Spring Pendulum (Oscillation Period)

Insert mass and spring constant into T = 2π·√(m/k). Example: a mass of 0.25 kg on a spring with k = 100 N/m oscillates with T = 2π·√(0.25/100) = 2π·0.05 ≈ 0.31 s, a good three oscillations per second. The mass sits in the numerator under the root: more mass makes the oscillation more sluggish and slower. The spring constant is in the denominator: a stiffer spring pulls back harder and shortens the period. Because of the root, both influences are damped; four times the mass only doubles T. The frequency is the reciprocal f = 1/T, here about 3.2 Hz.

This is the isochronism of harmonic oscillation, and it follows from the linear force law F = −k·x. With a larger displacement the path per oscillation gets longer, but the restoring force grows in exactly the same proportion, so the body moves correspondingly faster. Both effects compensate completely, and every oscillation takes the same time. Mathematically, the amplitude A appears in the solution x(t) = A·cos(ωt) only as a prefactor, while ω = √(k/m) depends solely on system properties. This property made springs and pendulums the timekeepers of clockmaking. It holds only while the spring stays in the linear Hooke range; overstretching makes the oscillation anharmonic.

Rearrange the period formula for k: k = 4π²·m/T². Measure the time for many oscillations and divide; ten oscillations in 10 s mean T = 1 s. Example: with m = 0.5 kg and T = 1 s, k = 4π² × 0.5 / 1 ≈ 19.7 N/m. Timing over many periods greatly reduces the stopwatch error, the most important practical tip in the experiment. Alternatively, the static extension gives the same k: attach a known mass and measure the stretch, k = m·g/x. If both methods agree, that is a nice confirmation of the model, a standard evaluation in physics labs and a popular exam task.

For the spring pendulum T = 2π·√(m/k) holds: the period depends on mass and spring constant but not on location; g does not appear, and the formula holds unchanged on the Moon. For the simple pendulum T = 2π·√(l/g) holds: the period depends on string length and local gravity but, surprisingly, not on mass. The reason: for the simple pendulum gravity provides the restoring force, which is itself proportional to the mass, so the mass cancels. Moreover, the simple pendulum is harmonic only for small angles (below about 10°), while the spring pendulum oscillates exactly harmonically throughout the Hooke range. Mixing up the two formulas is one of the most common exam mistakes in the oscillations chapter.

It swings losslessly between two forms: at the turning points everything is elastic energy of the spring, E = ½kA² with amplitude A, and the speed is zero. At the equilibrium crossing the spring is relaxed and all energy is kinetic, E = ½mv_max²; there the body is fastest, with v_max = ω·A. Equating the two gives the maximum speed directly: ½kA² = ½mv_max². Example: k = 100 N/m, m = 0.25 kg, A = 0.04 m gives v_max = A·√(k/m) = 0.04 × 20 = 0.8 m/s. Real pendulums lose a little energy per cycle to friction; the amplitude decays while the period stays nearly unchanged.