| l | d2ψ | = − g sin ψ. |
| dt2 |
(15)
If we multiply by 2dψ/dt and integrate, we obtain
| ( | dψ | )2 = | 2g | cos ψ + const., |
| dt | l |
(16)
which is seen to be equivalent to (14). If the pendulum oscillate between the limits ψ = ±α, we have
| ( | δψ | )2 = | 2g | (cos ψ − cos α) = | 4g | (sin2 1⁄2α − sin2 1⁄2ψ); |
| dt | l | l |
(17)
and, putting sin 1⁄2ψ = sin 1⁄2α. sin φ, we find for the period (τ) of a complete oscillation
| τ = 4 ∫1⁄2π0 | dt | dφ = 4√ | l | · ∫1⁄2π0 | dφ |
| dφ | g | √(1 − sin2 1⁄2α · sin2 φ) |