Periodic Motion & the SHM Model

Periodic motion is any motion whose state repeats after a fixed time: x(t + T) = x(t) for every t, and the smallest such T is the period. Its reciprocal f = 1/T counts cycles per second, in hertz, and ω = 2πf counts radians of phase per second, since one cycle is 2π rad. A heart beating 72 times a minute has T = 60 s ÷ 72 = 0.833 s, f = 1.20 Hz and ω = 7.54 rad s⁻¹. Measure T by timing many cycles: a 0.2 s reaction-time error spread over 20 cycles costs only 0.01 s per period. Amplitude is the largest displacement from the centre of the motion, half the peak-to-peak excursion, never the full swing. None of these numbers describes the shape of a cycle. A ball bouncing on a floor has a perfectly good period, yet spends each cycle in free fall and reverses in an abrupt impact. Periodicity is a description; simple harmonic motion is a claim about the force.

Every oscillation circles an equilibrium, a position x₀ where the net force is zero. It is stable if a small displacement produces a force pointing back: F < 0 just beyond x₀ on the positive side and F > 0 just below it, so dF/dx < 0 there. A marble in a bowl is stable; one balanced on an upturned bowl is pushed further away and never oscillates. The bookkeeping that makes SHM simple is to measure displacement from equilibrium, x = s − x₀, not from the end of a track or the zero of a ruler. If a glider's sensor reads s = 0.400 m at rest, then 0.430 m is x = +0.030 m and 0.370 m is x = −0.030 m. Measured this way, a restoring force always has the opposite sign to x. Measured from anywhere else, the same physics carries a hidden constant, a = −ω²(s − x₀), which is easy to misread as a failure of the model.

Simple harmonic motion is defined by one relation between acceleration and displacement, a = −ω²x, and it makes two separate claims. The minus sign says the acceleration always points toward equilibrium. The constant says a/x is the same at every displacement, so twice as far out the pull back is twice as strong. Newton's second law delivers exactly this whenever the net force is linear, F = −k_eff x: then a = −(k_eff/m)x and ω² = k_eff/m. Units confirm ω² is a rate squared, since N m⁻¹ ÷ kg = s⁻². For a 0.250 kg cart with k_eff = 40.0 N m⁻¹, ω² = 160 s⁻² and ω = 12.6 rad s⁻¹; at x = +3.0 cm its acceleration is −160 × 0.030 = −4.8 m s⁻². Solving the equation, the next lesson's job, turns ω into T = 2π/ω = 0.497 s. So a test for SHM is a test of a/x: if the ratio changes with x, the motion is not simple harmonic, however regular it looks.

Suppose x(t) satisfies a = −ω²x, and multiply the whole motion by 3. The displacement at every instant triples, so does the acceleration, and 3x(t) satisfies the same relation with the same ω: the tripled motion runs on exactly the same clock. Physically, a glider pulled three times as far feels three times the pull at every matching point, reaches three times the speed, and covers three times the distance in the same time. That is isochronism, and it belongs to linear forces alone. Give a puck a restoring acceleration of fixed size a₀ = 2.0 m s⁻², as on a smooth V-shaped trough, and each quarter-cycle is a constant-acceleration run of length A, so T = 4√(2A/a₀). Released from 0.10 m it has T = 1.26 s; from 0.40 m, T = 2.53 s. Four times the amplitude doubles the period. A force that stiffens faster than linearly does the reverse. Timing one system from two release distances is the cheapest SHM test there is.

Linear forces are common for a reason. Expand any smooth force about a stable equilibrium: F(x) = F(x₀) + F′(x₀)(x − x₀) + ½F″(x₀)(x − x₀)² + … . The first term is zero by the definition of equilibrium and F′(x₀) < 0 for stability, so near x₀ the force is F ≈ −k_eff(x − x₀) with k_eff = −F′(x₀), the curvature of the potential well at its bottom. Tie a 0.20 kg puck on an air table to the midpoint of a cord stretched between posts 1.00 m apart at a tension of 50 N, assumed to stay constant. Pulled sideways by x, the two halves give F = −100x/√(0.25 + x²), in newtons for x in metres. Differentiating at x = 0 gives k_eff = 100/0.50 = 200 N m⁻¹, so ω = 31.6 rad s⁻¹ and T = 0.199 s. At x = 5.0 cm the linear law overstates the force by 0.50%; at 15 cm, by 4.4%. SHM is a small-amplitude model, and the discarded terms tell you how small.

Because the model makes sharp predictions, data can refute it, and four observations each break SHM on their own: a/x changes across the range, so a plot of a against x bends; the period changes with release distance; the trace x(t) has flattened tops, corners or unequal halves, which no linear restoring law produces; or the amplitude shrinks cycle by cycle, which means a non-conservative force is acting. Suppose an accelerometer on a glider reads −0.79 m s⁻² at x = +2.0 cm and −2.37 m s⁻² at +6.0 cm. Both ratios are −39.5 s⁻², a straight line through the origin with slope −ω², so ω = 6.28 rad s⁻¹ and T = 1.00 s. A third reading of −4.10 m s⁻² at +10.0 cm gives −41.0 s⁻², 3.8% stiffer, a warning that wide swings will run slightly fast. A slow, steady loss of amplitude is usually damping, which a later lesson adds to the model rather than a reason to throw it away.