Bifilar suspension theory

Rotational inertia weights each small mass element by the square of its perpendicular distance from an axis. This experiment uses the vertical axis through the rod centre, perpendicular to its length. For a uniform slender rod of length D, integration gives I = MD squared/12. With M = 0.500 kg and D = 0.500 m, the model value is 0.0104167 kg m squared. A finite-width rectangular bar instead has I = M(D squared + w squared)/12; ignoring width is justified only for a slender rod.

Here a is half the string separation, so the two strings are 2a = 0.200 m apart. Both upper attachments must be directly above their corresponding lower attachments at equilibrium. L is their common equilibrium length, not the rod length D. Using full separation in a formula written for half-separation gives an inertia four times too large.

A gentle twist moves the lower attachment points sideways and lifts the rod slightly. Its increased gravitational potential energy supplies the restoring tendency. For small theta, the lift is approximately a squared theta squared/(2L). Differentiating Mg times this lift gives a restoring torque -(Mga squared/L)theta. Thin flexible strings are assumed to supply negligible material torsional stiffness.

Combining the restoring torque with I times angular acceleration gives a simple harmonic oscillator. A larger inertia slows the oscillation, longer strings weaken the restoring torque, and greater attachment separation strengthens it. Keep twists small and avoid sideways swinging or tilting so that one rotational mode dominates. A timing interval must include 20 full returns to the same orientation and direction.

Plot period squared vertically against string length horizontally. If T squared = mL + b, the gradient is 4pi squared I/(Mga squared). Thus I = mMga squared/(4pi squared); the gradient unit is seconds squared per metre and the resulting inertia unit is kg m squared. The ideal intercept is zero. Do not force it to zero before examining possible length offsets or effects omitted by the model.

Because I depends on a squared, a small fractional separation error contributes twice that fractional error to I. For independent inputs, combine relative standard uncertainties from gradient, mass, g and twice the half-separation uncertainty in quadrature. A physical comparison with MD squared/12 must also include rod dimension and mass uncertainty. Perfect virtual agreement is expected because the data generator uses that rod formula; it is a consistency exercise rather than independent experimental confirmation.