Last Updated: October 30, 2025
Calculate the period of a simple pendulum from length and gravitational acceleration using T = 2π √(L/g). Essential tool for physics oscillatory motion, timekeeping, and understanding harmonic motion principles. Perfect for students, educators, and engineers analyzing pendulum dynamics and small-angle approximations.
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The Simple Pendulum Calculator is a specialized physics tool that calculates the period of oscillation for a simple pendulum. A simple pendulum is an idealized model consisting of a point mass (bob) suspended by a massless, inextensible string under the influence of gravity. This system exhibits simple harmonic motion for small angular displacements.
This calculator is essential for analyzing oscillatory motion, understanding harmonic systems, and designing pendulum-based timing devices. It provides accurate calculations for the period of oscillation using the relationship between pendulum length and gravitational acceleration. For small angles (typically ≤ 10°), the period is independent of amplitude and mass, making it an ideal system for studying periodic motion.
Period (T): Time for one complete oscillation (swing from one side to the other and back), measured in seconds (s). Independent of mass and amplitude (for small angles), but depends on length and gravity. The period determines how fast the pendulum oscillates.
Length (L): Distance from the pivot point to the center of mass of the bob, measured in meters (m). Longer pendulums have longer periods. The period is proportional to the square root of length, so doubling length increases period by √2 ≈ 1.41.
Gravitational Acceleration (g): Acceleration due to gravity, measured in meters per second squared (m/s²). On Earth, g ≈ 9.81 m/s², but varies with location and altitude. Higher gravity produces shorter periods. The period is inversely proportional to the square root of gravity.
Angular Displacement (θ): Angle from vertical, measured in radians or degrees. For small angles (≤ 10°), the simple harmonic motion approximation is valid. Larger angles require corrections using elliptic integrals.
Amplitude: Maximum angular displacement from equilibrium. For small amplitudes, period is independent of amplitude. For large amplitudes, period increases slightly.
A simple pendulum oscillates because gravity provides a restoring force that tends to bring it back to equilibrium. When displaced, the bob experiences a component of gravitational force perpendicular to the string, creating a restoring torque. For small angles, this force is approximately proportional to the displacement, resulting in simple harmonic motion.
The formula T = 2π√(L/g) shows that period increases with length but decreases with gravitational acceleration. This explains why pendulums swing slower with longer strings and why pendulum clocks need different calibration for different locations (g varies with latitude and altitude). The period is independent of mass because both the restoring force and the inertia scale with mass, canceling out in the equation of motion.
The simple pendulum equation linearizes for small angles, where sin(θ) ≈ θ. This approximation is valid for angles typically ≤ 10°. For larger angles, the exact solution involves elliptic integrals, and the period increases with amplitude. The small-angle approximation makes the system analytically solvable and allows the period to be independent of amplitude—a key characteristic of simple harmonic motion.
The simple pendulum has a rich history, dating back to Galileo's observations of chandeliers in the 16th century. Galileo noted that the period is independent of amplitude (for small angles) and mass, and discovered the relationship between length and period. This led to the development of pendulum clocks, which were the most accurate timekeeping devices for centuries. The understanding of pendulum motion was crucial for navigation, physics education, and the development of classical mechanics.
Period: s (seconds) - typically ranges from fractions of a second to several seconds
Length: m (meters) - common values: 0.1 m to 10 m
Gravitational Acceleration: m/s² - Earth: 9.81 m/s², Moon: 1.62 m/s², Mars: 3.71 m/s²
Relationship: T = 2π√(L/g), where T is in seconds, L in meters, g in m/s²
Frequency: f = 1/T = (1/(2π))√(g/L) in Hz
T = 2π √(L/g)
Where:
Use case: Calculate the period of oscillation for a simple pendulum from its length and gravitational acceleration. This formula is valid for small angular displacements (typically ≤ 10°). The period is independent of mass and amplitude (for small angles). Essential for pendulum design, clock mechanisms, and physics education.
f = (1/(2π)) √(g/L)
Where:
Use case: Calculate the frequency of oscillation, which is the reciprocal of period (f = 1/T). Frequency represents the number of complete oscillations per second. Useful when analyzing periodic motion and when frequency measurements are available. Higher frequencies correspond to shorter periods.
L = g (T/(2π))²
Where:
Use case: Calculate the required pendulum length to achieve a specific period. Useful for designing pendulum clocks, timing devices, and experimental setups where a target period is desired. Rearranged from T = 2π√(L/g), this formula allows engineers to design pendulums with specific timing characteristics.
ω = √(g/L)
Where:
Use case: Calculate angular frequency, which relates to period by ω = 2π/T. Angular frequency measures the rate of phase change in radians per second and appears naturally in the differential equation of motion. Useful for mathematical analysis and when connecting to other oscillatory systems.
Simple pendulum calculations are essential across numerous physics, engineering, and timekeeping fields. Here's a comprehensive overview of practical applications:
| Industry | Applications | Importance |
|---|---|---|
| Timekeeping | Pendulum clocks, chronometers, timing devices, metronomes, historical timekeeping systems | Critical for accurate time measurement and clock design |
| Physics Education | Oscillatory motion demonstrations, simple harmonic motion studies, gravity measurements, physics experiments | Essential for teaching periodic motion and harmonic systems |
| Geophysics | Gravitational field measurements, gravity surveys, geodetic measurements, local gravity determination | Important for measuring local variations in gravitational acceleration |
| Mechanical Engineering | Vibration analysis, oscillation damping, structural dynamics, resonant system design | Fundamental for understanding oscillatory behavior in mechanical systems |
| Seismology | Seismometer design, ground motion analysis, vibration isolation, earthquake detection | Vital for earthquake detection and ground motion measurement |
| Entertainment & Design | Swing design, amusement park rides, kinetic sculptures, decorative pendulums, metronomes | Important for timing and rhythmic applications |
The equation m L θ¨ + m g sin θ = 0 linearizes to L θ¨ + g θ = 0 for small θ, giving SHM with ω = √(g/L) and T = 2π √(L/g). For large angles, the exact period involves elliptic integrals and exceeds the small-angle result by a factor that depends on amplitude.
Since T ∝ 1/√g, lower gravity (Moon, Mars) yields longer periods for the same L. Precision pendulum clocks historically corrected for temperature and local gravity variations to keep accurate time.
Total mechanical energy alternates between gravitational potential and kinetic forms. At maximum displacement, energy is all potential; at equilibrium crossing, it is all kinetic. In ideal conditions, the period is amplitude-independent for small angles.
Use small release angles to preserve the small-angle approximation. Minimize friction at the pivot and aerodynamic drag for consistent timing. If the string has significant mass or elasticity, corrections are needed.
Measure the time for N oscillations and divide by N to reduce reaction-time errors. Calibrate g by measuring T for a precisely known L, or find L for a measured T. Compare results across different amplitudes to observe the amplitude dependence.
Given:
Step 1: Use T = 2π √(L/g)
T = 2π √(1/9.81) ≈ 2.007 s
Final Answer
≈ 2.007 s
This represents the period of a 1-meter pendulum on Earth
Given:
T = 2π √(0.25/9.81) ≈ 1.002 s
Final Answer
≈ 1.002 s
Shorter pendulums oscillate faster with smaller periods
Given:
T = 2π √(2.25/9.81) ≈ 3.010 s
Final Answer
≈ 3.010 s
Longer pendulums have longer periods due to the square root dependence on length
Given:
T = 2π √(1/1.62) ≈ 4.935 s
Final Answer
≈ 4.935 s
Lower gravity on the Moon results in longer periods for the same pendulum length
Given:
T = 2π √(1/3.71) ≈ 3.273 s
Final Answer
≈ 3.273 s
Mars gravity produces intermediate periods between Earth and Moon values
Target:
Rearrange: L = g (T/2π)² = 9.81 × (2/6.283)^2 ≈ 0.994 m
Required Length
≈ 0.994 m
This is the pendulum length needed to achieve a 2-second period on Earth
Disclaimer: The calculators and tools are for educational purposes. Verify results independently before professional use.