2.006067
s
2.007022
s
0.0476
%
1
m
100
cm
9.81
m/s²
0.498488
Hz
1.003033
s
2.006067
s
2.007022
s
0.0476
%
1
m
100
cm
9.81
m/s²
0.498488
Hz
1.003033
s
The Pendulum Period Calculator is a focused tool for computing the period of a simple pendulum, or inversely determining the required length or local gravity from a measured period. It includes an amplitude correction for improved accuracy with larger swing angles.
The period of a simple pendulum—the time for one complete back-and-forth oscillation—is given by the classic formula: $$T = 2\pi\sqrt{\frac{L}{g}}$$ This equation, derived from Newton’s second law under the small-angle approximation, reveals that the period depends only on the string length L and the local gravitational acceleration g. It is independent of both the mass of the bob and the amplitude of oscillation (for small angles).
The versatility of this formula makes it useful in three directions. Forward calculation: given L and g, find the period. This is the standard physics problem. Length determination: given a desired period and known gravity, calculate the required length. This is essential for clock design. Gravity measurement: given a measured period and known length, extract the local value of g. This is one of the oldest and most elegant methods of measuring gravitational acceleration.
For larger amplitudes, the small-angle approximation introduces error. The exact period involves a complete elliptic integral of the first kind, but an excellent approximation uses the series expansion: $$T \approx T_0\left(1 + \frac{1}{4}\sin^2\frac{\theta_0}{2} + \frac{9}{64}\sin^4\frac{\theta_0}{2} + \cdots\right)$$ At 15°, the correction is about 0.43%. At 30°, it’s about 1.7%. At 60°, it’s about 7.3%. This calculator displays both the small-angle and corrected periods so you can assess the impact of amplitude.
The half-period (the time from one extreme to the other) is also provided, as this is the “tick” interval relevant for pendulum clocks. The seconds pendulum, with a half-period of exactly 1 second (full period of 2 seconds), has a length of approximately 0.994 m and played a historic role in the definition of the meter.
This calculator is invaluable for physics students, clock makers, geophysicists measuring local gravity variations, and engineers designing pendulum-based instruments such as seismometers and accelerometers.
The calculator uses three rearrangements of the pendulum period formula:
Solve for Period:
$$T = 2\pi\sqrt{\frac{L}{g}}$$
Solve for Length:
$$L = g\left(\frac{T}{2\pi}\right)^2 = \frac{gT^2}{4\pi^2}$$
Solve for Gravity:
$$g = \frac{4\pi^2 L}{T^2}$$
Amplitude Correction:
$$T_{corrected} = T_0\left(1 + \frac{1}{4}\sin^2\frac{\theta_0}{2} + \frac{9}{64}\sin^4\frac{\theta_0}{2}\right)$$
The correction percentage shows how much the true period exceeds the small-angle estimate.
The small-angle period is the textbook value valid for amplitudes below about 15°. The corrected period is more accurate for larger swings. The correction percentage quantifies the error of the simple formula. When solving for gravity, the result should be close to 9.81 m/s² at sea level; significant deviations indicate measurement error or unusual local geology.
Inputs
Results
For a clock with 1-second ticks (2 s period) at standard gravity, the pendulum must be 99.37 cm long. At 3° amplitude, the correction is negligible (0.034%).
Inputs
Results
A 75 cm pendulum with a measured period of 1.738 s gives g ≈ 9.806 m/s², very close to the standard value. The 10° amplitude introduces a 0.38% period correction.
The formula T = 2π√(L/g) is accurate to within 0.5% for amplitudes up to about 15°. At 23°, the error reaches 1%. At 45°, it’s about 4%. At 90°, the error is about 18%. For most educational and practical purposes, amplitudes below 15° are considered “small.” The amplitude correction in this calculator extends useful accuracy to about 70°.
The gravitational force on the pendulum is F = mg·sinθ, and by Newton’s second law, acceleration is a = F/m = g·sinθ. The mass cancels, so the motion (and hence the period) is the same for any mass. This is a consequence of the equivalence of gravitational and inertial mass, a principle that Einstein elevated to a foundation of general relativity.
At higher altitude, gravitational acceleration decreases (g ≈ 9.81·(1 − 2h/R₂) approximately), so the pendulum period increases. At 1000 m above sea level, g decreases by about 0.03%, lengthening the period by about 0.015%. A pendulum clock at high altitude runs slightly slow compared to one at sea level.
Yes. Simply enter the appropriate gravitational acceleration: 1.62 m/s² for the Moon, 3.72 m/s² for Mars, 24.79 m/s² for Jupiter. A 1 m pendulum that has a 2 s period on Earth would have a period of about 4.93 s on the Moon.
The half-period is the time for the pendulum to swing from one extreme to the other—one “tick” or “tock” of a pendulum clock. A seconds pendulum has a half-period of exactly 1 second. Clock mechanisms typically advance the gear train once per half-period, so this is the fundamental timing interval in horology.
Air resistance (damping) causes the amplitude to decrease over time but has very little effect on the period of a simple pendulum. The period of a lightly damped pendulum is T_damped = T₀/√(1 − γ²/ω²), which is extremely close to T₀ for typical damping levels. The amplitude decay, not the period change, is the main practical effect of air resistance.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
Simple Harmonic Motion Calculator
Oscillations & Waves Calculators
Spring Calculator
Oscillations & Waves Calculators
Spring Constant Calculator
Oscillations & Waves Calculators
Hooke's Law Calculator
Oscillations & Waves Calculators
Pendulum Calculator
Oscillations & Waves Calculators
Pendulum Frequency Calculator
Oscillations & Waves Calculators
