The Physics of Pendulums — Simple Harmonic Motion
A pendulum's period depends only on its length and local gravity — not on mass or amplitude (for small angles). This period independence is what made pendulums the basis of accurate clocks for 300 years. Pendulum Master puts length, mass, and release angle under your control to discover these relationships through direct experimentation.
Period Formula: T = 2π√(L/g)
For small angles (below ~15°): T = 2π√(L/g), where L is string length (m) and g is gravitational acceleration (m/s²). At g = 9.81 m/s²: a 1.0 m pendulum has T = 2.0 s (grandfather clock tick); a 0.25 m pendulum has T = 1.0 s. Temperature compensation is needed in precision clocks because thermal expansion changes L — a warmer clock runs slow as L increases.
Why Mass Doesn't Affect Period
The restoring force is mg sin θ ≈ mgθ (for small θ). Newton's second law: ma = −mgθ = −mg(s/L). The m cancels, giving a = −(g/L)s — SHM with ω² = g/L, period T = 2π/ω = 2π√(L/g). Mass appears in both force and inertia identically, so it cancels. This is the equivalence principle: gravitational mass (in mg) equals inertial mass (in ma).
Energy in a Pendulum
At the bottom: all KE = ½mv_max². At maximum displacement (height h above bottom): all PE = mgh. Conserving energy: v_max = √(2gh), where h = L(1 − cos θ). For L = 1 m, θ = 20°: h = 1(1 − cos 20°) = 0.0603 m → v_max = √(2 × 9.81 × 0.0603) = 1.087 m/s. Speed at any intermediate displacement x from bottom: v = √(v_max² − ω²x²).
Resonance
When a periodic driving force matches the natural frequency f₀ = (1/2π)√(g/L), amplitude grows dramatically — resonance. This is how a small child can build up large swinging amplitude by pushing at the right moment each cycle. The Tacoma Narrows Bridge collapse (1940) is partially attributed to aeroelastic resonance. Resonance is exploited in quartz watch crystals (piezoelectric resonance at ~32,768 Hz for precise timekeeping).
Frequently Asked Questions
Does a heavier pendulum swing faster?
No. Period T = 2π√(L/g) is independent of mass. A 100 g and 1 kg bob on the same 1 m string have identical periods. This follows from the equivalence of gravitational and inertial mass — both the driving force (mg sin θ) and the resistance (ma) are proportional to m, so m cancels from the equation of motion.
What is simple harmonic motion?
SHM is oscillatory motion where the restoring force is proportional to displacement from equilibrium and directed toward it: F = −kx → a = −ω²x. This produces sinusoidal motion x(t) = A cos(ωt + φ). Period T = 2π/ω is independent of amplitude A — the defining characteristic of SHM. Mass-spring systems, pendulums (small angles), and LC circuits all exhibit SHM.
What happens when amplitude is large?
The small-angle approximation sin θ ≈ θ (radians) fails at large angles. The true equation of motion is non-linear: T(θ₀) = T₀ × (1 + θ₀²/16 + …) for initial angle θ₀. At θ₀ = 90°, the period is about 18% longer than the small-angle formula predicts. At θ₀ = 180° (horizontal start), T → ∞ — the pendulum takes infinitely long in theory (a separatrix orbit).