A simple pendulum released from 8° completes one full oscillation in 2.0 seconds. An identical pendulum is released from 4°. How long does the second pendulum take to complete one oscillation?
A1.0 seconds — halving the angle halves the period
B2.0 seconds — within the small-angle regime, the period is independent of amplitude
C1.4 seconds — the period scales with the square root of the amplitude
D2.8 seconds — the smaller angle means less restoring force, so the oscillation is slower
This is isochrony — a direct consequence of the small-angle approximation. When sin(θ) ≈ θ, the pendulum equation becomes d²θ/dt² = −(g/L)θ, a simple harmonic oscillator with period T = 2π√(L/g). The period depends only on length and gravitational acceleration, NOT on amplitude. Both 4° and 8° are well within the approximation's valid range (~15°), so both pendulums have essentially the same period. Options A, C, and D all incorrectly assume the period depends on how far the pendulum swings.
Question 2 Multiple Choice
Why does the small-angle approximation sin(θ) ≈ θ only work when θ is measured in radians?
ARadians are more precise than degrees for small angles
BThe Taylor series sin(θ) = θ − θ³/6 + ⋯ requires θ in radians; in degrees, sin(1°) ≈ 0.0175, not 1
CPhysicists prefer radians by convention, and the convention must be applied consistently
DDegrees introduce rounding errors that accumulate when angles are small
The approximation sin(θ) ≈ θ means the numerical value of sin(θ) equals the numerical value of θ. For θ = 0.1 rad, sin(0.1) ≈ 0.0998 ≈ 0.1 — the numbers match closely. For θ = 0.1° (a tiny angle), sin(0.1°) ≈ 0.00175, which is far from 0.1. The Taylor series gives sin(θ) ≈ θ only when θ is in radians because the calculus result d/dθ[sin(θ)] = cos(θ) assumes radians. The approximation is mathematically meaningless when θ is expressed in degrees.
Question 3 True / False
Applying the small-angle approximation to the pendulum equation of motion converts it into the simple harmonic oscillator equation, which has a sinusoidal solution.
TTrue
FFalse
Answer: True
This is the whole point of the approximation. Starting from d²θ/dt² = −(g/L)sin(θ), substituting sin(θ) ≈ θ gives d²θ/dt² = −(g/L)θ. This is exactly the SHM equation d²x/dt² = −ω²x with ω = √(g/L), whose solution is θ(t) = A·cos(ωt + φ). The approximation converts an analytically intractable nonlinear equation into one with a known closed-form solution.
Question 4 True / False
A pendulum swinging through a 15° arc takes significantly longer to complete one oscillation than an identical pendulum swinging through a 3° arc.
TTrue
FFalse
Answer: False
Within the valid range of the small-angle approximation (roughly θ < 15°), the period depends only on pendulum length, not amplitude — this is isochrony. A 15° pendulum actually has a period only about 0.5% longer than a 3° pendulum, a difference too small to detect without precision instruments. The common misconception is that the pendulum swinging farther must take longer because it travels more distance. But the increased restoring force at larger angles compensates almost exactly, keeping the period nearly constant.
Question 5 Short Answer
The small-angle approximation is described as an instance of 'linearization.' Explain what this means and give one other example from physics where the same strategy is applied.
Think about your answer, then reveal below.
Model answer: Linearization means replacing a nonlinear function with its linear (first-order Taylor) approximation near an equilibrium point, converting an analytically intractable nonlinear equation into a linear one with known solutions. For the pendulum, sin(θ) is replaced by θ, turning a nonlinear ODE into the SHM equation. The same strategy appears in paraxial optics (sin(θ) ≈ θ to derive the thin-lens equation), perturbation theory in quantum mechanics (treating small Hamiltonians as linear corrections), and fluid mechanics (linearizing Euler's equations to analyze small-amplitude sound waves).
The broader lesson is that linearization is a foundational strategy throughout physics. Nonlinear equations are generally hard or impossible to solve analytically; linear equations have well-understood solutions. The question is always whether the system operates close enough to equilibrium that the linear approximation captures the essential physics. When it does, the analytical solution reveals qualitative behavior — like isochrony — that numerical solutions alone would not make obvious.