U(x) = mgh(x) = mgH [1 − cos(\frac{2πx}{L})]

cos(θ) = cos(0) + \frac{θ^{1}}{1!} (− sin(0)) + \frac{θ^{2}}{2!} (− cos(0)) + \frac{θ^{3}}{3!} (sin(0)) + \frac{θ^{4}}{4!} (cos(0)) + ...

= 1 + 0 − \frac{θ^{2}}{2!} − 0 + \frac{θ^{4}}{4!} + ...

sin(θ) = sin(0) + \frac{θ^{1}}{1!} (cos(0)) + \frac{θ^{2}}{2!} (− sin(0)) + \frac{θ^{3}}{3!} (− cos(0)) + \frac{θ^{4}}{4!} (sin(0)) + ...

= 0 + \frac{θ^{1}}{1!} − 0 − \frac{θ^{3}}{3!} + 0 + \frac{θ^{5}}{5!} + ...

\begin{aligned}e^{iθ} & = e^{0} + \frac{iθ^{1}}{1!} (e^{0}) + \frac{i^{2}θ^{2}}{2!} (e^{0}) + \frac{i^{3}θ^{3}}{3!} (e^{0}) + \frac{i^{4}θ^{4}}{4!} (e^{0}) + ... \\ & = 1 + \frac{iθ^{1}}{1!} − \frac{θ^{2}}{2!} − \frac{iθ^{3}}{3!} + \frac{θ^{4}}{4!} + \frac{iθ^{5}}{5!} + ... \end{aligned}

\frac{d^{2}y}{dt^{2}} = − (\frac{k}{m}) y = − ω^{2}y

ω ≡ \sqrt{\frac{k}{m}}

y(t) = A cos(ωt + ϕ)

y(t) = B cos(ωt) + C sin(ωt)

y(0) = 0 m

\frac{dy}{dt} |_{t=0} = v_{y}(0) = 0.12 m/s

y(t) = (−0.04 m) cos(ωt + \frac{π}{2} )

y(t) = (0) cos(ωt) + (0.04 m) sin(ωt)

y(t) = Ae^{i(ωt+ϕ)}

e^{iθ} = cos(θ) + i sin(θ)

y(t) = Ae^{i(ωt+ϕ)} = A cos(ωt + ϕ) + i A sin(ωt + ϕ)

\frac{dy}{dt} = v_{y}(t) = iωAe^{i(ωt+ϕ)} = iωy(t)

\frac{d^{2}y}{dt^{2}} = a_{y}(t) = −ω^{2}Ae^{i(ωt+ϕ)} = −ω^{2}y(t)

y(t) = Ae^{−i(ωt+ϕ)} = A cos(ωt + ϕ) − i A sin(ωt + ϕ)

y(t) = Ae^{i(ωt+ϕ)}

y(t) = Be^{i(ωt)} + Ce^{−i(ωt)}