F⃗ = −kx⃗

m\frac{d^{2}x⃗}{dt^{2}} = −kx⃗

m\frac{d^{2}x⃗}{dt^{2}} + kx⃗ = 0

x⃗(t) = A⃗ cos(ωt + ϕ)

ω = \sqrt{\frac{k}{m}}

F⃗_{resist} = −bv⃗

F⃗ = −kx⃗ − bv⃗

m\frac{d^{2}x⃗}{dt^{2}} = −kx⃗ − b\frac{dx⃗}{dt}

m\frac{d^{2}x⃗}{dt^{2}} + b\frac{dx⃗}{dt} + kx⃗ = 0

x(t) = A cos(ωt + ϕ)e^{−t/τ} ???

\frac{dx}{dt} = −\frac{1}{τ} A cos(ωt + ϕ)e^{−t/τ} − ωA sin(ωt + ϕ)e^{−t/τ}

\frac{d^{2}x}{dt^{2}} = \frac{1}{τ^{2}} A cos(ωt + ϕ)e^{−t/τ}

+ \frac{ω}{τ} A sin(ωt + ϕ)e^{−t/τ}

+ \frac{ω}{τ} A sin(ωt + ϕ)e^{−t/τ}

− ω^{2}A cos(ωt + ϕ)e^{−t/τ}

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

\frac{2mω}{τ} − bω = 0

τ = \frac{2m}{b}

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

\frac{k}{m} − \frac{b^{2}}{4m^{2}} = 0

max θ(t) = A e^{−t/τ} + 0

max θ(t) = A e^{−t/τ}