What Is a Simple Pendulum?
A simple pendulum is a mass (bob) suspended from a pivot by a massless, inextensible string. When displaced from its equilibrium position, it swings back and forth under gravity. For small angles, the motion is approximately simple harmonic.
T = 2π√(L/g)
The period T depends only on the string length L and gravitational acceleration g. Remarkably, it does not depend on the bob’s mass or the amplitude (for small oscillations).
Pendulum Diagram
A simple pendulum: mass m hangs from a fixed pivot by a string of length L. The angle θ measures the displacement from vertical.
Pendulum on Other Planets
| Body | g (m/s²) | 1 m pendulum T |
|---|---|---|
| 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 |
How to Use the Calculator
- Select the calculation mode.
- Enter length, period, gravity, or oscillation count.
- Click Calculate.
- Review period, frequency, and related values.
Example Calculations
1 m pendulum on Earth
T = 2π√(1/9.81) = 2.006 s
Seconds pendulum (T = 2 s)
L = 9.81 × 4 / (4π²) = 0.994 m
1 m on Moon (g = 1.62)
T = 2π√(1/1.62) = 4.93 s (2.46× Earth)
Common Mistakes
- Using the formula for large angles (> 15°) without correction.
- Measuring to the bottom of the bob instead of its centre of mass.
- Thinking mass or amplitude changes the period.
- Confusing period (time for full swing) with half-period (time to one side).
Accuracy and Limitations
T = 2π√(L/g) is the small-angle approximation. For angles above 15°, correction terms are needed: T ≈ T₀(1 + θ²/16 + ...). The formula also assumes a point mass, rigid support, massless string, and no air drag.
FAQ
Does mass affect pendulum period?›
No. For a simple pendulum with small oscillations, the period depends only on length and gravity, not mass. This was first noted by Galileo.
Does amplitude affect the period?›
For small angles (< ~15°), the period is nearly constant. For large angles, the period increases slightly. This is the ‘small-angle approximation’.
What is a seconds pendulum?›
A pendulum with a 2-second period (1 second each way). On Earth, its length is approximately 0.994 m (~1 metre).
How does a pendulum measure gravity?›
Measure the pendulum’s length and period, then compute g = 4π²L/T². This method was historically used to measure g before modern accelerometers.
Would a pendulum work in space?›
No. A pendulum requires gravity to provide the restoring force. In microgravity (ISS), a pendulum would not swing.
Why is the period longer on the Moon?›
The Moon’s gravity (1.62 m/s²) is weaker than Earth’s (9.81 m/s²). Since T ∝ 1/√g, weaker gravity means longer periods.
What about a physical (compound) pendulum?›
For a real pendulum with distributed mass, the formula uses the moment of inertia: T = 2π√(I/(mgh)). The simple pendulum formula is a special case.
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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