Question

2.) Explain why the starting angle doesnt impact the time it takes the pendulum to swing back and forth?​

Answer 1

The starting angle θθ of a pendulum does not affect its period for θ<<1θ<<1. At higher angles, however, the period TT increases with increasing θθ.

The relation between TT and θθ can be derived by solving the equation of motion of the simple pendulum (from F=ma)

−gsinθ=lθ¨−gain⁡θ=lθ¨

For small angles, θ≪1,θ≪1, and hence sinθ≈θsin⁡θ≈θ. Hence,

θ¨=−glθθ¨=−glθ

This second-order differential equation can be solved to get θ=θ0cos(ωt),ω=gl−−√θ=θ0cos⁡(ωt),ω=gl. The period is thus T=2πω=2πlg−−√T=2πω=2πlg, which is independent of the starting angle θ0θ0.

For large angles, however, the above derivation is invalid. Without going into the derivation, the general expression of the period is T=2πlg−−√(1+θ2016+...)T=2πlg(1+θ0216+...). At large angles, the θ2016θ0216 term starts to grow big and cause