Spring and mathematical pendulums
A spring pendulum is a load of mass m hanging on a spring of stiffness k. Its angular frequency is ω = √(k/m) and its period is T = 2π√(m/k): the period is directly proportional to the square root of the mass and inversely proportional to the square root of the stiffness. A mathematical pendulum is a point mass on an inextensible weightless string; for small deflections (α ≲ 6–8°) its period is given by Huygens’ formula: T = 2π√(l/g). Hence the period does not depend on the amplitude (for small α) or on the mass of the bob; it is proportional to the square root of the length and inversely proportional to the square root of g. In a spring pendulum the total energy E = kA²/2 = mυ²/2 + kx²/2 is constant: at the greatest displacement it is all potential, at the equilibrium position all kinetic (mυₘₐₓ²/2 = kA²/2). At large angles the oscillation of a mathematical pendulum is no longer harmonic, because the approximation sinα ≈ α fails.
In class with the teacher: with a small ball on a string measure the time of 10 oscillations at different lengths (25 cm, 100 cm) and check that a 4 times longer pendulum has a period 2 times larger. Use a small light ball and keep everyone clear.