linguistic:translation
Simple pendulum

Assuming no damping, the differential equation governing a simple pendulum of length l {\displaystyle l}, where g {\displaystyle g} is the local acceleration of gravity, is d 2 θ d t 2 + g l sin θ = 0. {\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{l}}\sin \theta =0.}

If the maximal displacement of the pendulum is small, we can use the approximation sin θ ≈ θ {\displaystyle \sin \theta \approx \theta } and instead consider the equation d 2 θ d t 2 + g l θ = 0. {\displaystyle {\frac {d^{2}\theta }{dt^{2}}}+{\frac {g}{l}}\theta =0.}

The general solution to this differential equation is θ (t) = A cos (g l t + φ), {\displaystyle \theta (t)=A\cos \left({\sqrt {\frac {g}{l}}}t+\varphi \right),} where A {\displaystyle A} and φ {\displaystyle \varphi } are constants that depend on the initial conditions.Using as initial conditions θ (0) = θ 0 {\displaystyle \theta (0)=\theta _{0}} and θ ˙ (0) = 0 {\displaystyle {\dot {\theta }}(0)=0}, the solution is given by θ (t) = θ 0 cos (g l t), {\displaystyle \theta (t)=\theta _{0}\cos \left({\sqrt {\frac {g}{l}}}t\right),} where θ 0 {\displaystyle \theta _{0}} is the largest angle attained by the pendulum (that is, θ 0 {\displaystyle \theta _{0}} is the amplitude of the pendulum). The period, the time for one complete oscillation, is given by the expression τ = 2 π l g = 2 π ω, {\displaystyle \tau =2\pi {\sqrt {\frac {l}{g}}}={\frac {2\pi }{\omega }},} which is a good approximation of the actual period when θ 0 {\displaystyle \theta _{0}} is small. Notice that in this approximation the period τ {\displaystyle \tau } is independent of the amplitude θ 0 {\displaystyle \theta _{0}}. In the above equation, ω {\displaystyle \omega } represents the angular frequency.