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)} + Be^{−i(ωt)}

F⃗ = − kx⃗

F⃗ = − kx⃗ − bv⃗

m \frac{d^{2}x}{dt^{2}} = − kx − bv

m \frac{d^{2}x}{dt^{2}} + b\frac{dx}{dt} + kx = 0

x(t) = A cos(ωt + ϕ) e^{−t/τ}

x(t) = Ae^{i(ωt+ϕ)−(t/τ)}

v(t) = \frac{dx}{dt} = [iω − \frac{1}{τ} ] Ae^{i(ωt+ϕ)−(t/τ)} = [iω − \frac{1}{τ} ] x(t)

a(t) = \frac{d^{2}x}{dt^{2}} = [iω − \frac{1}{τ} ]^{2}Ae^{i(ωt+ϕ)−(t/τ)} = ([\frac{1}{τ^{2}} − ω^{2}] − i(\frac{2ω}{τ})) x(t)

([\frac{m}{τ^{2}} − mω^{2}] − i[\frac{2mω}{τ}]) x(t) + (−\frac{b}{τ} + i bω) x(t) + (k) x(t) = 0

−\frac{2mω}{τ} + bω = 0

τ = \frac{2m}{b}

\frac{m}{τ^{2}} − mω^{2} − \frac{b}{τ} + k = 0

ω = \sqrt{\frac{k}{m} − \frac{b^{2}}{4m^{2}}}

ω = \sqrt{ω_{0}^{2} − \frac{b^{2}}{4m^{2}}}

x(t) = Ae^{i(ωt+ϕ)−(t/∞)} = Ae^{i(ωt+ϕ)}

x(t) = Ae^{i(0+ϕ)−(t/τ)} = ACe^{−t/τ}

x(t) = Bte^{−t/τ}

x(t) = Ae^{−t/τ} + Bte^{−t/τ}

γ = \sqrt{\frac{b^{2}}{4m^{2}} − \frac{k}{m}}

ω = ±iγ

x(t) = Ae^{−(γ+ \frac{1}{τ})t} + Be^{−( \frac{1}{τ} −γ)t}