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)?
- T = 2π√(1 / 9.81) = 2π × 0.3193
- T ≈ 2.006 s — almost exactly a 2-second period, which is by design.
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.
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
Related Calculators
- Gravity Equations Calculator — calculate gravitational acceleration at different altitudes.
- Hooke's Law Calculator — analyze another oscillating system (springs).
- Circular Motion Calculator — study periodic motion in a circular path.
- Potential Energy Calculator — calculate the gravitational PE at the pendulum's peak.
- Kinetic Energy Calculator — find the KE at the pendulum's lowest point.
- Time Converter — convert period between seconds, minutes, and hours.
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