A Interface circuit diagram

B
Derivation of the period of a non-linear pendulum
The equation of motion of a pendulum is
¨θ + ω 2 0 sin θ = 0
(B.1)
where ω 2 = g/L. For small oscillations, sin θ ≈ θ and the motion is harmonic with angular
0
frequency ω 0 . If the angle of swing is not small, the angular frequency will become ω, but
as a first approximation the motion can still be assumed to be sinusoidal, so θ ≈ Asin ωt.
Going back to the equation of motion (B.1), the sin θ term may be expanded to give
¨θ + ω 2 θ − θ
3
0
6 + . . . = 0 (B.2)
−Aω 2 sin ωt + ω 2 Asin ωt − A3 sin 3 ωt
+ . . . = 0 (B.3)
0
6
sin 3 ωt may be expanded as a Fourier series:
sin 3 ωt = a 1 sin ωt + a 2 sin 2ωt + . . .
where the Fourier coefficients a n are given by
a n = 2 π
ˆ π
0
f (θ) sin(nθ) dθ
∴
a 1 = 2 π
ˆ π
0
sin 4 ωt dt = 2 π · 3π 8 = 3 4
∴ sin 3 ωt = 3 sin ωt + . . . (B.4)
4
So, putting (B.4) into (B.3), the approximate equation of motion is
−Aω 2 sin ωt + ω 2 Asin ωt − A3 3 sin ωt · ≈ 0
0
6 4
And hence, since T = 2π/ω,
∴ ω 2 ≈ ω 2 0
1 − A2
8
T ≈ T 0 1 + A2
16
(B.5)
(B.6)
43