Simple pendulum theory

A period is the time to return to the same position with the same direction of motion. Starting at the centre moving right, the next centre crossing moving left is only half a period. Count full cycles by watching one centre crossing direction. Frequency is the number of full cycles per second; angular frequency measures phase change in radians per second.

The component of gravity along the bob's arc pulls it toward equilibrium. Tension acts along the string, keeping the motion on the arc. Dividing tangential acceleration by length gives theta-double-dot = -(g/L)sin(theta). For small angles in radians, replace sin(theta) by theta; this creates the linear equation of simple harmonic motion. Use an initial angle at most 5 degrees so this approximation is appropriate. The simulation uses this limiting equation rather than modelling the small finite-amplitude correction.

The effective length L runs from the pivot to the bob's centre of mass, not just along the exposed string. In the point-bob model the same bob mass appears in the gravitational force and inertia, so it cancels. Increasing bob mass alone therefore does not change the model period. A large bob, a heavy string, or a flexible support changes the physical system and may require a more complete description.

A single stopwatch start-and-stop interval has endpoint reaction error. Spreading that interval over N = 20 cycles reduces its contribution to the period uncertainty by the factor N. It does not remove a miscount, a biased timer, or a length error. Record repeated 20-cycle runs at each length, average their elapsed times, then divide by 20. The virtual timer is a teaching tool and has no human reaction time unless an explicit noise model is added.

Squaring the period formula gives T squared proportional to L. Plot L in metres on the horizontal axis and T squared in seconds squared on the vertical axis. Fit y = mL + b using several well-spaced lengths. The gradient has units s squared per metre, and g is 4pi squared divided by the gradient. The ideal intercept is zero. Fit an intercept initially: a persistent nonzero value can flag a fixed length offset or another systematic effect, although a zero intercept alone does not prove accuracy.

For a timing-derived period, the first-order uncertainty in T squared is 2T u(T). If the fitted gradient has uncertainty u(m), the relative gradient contribution to u(g) equals u(m)/m. Length uncertainty also matters: an ordinary least-squares fit treats horizontal coordinates as exact, so use a suitable fit or discuss that limitation when length errors are significant. The model uses g = 9.80665 m/s squared as a reference parameter, not a measured local value. Perfect virtual agreement tests the algebra and software; it does not establish laboratory accuracy.