Let the equilibrium position be given by the angle ϕ (Fig. 1.16). In this position the force on the mass m along the horizontal axis is equal to ma. The angle ϕ is determined by the equations.
When the pendulum is displaced by a small amount θ, it will perform simple harmonic motion around the equilibrium position. Its equation of motion is
m &&x = –T sin(θ + ϕ)
where x is the distance from the vertical OA
For small θ, sin(θ + ϕ) ≈ θ cos ϕ + sin ϕ
θ and l are related geometrically as
with time period of oscillation
