Hooke's Law Calculator
Calculate spring force, stiffness, or displacement using Hooke's Law F = kx. Supports series and parallel spring configurations with oscillation frequency.
Pendulum Frequency Calculator
Calculate pendulum frequency, period, and angular velocity for simple and physical pendulums. Includes large-angle correction, energy analysis, and period vs length chart.
m
°
kg
m/s²
Period⧉
2.00989 s
Small-angle approx: 2.00607 s
Frequency⧉
0.4975 Hz
29.9 beats/min
Angular Frequency⧉
3.1261 rad/s
ω = 2πf
Large-Angle Correction⧉
+0.191%
Difference from small-angle T
Max Speed at Bottom⧉
0.546 m/s
From energy conservation
Max Height⧉
1.52 cm
Vertical rise from lowest point
Max PE / Max KE⧉
0.1490 J
Energy alternates between PE and KE
Equivalent Simple Length⧉
1.0000 m
Same as length
Period vs Length (small angle)
0.25 m
1.003 s
0.50 m
1.419 s
0.75 m
1.737 s
1.00 m
2.006 s
1.25 m
2.243 s
1.50 m
2.457 s
1.75 m
2.654 s
2.00 m
2.837 s
2.25 m
3.009 s
2.50 m
3.172 s
T = 2π√(L/g) — period grows as √L
Pendulum Milestones
| Item | Year | Significance |
|---|---|---|
| Galileo's observation | 1602 | Discovered isochronism of small swings |
| Huygens' pendulum clock | 1656 | First accurate timekeeping device |
| Seconds pendulum (994 mm) | 1660s | 1 s half-period, proposed length standard |
| Newton's experiments | 1687 | Used pendulums to verify equivalence of inertial/gravitational mass |
| Foucault's pendulum | 1851 | Proved Earth rotation directly |
| National Bureau of Standards | 1920s | Quartz clocks supersede pendulum clocks |
Planning notes, formulas, and examples
About the Pendulum Frequency Calculator
A simple pendulum is a mass swinging on a string under gravity, and its frequency depends mainly on length and local gravitational acceleration: f = (1/2π)√(g/L). That relationship makes pendulums a clean way to study periodic motion and to estimate gravity from timing measurements.
This Pendulum Frequency Calculator handles both simple and physical pendulums. It computes period, frequency, angular frequency, large-angle correction, maximum speed, and the equivalent simple-pendulum length for rigid bodies. The presets are there to show how a short lab pendulum, a clock pendulum, and a longer pendulum all change with length and angle.
When This Page Helps
Pendulum problems are useful whenever timing depends on length, gravity, or pivot location. This calculator helps with lab reports, clock design, and rigid-body pendulum examples by showing the period and frequency directly instead of leaving you to rearrange the formula by hand. It is also helpful for checking when the small-angle approximation is still good enough.
How to Use the Inputs
- Choose a pendulum type: simple or physical (compound).
- Select a preset or enter the pendulum length, initial angle, mass, and gravity.
- For physical pendulums, also enter the moment of inertia and pivot-to-COM distance.
- Read the period, frequency, angular frequency, and large-angle correction.
- Check max speed at the bottom and energy values for the given amplitude.
- Explore the period-vs-length chart and pendulum milestones table.
Formula used
Simple Pendulum (small angle):
T₀ = 2π √(L / g)
f = 1 / T = (1/2π) √(g / L)
Large-Angle Correction:
T ≈ T₀ [1 + ¼ sin²(θ₀/2) + 9/64 sin⁴(θ₀/2) + …]
Physical Pendulum:
T = 2π √(I / (m g d))
Equivalent simple length: L_eq = I / (m d)
Max Speed at Bottom:
v_max = √(2gL(1 − cos θ₀))
Where:
L = string length (m)
g = gravitational acceleration (m/s²)
θ₀ = initial angle (rad)
I = moment of inertia (kg·m²)
d = pivot-to-center-of-mass distance (m)Example Calculation
Result: T ≈ 2.00709 s, f ≈ 0.4982 Hz
A 1-meter pendulum at 10° amplitude has a small-angle period of 2.00607 s. The large-angle correction adds about 0.05% (0.001 s), giving T ≈ 2.00709 s. The frequency is just under 0.5 Hz — very close to the "seconds pendulum" used in grandfather clocks.
Tips & Best Practices
- For quick estimates, a 1-meter pendulum has a period of about 2 seconds.
- Period scales as √L: to double the period, you need about four times the length.
- The large-angle correction matters most once the swing angle gets beyond roughly 15°.
- Measured period and length can be used to estimate local gravity with g = 4π²L/T².
- Air resistance and pivot friction cause damping, which shortens the swing over time.
History of the Pendulum Clock
Christiaan Huygens built the first pendulum clock in 1656, achieving accuracy of about 15 seconds per day — a 60× improvement over the best spring-driven clocks. He also discovered that (for small angles) the pendulum period is independent of amplitude, a property called isochronism. Pendulum clocks dominated timekeeping until quartz oscillators replaced them in the 1930s.
Damped and Driven Pendulums
Real pendulums lose energy to air resistance and pivot friction. A clock maintains the swing by providing small energy impulses from an escapement mechanism. The quality factor Q quantifies how many oscillations occur before the amplitude decays to 1/e. A good clock pendulum has Q ~ 300–500; the Shortt free-pendulum clock achieved Q ~ 100,000.
Chaos and the Double Pendulum
A single pendulum is one of the simplest dynamical systems. Adding a second pendulum to the end of the first creates the double pendulum — a classic example of deterministic chaos. Despite following simple equations, the double pendulum exhibits unpredictable, wildly different trajectories from nearly identical initial conditions, making it a favorite demonstration in nonlinear dynamics.
Sources & Methodology
Last updated:
Frequently Asked Questions
For a simple pendulum, no. Period depends only on length and gravity, not mass. This is why Galileo observed that pendulums of different masses swing at the same rate. For a physical pendulum, mass appears in the formula but is coupled with the moment of inertia.
At 10° amplitude, the error is about 0.05%. At 30°, it is about 1.7%. At 60°, the error exceeds 7%. The correction series converges quickly for angles below 20°.
A seconds pendulum has a half-period of exactly 1 second (full period of 2 seconds), requiring a length of about 99.4 cm at sea level. It was historically proposed as a natural length standard.
The pendulum's plane of swing appears to rotate slowly (in Paris, about 11°/hour). This is because the pendulum maintains its oscillation plane while Earth rotates beneath it. At the poles, the rotation is 360°/day; at the equator, there is no apparent rotation.
A pendulum needs gravity to produce the restoring force. In microgravity it will not swing normally, while on the Moon it will oscillate more slowly because g is smaller.
A simple pendulum is an idealized point mass on a massless string. A physical pendulum is a rigid body pivoted at a point other than its center of mass. The period of a physical pendulum depends on the moment of inertia and the distance from the pivot to the center of mass.
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