ENERGY AMPLITUDE RELATION

Energy methods are often cleaner than force methods for finding speeds at specific points. Step 1 — height at amplitude: the bob sweeps along a circular arc of radius L. When the string makes angle θ₀ with the vertical, the bob sits at a height h = L − L cos θ₀ = L(1 − cos θ₀) above the lowest point. For θ₀ = 0.15 rad: h = 0.8 × (1 − cos 0.15) = 0.8 × (1 − 0.98877) = 0.8 × 0.01123 ≈ 0.008983 m ≈ 8.98 mm. Note the small-angle approximation would give h ≈ L θ₀²/2 = 0.8 × 0.01125 ≈ 9.00 mm — nearly identical at this amplitude, confirming the regime. Step 2 — PE_max: at the amplitude, the bob is momentarily at rest, so all mechanical energy is potential. PE_max = mgh = 0.5 × 9.80665 × 0.008983 ≈ 0.04405 J. Step 3 — KE_max: by conservation of energy (frictionless pivot, massless string), the total mechanical energy E = PE + KE is constant. At the bottom, h = 0 so PE = 0 and KE = KE_max = E = PE_max ≈ 0.04405 J. Step 4 — v_max: from KE_max = (1/2)mv_max², we get v_max = √(2 KE_max/m) = √(2 × 0.04405/0.5) = √0.17620 ≈ 0.420 m/s. Step 5 — period: T = 2π√(0.8/9.80665) ≈ 1.795 s. Mass and amplitude do not appear — Galileo's isochronism in action.