What Is a Simple Pendulum?
A simple pendulum consists of a point mass (the bob) suspended from a fixed point by a massless, inextensible string. When displaced and released, it swings back and forth in a periodic arc. For small angles, the restoring force is approximately proportional to displacement, producing simple harmonic motion.
The simple pendulum is one of the most important models in physics. It demonstrates periodic motion, energy conservation, and the independence of period from mass — concepts foundational to mechanics, timekeeping, and wave theory.
Period Formula
The period T is the time for one complete back-and-forth oscillation. It depends only on the pendulum length L and the local gravitational acceleration g. Mass does not appear — heavy and light bobs swing at the same rate.
For larger amplitudes, the period increases. The series correction above (with θ₀ in radians) gives the exact period to better than 0.1% for angles up to 50°. At 15° the small-angle formula error is only ~0.5%.
Frequency and Angular Frequency
Frequency f counts how many oscillations occur per second (Hz). Angular frequency ω measures the rate in radians per second and appears directly in the SHM displacement equation x(t) = A cos(ωt).
Pendulum Energy
At maximum displacement, all energy is gravitational potential. At the lowest point, it is all kinetic. Total mechanical energy is conserved (ignoring friction). The height rise is h = L(1 − cos θ₀) and the maximum speed is v = √(2gh).
Effect of Length and Gravity
Since T ∝ √L, doubling the length multiplies the period by √2 ≈ 1.414. A seconds pendulum (T = 2.000 s) is approximately 0.994 m on Earth.
Since T ∝ 1/√g, stronger gravity shortens the period. On Jupiter (g = 24.79), a 1 m pendulum swings about 2.5× faster than on the Moon (g = 1.62).
Pendulum on Other Planets
| Body | g (m/s²) | T for 1 m |
|---|---|---|
| Earth | 9.81 | 2.01 s |
| Moon | 1.62 | 4.93 s |
| Mars | 3.71 | 3.27 s |
| Jupiter | 24.79 | 1.26 s |
| Pluto | 0.62 | 7.99 s |
Real-World Applications
- Grandfather clocks — a seconds pendulum (L ≈ 1 m) ticks once per second.
- Metronomes — adjustable weight varies effective length to change tempo.
- Seismic instruments — long-period pendulums detect slow ground motion.
- Measuring g — Galileo’s experiment: time many swings and solve for g.
- Foucault pendulum — demonstrates Earth’s rotation.
How to Use the Calculator
- Select a calculation mode (period, frequency, length, gravity, energy, etc.).
- Enter known values with units. Use gravity presets for other planets.
- Click Calculate.
- Review results, formula steps, and large-angle corrections.
Example Calculations
1 m on Earth
T = 2π√(1/9.807) ≈ 2.006 s, f ≈ 0.499 Hz
2 m on Earth
T = 2π√(2/9.807) ≈ 2.837 s
1 m on the Moon (g=1.62)
T = 2π√(1/1.62) ≈ 4.93 s
Seconds pendulum
L = 9.807 × (2/2π)² ≈ 0.994 m
Common Mistakes
- Confusing period (one full cycle) with half-period (one swing).
- Assuming mass affects the period — it does not.
- Using large angles (> 15°) without the amplitude correction.
- Mixing units (e.g., length in cm but expecting seconds).
- Forgetting that gravity differs by location and planet.
Accuracy and Limitations
The simple pendulum model assumes: (1) a point mass, (2) a massless inextensible string, (3) no air resistance, (4) small swing angles. Real pendulums deviate due to air drag, string mass and elasticity, large amplitudes, and pivot friction. For amplitudes above 15°, use the large-angle correction provided. This tool is for education and estimation, not precision engineering.
FAQ
What is a simple pendulum?›
A simple pendulum is an idealized model: a point mass (bob) on a massless, inextensible string swinging under gravity. Real pendulums approximate this when the string is light and the bob is compact.
How do you calculate pendulum period?›
Use T = 2π√(L/g), where L is the length from pivot to the centre of mass, and g is gravitational acceleration. This formula assumes small oscillation angles.
Does mass affect the pendulum period?›
No. Mass cancels out in the derivation. A 1 kg bob and a 10 kg bob on identical strings have identical periods. This was demonstrated by Galileo.
Why does pendulum length matter?›
Period is proportional to √L. Doubling the length increases the period by a factor of √2 ≈ 1.414. A seconds pendulum (T = 2 s) is about 0.994 m long on Earth.
What is angular frequency?›
ω = 2πf = √(g/L). It describes how many radians per second the pendulum traverses. It’s used in the SHM equation x(t) = A cos(ωt).
What is pendulum amplitude?›
Amplitude is the maximum displacement from equilibrium — typically measured as the arc length or the angle θ₀ from vertical. For the small-angle formula to be accurate, keep θ₀ below about 15°.
How does gravity affect a pendulum?›
Stronger gravity makes the pendulum swing faster (shorter T). On the Moon (g = 1.62 m/s²), a 1 m pendulum has T ≈ 4.93 s vs 2.01 s on Earth.
Why are small angles important?›
The derivation linearises sinθ ≈ θ (in radians), which is only accurate for small θ. At 15° the error is about 0.5%; at 45° it exceeds 5%.
What happens on the Moon?›
The Moon’s weaker gravity (1.62 m/s²) makes pendulums swing much more slowly. A 1 m pendulum takes about 4.93 seconds per swing.
How accurate is the simple pendulum model?›
Very accurate for small angles in vacuum. Real-world deviations come from air resistance, string mass, large amplitudes, and friction at the pivot.
Sources
Author & technical reviewer
Manish Kumar
PhysicsCalcs tools are reviewed with an educational focus: clear formulas, transparent assumptions, and practical context for students and science learners.
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