Small Angle Approximation in Mechanics

Core Idea

For small angles θ, sin(θ) ≈ θ and cos(θ) ≈ 1 convert nonlinear equations into linear ones. This approximation is essential for treating oscillating systems as simple harmonic, enabling analytical solutions.

Explainer

From your study of the physical pendulum, you arrived at the equation of motion by applying Newton's second law for rotation: the restoring torque is τ = -mg·L·sin(θ), giving d²θ/dt² = -(g/L)·sin(θ). This is a nonlinear differential equation — the presence of sin(θ) rather than θ makes it analytically intractable for large amplitudes. No closed-form solution exists. The small-angle approximation is the key that transforms this equation into one you already know how to solve.

The approximation rests on the Taylor expansion of sin(θ) around θ = 0 (with θ in radians): sin(θ) = θ − θ³/6 + θ⁵/120 − ⋯. For small θ, each successive term is far smaller than the previous one. At θ = 0.1 rad (about 5.7°), the first dropped term θ³/6 ≈ 0.000167 — less than 0.2% of θ itself. We simply discard all terms beyond first order, leaving sin(θ) ≈ θ. Similarly, cos(θ) = 1 − θ²/2 + ⋯, so cos(θ) ≈ 1 for small θ. The entire approximation depends on working in radians, where the small-angle series has this clean form — in degrees, none of this works.

Substituting sin(θ) ≈ θ into the pendulum equation gives d²θ/dt² = −(g/L)·θ. This is precisely the simple harmonic oscillator equation d²x/dt² = −ω²x, with ω = √(g/L). The solution is θ(t) = A·cos(ωt + φ), a sinusoidal oscillation with period T = 2π/ω = 2π√(L/g). Two things follow immediately. First, the pendulum oscillates sinusoidally for small amplitudes — which you already knew from SHM. Second, and crucially, the period is independent of amplitude A. This is isochrony: a pendulum swinging through a 5° arc and one swinging through a 10° arc have the same period (approximately). This is why pendulum clocks work — as the clock winds down and the swing becomes smaller, the period barely changes, keeping accurate time.

The approximation's domain of validity is roughly θ < 15° (about 0.26 rad), where the error in sin(θ) ≈ θ stays below 1%. Beyond this, the period begins to depend on amplitude and corrections are needed. More broadly, the small-angle technique is an instance of linearization — replacing a nonlinear function with its linear approximation near an equilibrium point. This pattern recurs throughout physics: paraxial optics replaces sin(θ) ≈ θ to analyze lens systems; perturbation theory in quantum mechanics treats small Hamiltonians linearly; sound waves in a fluid are analyzed by linearizing the nonlinear fluid equations around equilibrium. Whenever you encounter a nonlinear system, the first question is always: is there a regime where nonlinearity is small, so I can linearize and get analytic results? The small-angle approximation is the simplest and most transparent example of this fundamental physical strategy.