.9) ≈ (1 / 2π) × 3.302

f ≈ 0.526 Hz (about 1 step every 1.9 s per leg, ~63 steps/min total)

Measured human walking cadence at preferred speed clusters around 110–120 steps/min total — higher than the pure-pendulum prediction because muscles add active force. The pendulum analysis explains why walking with weights on your ankles changes your natural cadence so much.

Clockmaking

How long is the pendulum in a seconds-tick mantel clock?

A traditional seconds pendulum is one that ticks once per second — half-period of 1 s, so full period T = 2 s. On Earth (g = 9.81 m/s²), what length L = g × (T/(2π))² does that require? Pre-metric the result was tantalizingly close to 1 m, and was even briefly proposed in 1791 as the definition of the meter.

  • Knowns: T = 2 s, g = 9.81 m/s²
  • L = g × (T / (2π))²
  • L = 9.81 × (2 / (2π))² = 9.81 × (0.31831)²
  • L = 9.81 × 0.10132

L ≈ 0.994 m (≈ 994 mm)

The Académie des sciences proposed exactly this length as the metric unit of length before the meridian-arc definition won out. Every astronomical regulator clock built in the 19th century has a rod tuned to within a few mm of this value.

Pendulum Formulas

The pendulum calculator covers three related equations that describe oscillating swinging motion under gravity:

T = 2π × √(L / g)Simple pendulum period
f = (1 / 2π) × √(g / L)Simple pendulum frequency
T = 2π × √(I / (M·g·d))Physical (compound) pendulum period

Where:

  • T — period, time for one complete swing (seconds)
  • f — frequency, swings per second (hertz)
  • L — pendulum length, pivot to center of mass of the bob (meters)
  • g — local gravitational acceleration (m/s²); ≈ 9.81 m/s² on Earth's surface
  • I — moment of inertia of the rigid body about the pivot (kg·m²)
  • M — total mass of the rigid body (kg)
  • d — distance from pivot to the body's center of mass (meters)

The simple pendulum formula assumes a point-mass bob on a massless string with small swing angles (under ~15°). At larger amplitudes the period increases and an elliptic-integral correction is needed. The physical pendulum formula applies to any rigid body — rods, plates, swinging limbs — and reduces to the simple pendulum when I = ML² and d = L.

Related Calculators

Related Sites