whence
| 4π2 | = | 1⁄2 (τ12 + τ22) | + | 1⁄2 (τ12 − τ22) | . |
| g | h1 + h2 | h1 − h2 |
(5)
The distance h1 + h2, which occurs in the first term on the right hand can be measured directly. For the second term we require the values of h1, h2 separately, but if τ1, τ2 are nearly equal whilst h1, h2 are distinctly unequal this term will be relatively small, so that an approximate knowledge of h1, h2 is sufficient.
As a final example we may note the arrangement, often employed in physical measurements, where a body performs small oscillations about a vertical axis through its mass-centre G, under the influence of a couple whose moment varies as the angle of rotation from the equilibrium position. The equation of motion is of the type
I θ̈ = −Kθ,
(6)
and the period is therefore τ = 2π√(I/K). If by the attachment of another body of known moment of inertia I′, the period is altered from τ to τ′, we have τ′ = 2π√{ (I + I′)/K }. We are thus enabled to determine both I and K, viz.
I / I′ = τ2 / (τ′2 − τ2), K = 4π2τ2I / (τ′2 − τ2).
(7)