Calculate pendulum frequency, period, and angular velocity for simple and physical pendulums. Includes large-angle correction, energy analysis, and period vs length chart.
A simple pendulum — a mass on a string swinging under gravity — produces periodic motion with a frequency determined almost entirely by its length and the local gravitational acceleration: f = (1/2π)√(g/L). This elegant relationship, discovered by Galileo and formalized by Huygens, was the basis of accurate timekeeping for over two centuries and remains a cornerstone of introductory physics.
This Pendulum Frequency Calculator handles both simple and physical (compound) pendulums, computing period, frequency, angular frequency, large-angle corrections, maximum speed, energy exchange, and the equivalent simple pendulum length. Presets range from a classic 1-meter pendulum to the famous 67-meter Foucault pendulum and a hypothetical Moon pendulum. The period-vs-length chart illustrates the square-root relationship, and a historical milestones table traces the pendulum's role in science.
Whether you're a physics student verifying lab results, an engineer designing a pendulum mechanism, or a clock enthusiast, this calculator provides a complete pendulum analysis toolkit.
The pendulum is the simplest periodic oscillator in mechanics and appears in clocks, seismometers, accelerometers, and physics demonstrations. This calculator goes beyond the basic formula to show large-angle effects, energy conservation, and physical pendulum analysis — making it useful for real-world applications. Keep these notes focused on your operational context.
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)
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.
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.
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.
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.
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 work. In Earth orbit (microgravity), it would not oscillate. On the Moon, with g ≈ 1.62 m/s², a pendulum would swing about 2.5 times slower than on Earth.
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.